KILLED
proof of input_KWnKrN11ph.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (10) TRS for Loop Detection
    (11) RenamingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (12) CpxRelTRS
        (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (14) typed CpxTrs
        (15) OrderProof [LOWER BOUND(ID), 0 ms]
        (16) typed CpxTrs
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (20) CdtProblem
    (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxRelTRS
        (29) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (30) CpxTRS
        (31) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (32) CpxWeightedTrs
            (33) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (34) CpxTypedWeightedTrs
            (35) CompletionProof [UPPER BOUND(ID), 0 ms]
            (36) CpxTypedWeightedCompleteTrs
                (37) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (38) CpxTypedWeightedCompleteTrs
                (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (40) CpxRNTS
                (41) InliningProof [UPPER BOUND(ID), 506 ms]
                (42) CpxRNTS
                (43) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (44) CpxRNTS
                (45) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (46) CpxRNTS
                (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (48) CpxRNTS
                (49) IntTrsBoundProof [UPPER BOUND(ID), 419 ms]
                (50) CpxRNTS
                (51) IntTrsBoundProof [UPPER BOUND(ID), 38 ms]
                (52) CpxRNTS
                (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (54) CpxRNTS
                (55) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
                (56) CpxRNTS
                (57) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
                (58) CpxRNTS
                (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (60) CpxRNTS
                (61) IntTrsBoundProof [UPPER BOUND(ID), 171 ms]
                (62) CpxRNTS
                (63) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
                (64) CpxRNTS
                (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (66) CpxRNTS
                (67) IntTrsBoundProof [UPPER BOUND(ID), 332 ms]
                (68) CpxRNTS
                (69) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
                (70) CpxRNTS
                (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (72) CpxRNTS
                (73) IntTrsBoundProof [UPPER BOUND(ID), 10.8 s]
                (74) CpxRNTS
                (75) IntTrsBoundProof [UPPER BOUND(ID), 3394 ms]
                (76) CpxRNTS
            (77) CompletionProof [UPPER BOUND(ID), 0 ms]
            (78) CpxTypedWeightedCompleteTrs
                (79) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms]
                (80) CpxRNTS
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
        (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (88) CdtProblem
        (89) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (96) CdtProblem
        (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (98) CdtProblem
        (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms]
        (100) CdtProblem
        (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem
        (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem
        (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms]
        (116) CdtProblem
        (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (118) CdtProblem
        (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
        (120) CdtProblem
        (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CdtProblem
        (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (126) CdtProblem
        (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (130) CdtProblem
        (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms]
        (134) CdtProblem
        (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms]
        (136) CdtProblem
        (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (140) CdtProblem
        (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (142) CdtProblem
        (143) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (144) CdtProblem
        (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (148) CdtProblem
        (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (150) CdtProblem
        (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (152) CdtProblem
        (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
        (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (156) CdtProblem
        (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (158) CdtProblem
        (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (160) CdtProblem
        (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (162) CdtProblem
        (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (164) CdtProblem
        (165) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (166) CdtProblem
        (167) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (168) CdtProblem
        (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (170) CdtProblem
        (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (172) CdtProblem
        (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (174) CdtProblem
        (175) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (176) CdtProblem
        (177) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (178) CdtProblem
        (179) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CdtProblem
        (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CdtProblem
        (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (184) CdtProblem
        (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (186) CdtProblem
        (187) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
        (188) CdtProblem
        (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (190) CdtProblem
        (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (192) CdtProblem
        (193) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (194) CdtProblem
        (195) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (196) CdtProblem
        (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (198) CdtProblem
        (199) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (200) CdtProblem
        (201) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (202) CdtProblem
        (203) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (204) CdtProblem
        (205) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (206) CdtProblem
(207) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(208) CpxWeightedTrs
    (209) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) CpxTypedWeightedTrs
    (211) CompletionProof [UPPER BOUND(ID), 0 ms]
    (212) CpxTypedWeightedCompleteTrs
        (213) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (214) CpxTypedWeightedCompleteTrs
        (215) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (216) CpxRNTS
        (217) InliningProof [UPPER BOUND(ID), 429 ms]
        (218) CpxRNTS
        (219) SimplificationProof [BOTH BOUNDS(ID, ID), 15 ms]
        (220) CpxRNTS
        (221) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (222) CpxRNTS
        (223) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (224) CpxRNTS
        (225) IntTrsBoundProof [UPPER BOUND(ID), 523 ms]
        (226) CpxRNTS
        (227) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
        (228) CpxRNTS
        (229) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (230) CpxRNTS
        (231) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
        (232) CpxRNTS
        (233) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
        (234) CpxRNTS
        (235) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (236) CpxRNTS
        (237) IntTrsBoundProof [UPPER BOUND(ID), 184 ms]
        (238) CpxRNTS
        (239) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (240) CpxRNTS
        (241) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (242) CpxRNTS
        (243) IntTrsBoundProof [UPPER BOUND(ID), 285 ms]
        (244) CpxRNTS
        (245) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
        (246) CpxRNTS
        (247) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (248) CpxRNTS
        (249) IntTrsBoundProof [UPPER BOUND(ID), 201 ms]
        (250) CpxRNTS
        (251) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
        (252) CpxRNTS
        (253) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (254) CpxRNTS
        (255) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
        (256) CpxRNTS
        (257) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
        (258) CpxRNTS
        (259) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (260) CpxRNTS
        (261) IntTrsBoundProof [UPPER BOUND(ID), 635 ms]
        (262) CpxRNTS
        (263) IntTrsBoundProof [UPPER BOUND(ID), 252 ms]
        (264) CpxRNTS
        (265) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (266) CpxRNTS
        (267) IntTrsBoundProof [UPPER BOUND(ID), 2117 ms]
        (268) CpxRNTS
        (269) IntTrsBoundProof [UPPER BOUND(ID), 435 ms]
        (270) CpxRNTS
    (271) CompletionProof [UPPER BOUND(ID), 0 ms]
    (272) CpxTypedWeightedCompleteTrs
        (273) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (274) CpxRNTS


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2))
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue')
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue')
   breadth#2(nil, @queue') -> nil
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue')))
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6)
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8)
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2))
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs)
   children#1(nil, @b, @l2) -> children#2(@l2, @b)
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil)
   children#2(nil, @b) -> nil
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil))
   children#3(nil, @b, @x, @xs) -> nil
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2))
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq)
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq))
   copyover#2(nil, @outq) -> tuple#2(nil, @outq)
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2))
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq)
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil))
   dequeue#2(nil, @inq) -> dequeue#3(@inq)
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4)
   empty(@x) -> tuple#2(nil, nil)
   enqueue(@x, @queue) -> enqueue#1(@queue, @x)
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq)
   enqueues(@l, @queue) -> enqueues#1(@l, @queue)
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue))
   enqueues#1(nil, @queue) -> @queue
   startBreadth(@xs) -> startBreadth#1(@xs)
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2))
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue')
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue')
   breadth#2(nil, @queue') -> nil
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue')))
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6)
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8)
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2))
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs)
   children#1(nil, @b, @l2) -> children#2(@l2, @b)
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil)
   children#2(nil, @b) -> nil
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil))
   children#3(nil, @b, @x, @xs) -> nil
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2))
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq)
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq))
   copyover#2(nil, @outq) -> tuple#2(nil, @outq)
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2))
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq)
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil))
   dequeue#2(nil, @inq) -> dequeue#3(@inq)
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4)
   empty(@x) -> tuple#2(nil, nil)
   enqueue(@x, @queue) -> enqueue#1(@queue, @x)
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq)
   enqueues(@l, @queue) -> enqueues#1(@l, @queue)
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue))
   enqueues#1(nil, @queue) -> @queue
   startBreadth(@xs) -> startBreadth#1(@xs)
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2))
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue')
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue')
   breadth#2(nil, @queue') -> nil
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue')))
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6)
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8)
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2))
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs)
   children#1(nil, @b, @l2) -> children#2(@l2, @b)
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil)
   children#2(nil, @b) -> nil
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil))
   children#3(nil, @b, @x, @xs) -> nil
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2))
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq)
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq))
   copyover#2(nil, @outq) -> tuple#2(nil, @outq)
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2))
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq)
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil))
   dequeue#2(nil, @inq) -> dequeue#3(@inq)
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4)
   empty(@x) -> tuple#2(nil, nil)
   enqueue(@x, @queue) -> enqueue#1(@queue, @x)
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq)
   enqueues(@l, @queue) -> enqueues#1(@l, @queue)
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue))
   enqueues#1(nil, @queue) -> @queue
   startBreadth(@xs) -> startBreadth#1(@xs)
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)
Tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
S tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
K tuples:none
Defined Rule Symbols:   breadth_2, breadth#1_1, breadth#2_2, breadth#3_2, breadth#4_1, breadth#5_1, children_4, children#1_3, children#2_2, children#3_4, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, empty_1, enqueue_2, enqueue#1_2, enqueues_2, enqueues#1_2, startBreadth_1, startBreadth#1_1, startBreadth#2_1

Defined Pair Symbols:   BREADTH_2, BREADTH#1_1, BREADTH#2_2, BREADTH#3_2, BREADTH#4_1, BREADTH#5_1, CHILDREN_4, CHILDREN#1_3, CHILDREN#2_2, CHILDREN#3_4, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, EMPTY_1, ENQUEUE_2, ENQUEUE#1_2, ENQUEUES_2, ENQUEUES#1_2, STARTBREADTH_1, STARTBREADTH#1_1, STARTBREADTH#2_1

Compound Symbols:   c_2, c1_1, c2_2, c3, c4_2, c5_1, c6_1, c7_1, c8_1, c9_1, c10, c11, c12, c13, c14_1, c15_1, c16_1, c17, c18_1, c19_1, c20, c21_1, c22_2, c23, c24_1, c25, c26_1, c27, c28_1, c29_2, c30, c31_1, c32_3, c33, c34_1


----------------------------------------

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))

The (relative) TRS S consists of the following rules:

   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(10)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))

The (relative) TRS S consists of the following rules:

   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(11) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))

The (relative) TRS S consists of the following rules:

   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
BREADTH#2(nil, z0) -> c3
BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
CHILDREN#2(::(z0, z1), z2) -> c10
CHILDREN#2(nil, z0) -> c11
CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
CHILDREN#3(nil, z0, z1, z2) -> c13
COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
COPYOVER#2(nil, z0) -> c17
DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
DEQUEUE#2(::(z0, z1), z2) -> c20
DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
DEQUEUE#3(nil) -> c23
DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
EMPTY(z0) -> c25
ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
ENQUEUES#1(nil, z0) -> c30
STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
STARTBREADTH#1(nil) -> c33
STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
breadth#2(nil, z0) -> nil
breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
children#1(nil, z0, z1) -> children#2(z1, z0)
children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
children#2(nil, z0) -> nil
children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
children#3(nil, z0, z1, z2) -> nil
copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
copyover#2(nil, z0) -> tuple#2(nil, z0)
dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
dequeue#2(nil, z0) -> dequeue#3(z0)
dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
empty(z0) -> tuple#2(nil, nil)
enqueue(z0, z1) -> enqueue#1(z1, z0)
enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
enqueues(z0, z1) -> enqueues#1(z0, z1)
enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
enqueues#1(nil, z0) -> z0
startBreadth(z0) -> startBreadth#1(z0)
startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
startBreadth#1(nil) -> nil
startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)

Types:
BREADTH :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c
c :: c1 -> c18 -> c
BREADTH#1 :: tuple#2::::nil:tuple#4 -> c1
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
DEQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c18
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c1 :: c2:c3 -> c1
BREADTH#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c2:c3
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c2 :: c4 -> c5 -> c2:c3
BREADTH#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c4
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
BREADTH#4 :: tuple#2::::nil:tuple#4 -> c5
nil :: tuple#2::::nil:tuple#4
c3 :: c2:c3
c4 :: c6 -> c28 -> c4
BREADTH#5 :: tuple#2::::nil:tuple#4 -> c6
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
ENQUEUES :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c28
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c5 :: c7 -> c5
CHILDREN :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c7
c6 :: c -> c6
c7 :: c8:c9 -> c7
CHILDREN#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c8:c9
c8 :: c12:c13 -> c8:c9
CHILDREN#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c12:c13
c9 :: c10:c11 -> c8:c9
CHILDREN#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
c12 :: c12:c13
c13 :: c12:c13
COPYOVER :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c14
c14 :: c15 -> c14
COPYOVER#1 :: tuple#2::::nil:tuple#4 -> c15
c15 :: c16:c17 -> c15
COPYOVER#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c16:c17
c16 :: c14 -> c16:c17
c17 :: c16:c17
c18 :: c19 -> c18
DEQUEUE#1 :: tuple#2::::nil:tuple#4 -> c19
c19 :: c20:c21 -> c19
DEQUEUE#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c20:c21
c20 :: c20:c21
c21 :: c22:c23 -> c20:c21
DEQUEUE#3 :: tuple#2::::nil:tuple#4 -> c22:c23
c22 :: c24 -> c14 -> c22:c23
DEQUEUE#4 :: tuple#2::::nil:tuple#4 -> c24
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c23 :: c22:c23
c24 :: c18 -> c24
EMPTY :: #unit -> c25
c25 :: c25
ENQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c26
c26 :: c27 -> c26
ENQUEUE#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c27
c27 :: c27
c28 :: c29:c30 -> c28
ENQUEUES#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c29:c30
c29 :: c28 -> c26 -> c29:c30
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c30 :: c29:c30
STARTBREADTH :: tuple#2::::nil:tuple#4 -> c31
c31 :: c32:c33 -> c31
STARTBREADTH#1 :: tuple#2::::nil:tuple#4 -> c32:c33
c32 :: c34 -> c26 -> c25 -> c32:c33
STARTBREADTH#2 :: tuple#2::::nil:tuple#4 -> c34
empty :: #unit -> tuple#2::::nil:tuple#4
#unit :: #unit
c33 :: c32:c33
c34 :: c -> c34
breadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#5 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
hole_c1_35 :: c
hole_tuple#2::::nil:tuple#42_35 :: tuple#2::::nil:tuple#4
hole_c13_35 :: c1
hole_c184_35 :: c18
hole_c2:c35_35 :: c2:c3
hole_c46_35 :: c4
hole_c57_35 :: c5
hole_c68_35 :: c6
hole_c289_35 :: c28
hole_c710_35 :: c7
hole_c8:c911_35 :: c8:c9
hole_c12:c1312_35 :: c12:c13
hole_c10:c1113_35 :: c10:c11
hole_c1414_35 :: c14
hole_c1515_35 :: c15
hole_c16:c1716_35 :: c16:c17
hole_c1917_35 :: c19
hole_c20:c2118_35 :: c20:c21
hole_c22:c2319_35 :: c22:c23
hole_c2420_35 :: c24
hole_c2521_35 :: c25
hole_#unit22_35 :: #unit
hole_c2623_35 :: c26
hole_c2724_35 :: c27
hole_c29:c3025_35 :: c29:c30
hole_c3126_35 :: c31
hole_c32:c3327_35 :: c32:c33
hole_c3428_35 :: c34
gen_tuple#2::::nil:tuple#429_35 :: Nat -> tuple#2::::nil:tuple#4

----------------------------------------

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
BREADTH#1, dequeue, DEQUEUE, BREADTH#5, enqueues, ENQUEUES, COPYOVER, COPYOVER#1, DEQUEUE#1, DEQUEUE#3, DEQUEUE#4, copyover, ENQUEUES#1, breadth#1, breadth#5, copyover#1, dequeue#1, dequeue#3, dequeue#4, enqueues#1

They will be analysed ascendingly in the following order:
BREADTH#1 = BREADTH#5
enqueues < BREADTH#1
ENQUEUES < BREADTH#1
dequeue < BREADTH#5
dequeue < breadth#5
dequeue = dequeue#1
dequeue = dequeue#3
dequeue = dequeue#4
DEQUEUE < BREADTH#5
DEQUEUE = DEQUEUE#1
DEQUEUE = DEQUEUE#3
DEQUEUE = DEQUEUE#4
enqueues < breadth#1
enqueues = enqueues#1
ENQUEUES = ENQUEUES#1
COPYOVER = COPYOVER#1
COPYOVER < DEQUEUE#3
DEQUEUE#1 = DEQUEUE#3
DEQUEUE#1 = DEQUEUE#4
DEQUEUE#3 = DEQUEUE#4
copyover < DEQUEUE#3
copyover = copyover#1
copyover < dequeue#3
breadth#1 = breadth#5
dequeue#1 = dequeue#3
dequeue#1 = dequeue#4
dequeue#3 = dequeue#4

----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
BREADTH#2(nil, z0) -> c3
BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
CHILDREN#2(::(z0, z1), z2) -> c10
CHILDREN#2(nil, z0) -> c11
CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
CHILDREN#3(nil, z0, z1, z2) -> c13
COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
COPYOVER#2(nil, z0) -> c17
DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
DEQUEUE#2(::(z0, z1), z2) -> c20
DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
DEQUEUE#3(nil) -> c23
DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
EMPTY(z0) -> c25
ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
ENQUEUES#1(nil, z0) -> c30
STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
STARTBREADTH#1(nil) -> c33
STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
breadth#2(nil, z0) -> nil
breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
children#1(nil, z0, z1) -> children#2(z1, z0)
children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
children#2(nil, z0) -> nil
children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
children#3(nil, z0, z1, z2) -> nil
copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
copyover#2(nil, z0) -> tuple#2(nil, z0)
dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
dequeue#2(nil, z0) -> dequeue#3(z0)
dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
empty(z0) -> tuple#2(nil, nil)
enqueue(z0, z1) -> enqueue#1(z1, z0)
enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
enqueues(z0, z1) -> enqueues#1(z0, z1)
enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
enqueues#1(nil, z0) -> z0
startBreadth(z0) -> startBreadth#1(z0)
startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
startBreadth#1(nil) -> nil
startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)

Types:
BREADTH :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c
c :: c1 -> c18 -> c
BREADTH#1 :: tuple#2::::nil:tuple#4 -> c1
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
DEQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c18
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c1 :: c2:c3 -> c1
BREADTH#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c2:c3
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c2 :: c4 -> c5 -> c2:c3
BREADTH#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c4
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
BREADTH#4 :: tuple#2::::nil:tuple#4 -> c5
nil :: tuple#2::::nil:tuple#4
c3 :: c2:c3
c4 :: c6 -> c28 -> c4
BREADTH#5 :: tuple#2::::nil:tuple#4 -> c6
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
ENQUEUES :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c28
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c5 :: c7 -> c5
CHILDREN :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c7
c6 :: c -> c6
c7 :: c8:c9 -> c7
CHILDREN#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c8:c9
c8 :: c12:c13 -> c8:c9
CHILDREN#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c12:c13
c9 :: c10:c11 -> c8:c9
CHILDREN#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
c12 :: c12:c13
c13 :: c12:c13
COPYOVER :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c14
c14 :: c15 -> c14
COPYOVER#1 :: tuple#2::::nil:tuple#4 -> c15
c15 :: c16:c17 -> c15
COPYOVER#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c16:c17
c16 :: c14 -> c16:c17
c17 :: c16:c17
c18 :: c19 -> c18
DEQUEUE#1 :: tuple#2::::nil:tuple#4 -> c19
c19 :: c20:c21 -> c19
DEQUEUE#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c20:c21
c20 :: c20:c21
c21 :: c22:c23 -> c20:c21
DEQUEUE#3 :: tuple#2::::nil:tuple#4 -> c22:c23
c22 :: c24 -> c14 -> c22:c23
DEQUEUE#4 :: tuple#2::::nil:tuple#4 -> c24
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c23 :: c22:c23
c24 :: c18 -> c24
EMPTY :: #unit -> c25
c25 :: c25
ENQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c26
c26 :: c27 -> c26
ENQUEUE#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c27
c27 :: c27
c28 :: c29:c30 -> c28
ENQUEUES#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c29:c30
c29 :: c28 -> c26 -> c29:c30
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c30 :: c29:c30
STARTBREADTH :: tuple#2::::nil:tuple#4 -> c31
c31 :: c32:c33 -> c31
STARTBREADTH#1 :: tuple#2::::nil:tuple#4 -> c32:c33
c32 :: c34 -> c26 -> c25 -> c32:c33
STARTBREADTH#2 :: tuple#2::::nil:tuple#4 -> c34
empty :: #unit -> tuple#2::::nil:tuple#4
#unit :: #unit
c33 :: c32:c33
c34 :: c -> c34
breadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#5 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
hole_c1_35 :: c
hole_tuple#2::::nil:tuple#42_35 :: tuple#2::::nil:tuple#4
hole_c13_35 :: c1
hole_c184_35 :: c18
hole_c2:c35_35 :: c2:c3
hole_c46_35 :: c4
hole_c57_35 :: c5
hole_c68_35 :: c6
hole_c289_35 :: c28
hole_c710_35 :: c7
hole_c8:c911_35 :: c8:c9
hole_c12:c1312_35 :: c12:c13
hole_c10:c1113_35 :: c10:c11
hole_c1414_35 :: c14
hole_c1515_35 :: c15
hole_c16:c1716_35 :: c16:c17
hole_c1917_35 :: c19
hole_c20:c2118_35 :: c20:c21
hole_c22:c2319_35 :: c22:c23
hole_c2420_35 :: c24
hole_c2521_35 :: c25
hole_#unit22_35 :: #unit
hole_c2623_35 :: c26
hole_c2724_35 :: c27
hole_c29:c3025_35 :: c29:c30
hole_c3126_35 :: c31
hole_c32:c3327_35 :: c32:c33
hole_c3428_35 :: c34
gen_tuple#2::::nil:tuple#429_35 :: Nat -> tuple#2::::nil:tuple#4


Generator Equations:
gen_tuple#2::::nil:tuple#429_35(0) <=> nil
gen_tuple#2::::nil:tuple#429_35(+(x, 1)) <=> tuple#2(nil, gen_tuple#2::::nil:tuple#429_35(x))


The following defined symbols remain to be analysed:
copyover#1, BREADTH#1, dequeue, DEQUEUE, BREADTH#5, enqueues, ENQUEUES, COPYOVER, COPYOVER#1, DEQUEUE#1, DEQUEUE#3, DEQUEUE#4, copyover, ENQUEUES#1, breadth#1, breadth#5, dequeue#1, dequeue#3, dequeue#4, enqueues#1

They will be analysed ascendingly in the following order:
BREADTH#1 = BREADTH#5
enqueues < BREADTH#1
ENQUEUES < BREADTH#1
dequeue < BREADTH#5
dequeue < breadth#5
dequeue = dequeue#1
dequeue = dequeue#3
dequeue = dequeue#4
DEQUEUE < BREADTH#5
DEQUEUE = DEQUEUE#1
DEQUEUE = DEQUEUE#3
DEQUEUE = DEQUEUE#4
enqueues < breadth#1
enqueues = enqueues#1
ENQUEUES = ENQUEUES#1
COPYOVER = COPYOVER#1
COPYOVER < DEQUEUE#3
DEQUEUE#1 = DEQUEUE#3
DEQUEUE#1 = DEQUEUE#4
DEQUEUE#3 = DEQUEUE#4
copyover < DEQUEUE#3
copyover = copyover#1
copyover < dequeue#3
breadth#1 = breadth#5
dequeue#1 = dequeue#3
dequeue#1 = dequeue#4
dequeue#3 = dequeue#4

----------------------------------------

(17) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)
Tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
S tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#2(nil, z0) -> c3
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   CHILDREN#2(::(z0, z1), z2) -> c10
   CHILDREN#2(nil, z0) -> c11
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   CHILDREN#3(nil, z0, z1, z2) -> c13
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   COPYOVER#2(nil, z0) -> c17
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(::(z0, z1), z2) -> c20
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(nil) -> c23
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   EMPTY(z0) -> c25
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   ENQUEUES#1(nil, z0) -> c30
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#1(nil) -> c33
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
K tuples:none
Defined Rule Symbols:   breadth_2, breadth#1_1, breadth#2_2, breadth#3_2, breadth#4_1, breadth#5_1, children_4, children#1_3, children#2_2, children#3_4, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, empty_1, enqueue_2, enqueue#1_2, enqueues_2, enqueues#1_2, startBreadth_1, startBreadth#1_1, startBreadth#2_1

Defined Pair Symbols:   BREADTH_2, BREADTH#1_1, BREADTH#2_2, BREADTH#3_2, BREADTH#4_1, BREADTH#5_1, CHILDREN_4, CHILDREN#1_3, CHILDREN#2_2, CHILDREN#3_4, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, EMPTY_1, ENQUEUE_2, ENQUEUE#1_2, ENQUEUES_2, ENQUEUES#1_2, STARTBREADTH_1, STARTBREADTH#1_1, STARTBREADTH#2_1

Compound Symbols:   c_2, c1_1, c2_2, c3, c4_2, c5_1, c6_1, c7_1, c8_1, c9_1, c10, c11, c12, c13, c14_1, c15_1, c16_1, c17, c18_1, c19_1, c20, c21_1, c22_2, c23, c24_1, c25, c26_1, c27, c28_1, c29_2, c30, c31_1, c32_3, c33, c34_1


----------------------------------------

(19) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   STARTBREADTH(z0) -> c31(STARTBREADTH#1(z0))
Removed 17 trailing nodes:
   DEQUEUE#2(::(z0, z1), z2) -> c20
   CHILDREN#1(nil, z0, z1) -> c9(CHILDREN#2(z1, z0))
   BREADTH#2(nil, z0) -> c3
   COPYOVER#2(nil, z0) -> c17
   BREADTH#4(tuple#4(z0, z1, z2, z3)) -> c5(CHILDREN(z0, z1, z2, z3))
   CHILDREN(z0, z1, z2, z3) -> c7(CHILDREN#1(z2, z1, z3))
   CHILDREN#3(::(z0, z1), z2, z3, z4) -> c12
   ENQUEUE#1(tuple#2(z0, z1), z2) -> c27
   EMPTY(z0) -> c25
   CHILDREN#2(::(z0, z1), z2) -> c10
   ENQUEUE(z0, z1) -> c26(ENQUEUE#1(z1, z0))
   CHILDREN#3(nil, z0, z1, z2) -> c13
   CHILDREN#1(::(z0, z1), z2, z3) -> c8(CHILDREN#3(z3, z2, z0, z1))
   ENQUEUES#1(nil, z0) -> c30
   CHILDREN#2(nil, z0) -> c11
   STARTBREADTH#1(nil) -> c33
   DEQUEUE#3(nil) -> c23

----------------------------------------

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)
Tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
S tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2), BREADTH#4(z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)), ENQUEUE(z0, z2))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))), ENQUEUE(tuple#4(z0, z0, z1, z1), empty(#unit)), EMPTY(#unit))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
K tuples:none
Defined Rule Symbols:   breadth_2, breadth#1_1, breadth#2_2, breadth#3_2, breadth#4_1, breadth#5_1, children_4, children#1_3, children#2_2, children#3_4, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, empty_1, enqueue_2, enqueue#1_2, enqueues_2, enqueues#1_2, startBreadth_1, startBreadth#1_1, startBreadth#2_1

Defined Pair Symbols:   BREADTH_2, BREADTH#1_1, BREADTH#2_2, BREADTH#3_2, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, STARTBREADTH#1_1, STARTBREADTH#2_1

Compound Symbols:   c_2, c1_1, c2_2, c4_2, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c22_2, c24_1, c28_1, c29_2, c32_3, c34_1


----------------------------------------

(21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)
Tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
S tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
K tuples:none
Defined Rule Symbols:   breadth_2, breadth#1_1, breadth#2_2, breadth#3_2, breadth#4_1, breadth#5_1, children_4, children#1_3, children#2_2, children#3_4, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, empty_1, enqueue_2, enqueue#1_2, enqueues_2, enqueues#1_2, startBreadth_1, startBreadth#1_1, startBreadth#2_1

Defined Pair Symbols:   BREADTH_2, BREADTH#1_1, BREADTH#3_2, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1

Compound Symbols:   c_2, c1_1, c4_2, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c22_2, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1


----------------------------------------

(23) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))

----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)
Tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
S tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   breadth_2, breadth#1_1, breadth#2_2, breadth#3_2, breadth#4_1, breadth#5_1, children_4, children#1_3, children#2_2, children#3_4, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, empty_1, enqueue_2, enqueue#1_2, enqueues_2, enqueues#1_2, startBreadth_1, startBreadth#1_1, startBreadth#2_1

Defined Pair Symbols:   BREADTH_2, BREADTH#1_1, BREADTH#3_2, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1

Compound Symbols:   c_2, c1_1, c4_2, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c22_2, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1


----------------------------------------

(25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   breadth(z0, z1) -> breadth#1(dequeue(z0, z1))
   breadth#1(tuple#2(z0, z1)) -> breadth#2(z1, z0)
   breadth#2(::(z0, z1), z2) -> breadth#3(breadth#4(z0), z2)
   breadth#2(nil, z0) -> nil
   breadth#3(tuple#2(z0, z1), z2) -> ::(z0, breadth#5(enqueues(z1, z2)))
   breadth#5(tuple#2(z0, z1)) -> breadth(z0, z1)
   startBreadth(z0) -> startBreadth#1(z0)
   startBreadth#1(::(z0, z1)) -> startBreadth#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))
   startBreadth#1(nil) -> nil
   startBreadth#2(tuple#2(z0, z1)) -> breadth(z0, z1)

----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)
Tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
S tuples:
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, enqueues_2, enqueues#1_2, enqueue_2, enqueue#1_2, breadth#4_1, children_4, children#1_3, children#3_4, children#2_2, empty_1

Defined Pair Symbols:   BREADTH_2, BREADTH#1_1, BREADTH#3_2, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1

Compound Symbols:   c_2, c1_1, c4_2, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c22_2, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1


----------------------------------------

(27) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))

The (relative) TRS S consists of the following rules:

   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)

Rewrite Strategy: INNERMOST
----------------------------------------

(29) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(30)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1))
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(31) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(32)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1)) [1]
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0)) [1]
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2)) [1]
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1)) [1]
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1))) [1]
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1)) [1]
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2))) [1]
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1))) [1]
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0)) [1]
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0)) [1]
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil)) [1]
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1)) [1]
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1)) [1]
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2)) [1]
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2))) [1]
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1)) [0]
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))) [0]
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1)) [0]
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0) [0]
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil)) [0]
   dequeue#2(nil, z0) -> dequeue#3(z0) [0]
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil)) [0]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [0]
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1) [0]
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1)) [0]
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1) [0]
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2)) [0]
   copyover#2(nil, z0) -> tuple#2(nil, z0) [0]
   enqueues(z0, z1) -> enqueues#1(z0, z1) [0]
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2)) [0]
   enqueues#1(nil, z0) -> z0 [0]
   enqueue(z0, z1) -> enqueue#1(z1, z0) [0]
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1) [0]
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3) [0]
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)) [0]
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1) [0]
   children#1(nil, z0, z1) -> children#2(z1, z0) [0]
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil)) [0]
   children#3(nil, z0, z1, z2) -> nil [0]
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil) [0]
   children#2(nil, z0) -> nil [0]
   empty(z0) -> tuple#2(nil, nil) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(33) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(34)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1)) [1]
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0)) [1]
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2)) [1]
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1)) [1]
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1))) [1]
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1)) [1]
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2))) [1]
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1))) [1]
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0)) [1]
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0)) [1]
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil)) [1]
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1)) [1]
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1)) [1]
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2)) [1]
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2))) [1]
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1)) [0]
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))) [0]
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1)) [0]
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0) [0]
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil)) [0]
   dequeue#2(nil, z0) -> dequeue#3(z0) [0]
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil)) [0]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [0]
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1) [0]
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1)) [0]
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1) [0]
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2)) [0]
   copyover#2(nil, z0) -> tuple#2(nil, z0) [0]
   enqueues(z0, z1) -> enqueues#1(z0, z1) [0]
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2)) [0]
   enqueues#1(nil, z0) -> z0 [0]
   enqueue(z0, z1) -> enqueue#1(z1, z0) [0]
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1) [0]
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3) [0]
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)) [0]
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1) [0]
   children#1(nil, z0, z1) -> children#2(z1, z0) [0]
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil)) [0]
   children#3(nil, z0, z1, z2) -> nil [0]
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil) [0]
   children#2(nil, z0) -> nil [0]
   empty(z0) -> tuple#2(nil, nil) [0]

The TRS has the following type information:
BREADTH :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c
c :: c1 -> c18 -> c
BREADTH#1 :: tuple#2::::nil:tuple#4 -> c1
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
DEQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c18
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c1 :: c2 -> c1
BREADTH#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c2
BREADTH#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c4
c4 :: c6 -> c28 -> c4
BREADTH#5 :: tuple#2::::nil:tuple#4 -> c6
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
ENQUEUES :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c28
c6 :: c -> c6
COPYOVER :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c14
c14 :: c15 -> c14
COPYOVER#1 :: tuple#2::::nil:tuple#4 -> c15
c15 :: c16 -> c15
COPYOVER#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c16
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c16 :: c14 -> c16
c18 :: c19 -> c18
DEQUEUE#1 :: tuple#2::::nil:tuple#4 -> c19
c19 :: c21 -> c19
DEQUEUE#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c21
nil :: tuple#2::::nil:tuple#4
c21 :: c22 -> c21
DEQUEUE#3 :: tuple#2::::nil:tuple#4 -> c22
c22 :: c24 -> c14 -> c22
DEQUEUE#4 :: tuple#2::::nil:tuple#4 -> c24
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c24 :: c18 -> c24
c28 :: c29 -> c28
ENQUEUES#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c29
c2 :: c4 -> c2
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c29 :: c28 -> c29
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
STARTBREADTH#2 :: tuple#2::::nil:tuple#4 -> c34
c34 :: c -> c34
STARTBREADTH#1 :: tuple#2::::nil:tuple#4 -> c32
c32 :: c34 -> c32
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
empty :: #unit -> tuple#2::::nil:tuple#4
#unit :: #unit
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4

Rewrite Strategy: INNERMOST
----------------------------------------

(35) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   BREADTH_2
   BREADTH#1_1
   BREADTH#3_2
   BREADTH#5_1
   COPYOVER_2
   COPYOVER#1_1
   COPYOVER#2_2
   DEQUEUE_2
   DEQUEUE#1_1
   DEQUEUE#2_2
   DEQUEUE#3_1
   DEQUEUE#4_1
   ENQUEUES_2
   BREADTH#2_2
   ENQUEUES#1_2

(c) The following functions are completely defined:

   STARTBREADTH#2_1
   STARTBREADTH#1_1
   dequeue_2
   dequeue#1_1
   dequeue#2_2
   dequeue#3_1
   dequeue#4_1
   copyover_2
   copyover#1_1
   copyover#2_2
   enqueues_2
   enqueues#1_2
   enqueue_2
   enqueue#1_2
   breadth#4_1
   children_4
   children#1_3
   children#3_4
   children#2_2
   empty_1

Due to the following rules being added:

   STARTBREADTH#2(v0) -> const15 [0]
   STARTBREADTH#1(v0) -> const16 [0]
   dequeue(v0, v1) -> nil [0]
   dequeue#1(v0) -> nil [0]
   dequeue#2(v0, v1) -> nil [0]
   dequeue#3(v0) -> nil [0]
   dequeue#4(v0) -> nil [0]
   copyover(v0, v1) -> nil [0]
   copyover#1(v0) -> nil [0]
   copyover#2(v0, v1) -> nil [0]
   enqueues(v0, v1) -> nil [0]
   enqueues#1(v0, v1) -> nil [0]
   enqueue(v0, v1) -> nil [0]
   enqueue#1(v0, v1) -> nil [0]
   breadth#4(v0) -> nil [0]
   children(v0, v1, v2, v3) -> nil [0]
   children#1(v0, v1, v2) -> nil [0]
   children#3(v0, v1, v2, v3) -> nil [0]
   children#2(v0, v1) -> nil [0]
   empty(v0) -> nil [0]

And the following fresh constants:    const15, const16, const, const1, const2, const3, const4, const5, const6, const7, const8, const9, const10, const11, const12, const13, const14

----------------------------------------

(36)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1)) [1]
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0)) [1]
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2)) [1]
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1)) [1]
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1))) [1]
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1)) [1]
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2))) [1]
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1))) [1]
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0)) [1]
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0)) [1]
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil)) [1]
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1)) [1]
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1)) [1]
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2)) [1]
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2))) [1]
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1)) [0]
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))) [0]
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1)) [0]
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0) [0]
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil)) [0]
   dequeue#2(nil, z0) -> dequeue#3(z0) [0]
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil)) [0]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [0]
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1) [0]
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1)) [0]
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1) [0]
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2)) [0]
   copyover#2(nil, z0) -> tuple#2(nil, z0) [0]
   enqueues(z0, z1) -> enqueues#1(z0, z1) [0]
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2)) [0]
   enqueues#1(nil, z0) -> z0 [0]
   enqueue(z0, z1) -> enqueue#1(z1, z0) [0]
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1) [0]
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3) [0]
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)) [0]
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1) [0]
   children#1(nil, z0, z1) -> children#2(z1, z0) [0]
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil)) [0]
   children#3(nil, z0, z1, z2) -> nil [0]
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil) [0]
   children#2(nil, z0) -> nil [0]
   empty(z0) -> tuple#2(nil, nil) [0]
   STARTBREADTH#2(v0) -> const15 [0]
   STARTBREADTH#1(v0) -> const16 [0]
   dequeue(v0, v1) -> nil [0]
   dequeue#1(v0) -> nil [0]
   dequeue#2(v0, v1) -> nil [0]
   dequeue#3(v0) -> nil [0]
   dequeue#4(v0) -> nil [0]
   copyover(v0, v1) -> nil [0]
   copyover#1(v0) -> nil [0]
   copyover#2(v0, v1) -> nil [0]
   enqueues(v0, v1) -> nil [0]
   enqueues#1(v0, v1) -> nil [0]
   enqueue(v0, v1) -> nil [0]
   enqueue#1(v0, v1) -> nil [0]
   breadth#4(v0) -> nil [0]
   children(v0, v1, v2, v3) -> nil [0]
   children#1(v0, v1, v2) -> nil [0]
   children#3(v0, v1, v2, v3) -> nil [0]
   children#2(v0, v1) -> nil [0]
   empty(v0) -> nil [0]

The TRS has the following type information:
BREADTH :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c
c :: c1 -> c18 -> c
BREADTH#1 :: tuple#2::::nil:tuple#4 -> c1
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
DEQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c18
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c1 :: c2 -> c1
BREADTH#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c2
BREADTH#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c4
c4 :: c6 -> c28 -> c4
BREADTH#5 :: tuple#2::::nil:tuple#4 -> c6
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
ENQUEUES :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c28
c6 :: c -> c6
COPYOVER :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c14
c14 :: c15 -> c14
COPYOVER#1 :: tuple#2::::nil:tuple#4 -> c15
c15 :: c16 -> c15
COPYOVER#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c16
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c16 :: c14 -> c16
c18 :: c19 -> c18
DEQUEUE#1 :: tuple#2::::nil:tuple#4 -> c19
c19 :: c21 -> c19
DEQUEUE#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c21
nil :: tuple#2::::nil:tuple#4
c21 :: c22 -> c21
DEQUEUE#3 :: tuple#2::::nil:tuple#4 -> c22
c22 :: c24 -> c14 -> c22
DEQUEUE#4 :: tuple#2::::nil:tuple#4 -> c24
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c24 :: c18 -> c24
c28 :: c29 -> c28
ENQUEUES#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c29
c2 :: c4 -> c2
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c29 :: c28 -> c29
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
STARTBREADTH#2 :: tuple#2::::nil:tuple#4 -> c34:const15
c34 :: c -> c34:const15
STARTBREADTH#1 :: tuple#2::::nil:tuple#4 -> c32:const16
c32 :: c34:const15 -> c32:const16
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
empty :: #unit -> tuple#2::::nil:tuple#4
#unit :: #unit
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
const15 :: c34:const15
const16 :: c32:const16
const :: c
const1 :: c1
const2 :: c18
const3 :: c2
const4 :: c4
const5 :: c6
const6 :: c28
const7 :: c14
const8 :: c15
const9 :: c16
const10 :: c19
const11 :: c21
const12 :: c22
const13 :: c24
const14 :: c29

Rewrite Strategy: INNERMOST
----------------------------------------

(37) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(38)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1)) [1]
   BREADTH(z0, z1) -> c(BREADTH#1(nil), DEQUEUE(z0, z1)) [1]
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0)) [1]
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues#1(z1, z2)), ENQUEUES(z1, z2)) [1]
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(nil), ENQUEUES(z1, z2)) [1]
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1)) [1]
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1))) [1]
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1)) [1]
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2))) [1]
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1))) [1]
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0)) [1]
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0)) [1]
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(z0, z1), nil))), COPYOVER(::(z0, z1), nil)) [1]
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(nil), COPYOVER(::(z0, z1), nil)) [1]
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1)) [1]
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1)) [1]
   BREADTH#2(::(tuple#4(z0', z1', z2', z3'), z1), z2) -> c2(BREADTH#3(children(z0', z1', z2', z3'), z2)) [1]
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(nil, z2)) [1]
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue#1(z2, z0))) [1]
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, nil)) [1]
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1)) [0]
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), tuple#2(nil, nil)))) [0]
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), nil))) [0]
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1)) [0]
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0) [0]
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil)) [0]
   dequeue#2(nil, z0) -> dequeue#3(z0) [0]
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover#1(tuple#2(::(z0, z1), nil))) [0]
   dequeue#3(::(z0, z1)) -> dequeue#4(nil) [0]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [0]
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1) [0]
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1)) [0]
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1) [0]
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2)) [0]
   copyover#2(nil, z0) -> tuple#2(nil, z0) [0]
   enqueues(z0, z1) -> enqueues#1(z0, z1) [0]
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue#1(z2, z0)) [0]
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, nil) [0]
   enqueues#1(nil, z0) -> z0 [0]
   enqueue(z0, z1) -> enqueue#1(z1, z0) [0]
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1) [0]
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3) [0]
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)) [0]
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1) [0]
   children#1(nil, z0, z1) -> children#2(z1, z0) [0]
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil)) [0]
   children#3(nil, z0, z1, z2) -> nil [0]
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil) [0]
   children#2(nil, z0) -> nil [0]
   empty(z0) -> tuple#2(nil, nil) [0]
   STARTBREADTH#2(v0) -> const15 [0]
   STARTBREADTH#1(v0) -> const16 [0]
   dequeue(v0, v1) -> nil [0]
   dequeue#1(v0) -> nil [0]
   dequeue#2(v0, v1) -> nil [0]
   dequeue#3(v0) -> nil [0]
   dequeue#4(v0) -> nil [0]
   copyover(v0, v1) -> nil [0]
   copyover#1(v0) -> nil [0]
   copyover#2(v0, v1) -> nil [0]
   enqueues(v0, v1) -> nil [0]
   enqueues#1(v0, v1) -> nil [0]
   enqueue(v0, v1) -> nil [0]
   enqueue#1(v0, v1) -> nil [0]
   breadth#4(v0) -> nil [0]
   children(v0, v1, v2, v3) -> nil [0]
   children#1(v0, v1, v2) -> nil [0]
   children#3(v0, v1, v2, v3) -> nil [0]
   children#2(v0, v1) -> nil [0]
   empty(v0) -> nil [0]

The TRS has the following type information:
BREADTH :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c
c :: c1 -> c18 -> c
BREADTH#1 :: tuple#2::::nil:tuple#4 -> c1
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
DEQUEUE :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c18
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c1 :: c2 -> c1
BREADTH#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c2
BREADTH#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c4
c4 :: c6 -> c28 -> c4
BREADTH#5 :: tuple#2::::nil:tuple#4 -> c6
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
ENQUEUES :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c28
c6 :: c -> c6
COPYOVER :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c14
c14 :: c15 -> c14
COPYOVER#1 :: tuple#2::::nil:tuple#4 -> c15
c15 :: c16 -> c15
COPYOVER#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c16
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c16 :: c14 -> c16
c18 :: c19 -> c18
DEQUEUE#1 :: tuple#2::::nil:tuple#4 -> c19
c19 :: c21 -> c19
DEQUEUE#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c21
nil :: tuple#2::::nil:tuple#4
c21 :: c22 -> c21
DEQUEUE#3 :: tuple#2::::nil:tuple#4 -> c22
c22 :: c24 -> c14 -> c22
DEQUEUE#4 :: tuple#2::::nil:tuple#4 -> c24
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c24 :: c18 -> c24
c28 :: c29 -> c28
ENQUEUES#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> c29
c2 :: c4 -> c2
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
c29 :: c28 -> c29
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
STARTBREADTH#2 :: tuple#2::::nil:tuple#4 -> c34:const15
c34 :: c -> c34:const15
STARTBREADTH#1 :: tuple#2::::nil:tuple#4 -> c32:const16
c32 :: c34:const15 -> c32:const16
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
empty :: #unit -> tuple#2::::nil:tuple#4
#unit :: #unit
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
const15 :: c34:const15
const16 :: c32:const16
const :: c
const1 :: c1
const2 :: c18
const3 :: c2
const4 :: c4
const5 :: c6
const6 :: c28
const7 :: c14
const8 :: c15
const9 :: c16
const10 :: c19
const11 :: c21
const12 :: c22
const13 :: c24
const14 :: c29

Rewrite Strategy: INNERMOST
----------------------------------------

(39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   #unit => 0
   const15 => 0
   const16 => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0
   const5 => 0
   const6 => 0
   const7 => 0
   const8 => 0
   const9 => 0
   const10 => 0
   const11 => 0
   const12 => 0
   const13 => 0
   const14 => 0

----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z0 + z1)) + DEQUEUE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z2) :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' = z2, z2 >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z2)) + ENQUEUES(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z0) :|: z0 >= 0, z = 0, z' = z0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, enqueue#1(z2, z0)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue(1 + z0 + z0 + z1 + z1, 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue(1 + z0 + z0 + z1 + z1, 1 + 0 + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z0 + z1) + children#1(z2, z1, z3) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z5 = z3, z2 >= 0, z3 >= 0
   children#1(z, z', z'') -{ 0 }-> children#3(z3, z2, z0, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   children#1(z, z', z'') -{ 0 }-> children#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z2 + 0 + z1) + 0 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 0, z5 = z2, z' = z0, z2 >= 0, z'' = z1
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z3 + z2 + 0 + z4) + (1 + (1 + z3 + z0 + z4 + z1) + 0) :|: z1 >= 0, z5 = z4, z' = z2, z0 >= 0, z'' = z3, z4 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   copyover(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z0 :|: z0 >= 0, z = 0, z' = z0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z0) :|: z0 >= 0, z = 0, z' = z0
   dequeue#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z2 + z1) + (1 + z0 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z = z0, z0 >= 0
   enqueue(z, z') -{ 0 }-> enqueue#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueues#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, enqueue#1(z2, z0)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


----------------------------------------

(41) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 0, z5 = z2, z' = z0, z2 >= 0, z'' = z1
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z3 + z2 + 0 + z4) + (1 + (1 + z3 + z0 + z4 + z1) + 0) :|: z1 >= 0, z5 = z4, z' = z2, z0 >= 0, z'' = z3, z4 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z2 + 0 + z1) + 0 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue(z, z') -{ 0 }-> enqueue#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   children#1(z, z', z'') -{ 0 }-> children#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#1(z, z', z'') -{ 0 }-> children#3(z3, z2, z0, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z0 + z1)) + DEQUEUE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z2) :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' = z2, z2 >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z2)) + ENQUEUES(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z0) :|: z0 >= 0, z = 0, z' = z0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, v0 >= 0, v1 >= 0, z2 = v0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z1' >= 0, z0 = z2', z0' >= 0, z2 = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z0 + z1) + children#3(z3', z2', z0', z1') :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z5 = z3, z2 >= 0, z3 >= 0, z1' >= 0, z1 = z2', z0' >= 0, z3 = z3', z2 = 1 + z0' + z1', z2' >= 0, z3' >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z0 + z1) + children#2(z1', z0') :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z5 = z3, z2 >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z2 = 0, z1 = z0', z3 = z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z0 + z1) + 0 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z5 = z3, z2 >= 0, z3 >= 0, v0 >= 0, z3 = v2, v1 >= 0, z2 = v0, z1 = v1, v2 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z3 = 0, z1 = z2', z2 = z0', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0, z1 = v3, v0 >= 0, z0 = v2, v1 >= 0, z3 = v0, z2 = v1, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z0' >= 0, z1 = 0, z0 = z0'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, v1 >= 0, z1 = v0, z0 = v1
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z2 + 0 + z1') + 0 :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1' >= 0, z0 = z2, z0' >= 0, z1 = 1 + z0' + z1', z2 >= 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z2' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0, z1' >= 0, z1 = z4, z2 = z2', z0' >= 0, z0 = z3', z4 >= 0, z3 = 1 + z0' + z1', z2' >= 0, z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z2 + 0 + z1) + 0 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 0, z5 = z2, z' = z0, z2 >= 0, z'' = z1
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z3 + z2 + 0 + z4) + (1 + (1 + z3 + z0 + z4 + z1) + 0) :|: z1 >= 0, z5 = z4, z' = z2, z0 >= 0, z'' = z3, z4 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   copyover(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z0 :|: z0 >= 0, z = 0, z' = z0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z0) :|: z0 >= 0, z = 0, z' = z0
   dequeue#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z2 + z1) + (1 + z0 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z = z0, z0 >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue(z, z') -{ 0 }-> 0 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, v0 >= 0, v1 >= 0, z1 = v0, z0 = v1
   enqueue(z, z') -{ 0 }-> 1 + (1 + z2 + z0') + z1' :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z1' >= 0, z0 = z2, z0' >= 0, z1 = 1 + z0' + z1', z2 >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueues#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, v0 >= 0, v1 >= 0, z2 = v0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z1' >= 0, z0 = z2', z0' >= 0, z2 = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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(43) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


----------------------------------------

(45) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { children#1 }
   { empty }
   { enqueue }
   { children#3 }
   { COPYOVER#1, COPYOVER, COPYOVER#2 }
   { enqueue#1 }
   { children#2 }
   { ENQUEUES, ENQUEUES#1 }
   { enqueues#1, enqueues }
   { copyover#2, copyover#1, copyover }
   { children }
   { DEQUEUE#3, DEQUEUE#4, DEQUEUE#2, DEQUEUE, DEQUEUE#1 }
   { dequeue, dequeue#3, dequeue#1, dequeue#2, dequeue#4 }
   { BREADTH#1, BREADTH#2, BREADTH#3, BREADTH#5, BREADTH }
   { breadth#4 }
   { STARTBREADTH#2 }
   { STARTBREADTH#1 }

----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {children#1}, {empty}, {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}

----------------------------------------

(47) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {children#1}, {empty}, {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}

----------------------------------------

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: children#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 2*z + z' + z''

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {children#1}, {empty}, {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: ?, size: O(n^1) [1 + 2*z + z' + z'']

----------------------------------------

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: children#1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {empty}, {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']

----------------------------------------

(53) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {empty}, {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']

----------------------------------------

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {empty}, {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: ?, size: O(1) [1]

----------------------------------------

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]

----------------------------------------

(59) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]

----------------------------------------

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: enqueue
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {enqueue}, {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: enqueue
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']

----------------------------------------

(65) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']

----------------------------------------

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: children#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z + z' + 2*z'' + 2*z5

----------------------------------------

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {children#3}, {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  children#3: runtime: ?, size: O(n^1) [3 + z + z' + 2*z'' + 2*z5]

----------------------------------------

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: children#3
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#3(z5, z', z0', z1') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [0], size: O(n^1) [3 + z + z' + 2*z'' + 2*z5]

----------------------------------------

(71) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z5 + z' + 2 * z0' + 2 * z1' + 3, z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [0], size: O(n^1) [3 + z + z' + 2*z'' + 2*z5]

----------------------------------------

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: COPYOVER#1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: COPYOVER
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

Computed SIZE bound using CoFloCo for: COPYOVER#2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z5 + z' + 2 * z0' + 2 * z1' + 3, z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [0], size: O(n^1) [3 + z + z' + 2*z'' + 2*z5]
  COPYOVER#1: runtime: ?, size: O(1) [0]
  COPYOVER: runtime: ?, size: O(1) [1]
  COPYOVER#2: runtime: ?, size: O(1) [2]

----------------------------------------

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: COPYOVER#1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue#1(1 + z + z')) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(0) + DEQUEUE(z, z') :|: z' >= 0, z >= 0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(children(z0', z1', z2', z3'), z') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0' + z1' + z2' + z3') + z1, z3' >= 0, z1' >= 0, z2' >= 0, z' >= 0
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues#1(z1, z')) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(0) + ENQUEUES(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z + z') :|: z' >= 0, z >= 0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z + z') :|: z' >= 0, z >= 0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z') :|: z' >= 0, z = 0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover#1(1 + (1 + z0 + z1) + 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(0) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z, z') :|: z' >= 0, z >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 1 + 0 + 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue#1(z1', z0')) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, 1 + z0 + z0 + z1 + z1 = z0', z1' >= 0, 0 = z1', z0' >= 0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 1 + 0 + 0 = v1
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, 1 + z0 + z0 + z1 + z1 = v0, 0 = v1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: z >= 0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z5 + z' + 2 * z0' + 2 * z1' + 3, z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + children#2(z5, z') :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0, z'' = 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z + z') + 0 :|: z' >= 0, z >= 0, z'' >= 0, z5 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z'' = 0, z1 = z2', z2' >= 0, z0 = z1'
   children#1(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1 = v3, z0 = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z'' = 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z0' + z' + 0 + z1') + 0 :|: z'' >= 0, z' >= 0, z = 0, z1' >= 0, z0' >= 0, z'' = 1 + z0' + z1'
   children#1(z, z', z'') -{ 0 }-> 1 + (1 + z3' + z' + 0 + z4) + (1 + (1 + z3' + z0' + z4 + z1') + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, z1' >= 0, z1 = z4, z0' >= 0, z0 = z3', z4 >= 0, z'' = 1 + z0' + z1', z3' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z' + 0 + z1) + 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z'' >= 0, z' >= 0, z = 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z5 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z'' + z' + 0 + z5) + (1 + (1 + z'' + z0 + z5 + z1) + 0) :|: z1 >= 0, z0 >= 0, z5 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z' + z1) + (1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover#1(1 + (1 + z0 + z1) + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> dequeue#4(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue(z, z') -{ 0 }-> 1 + (1 + z + z0') + z1' :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z' + z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueues#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, v1 >= 0, z0 = v1
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, 1 + (1 + z2' + z0') + z1') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z1' >= 0, z0 = z2', z0' >= 0, z' = 1 + z0' + z1', z2' >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {COPYOVER#1,COPYOVER,COPYOVER#2}, {enqueue#1}, {children#2}, {ENQUEUES,ENQUEUES#1}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {DEQUEUE#3,DEQUEUE#4,DEQUEUE#2,DEQUEUE,DEQUEUE#1}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {BREADTH#1,BREADTH#2,BREADTH#3,BREADTH#5,BREADTH}, {breadth#4}, {STARTBREADTH#2}, {STARTBREADTH#1}
Previous analysis results are:
  children#1: runtime: O(1) [0], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [0], size: O(1) [1]
  enqueue: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [0], size: O(n^1) [3 + z + z' + 2*z'' + 2*z5]
  COPYOVER#1: runtime: INF, size: O(1) [0]
  COPYOVER: runtime: ?, size: O(1) [1]
  COPYOVER#2: runtime: ?, size: O(1) [2]

----------------------------------------

(77) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   STARTBREADTH#2(v0) -> null_STARTBREADTH#2 [0]
   STARTBREADTH#1(v0) -> null_STARTBREADTH#1 [0]
   dequeue(v0, v1) -> null_dequeue [0]
   dequeue#1(v0) -> null_dequeue#1 [0]
   dequeue#2(v0, v1) -> null_dequeue#2 [0]
   dequeue#3(v0) -> null_dequeue#3 [0]
   dequeue#4(v0) -> null_dequeue#4 [0]
   copyover(v0, v1) -> null_copyover [0]
   copyover#1(v0) -> null_copyover#1 [0]
   copyover#2(v0, v1) -> null_copyover#2 [0]
   enqueues(v0, v1) -> null_enqueues [0]
   enqueues#1(v0, v1) -> null_enqueues#1 [0]
   enqueue(v0, v1) -> null_enqueue [0]
   enqueue#1(v0, v1) -> null_enqueue#1 [0]
   breadth#4(v0) -> null_breadth#4 [0]
   children(v0, v1, v2, v3) -> null_children [0]
   children#1(v0, v1, v2) -> null_children#1 [0]
   children#3(v0, v1, v2, v3) -> null_children#3 [0]
   children#2(v0, v1) -> null_children#2 [0]
   empty(v0) -> null_empty [0]
   BREADTH#1(v0) -> null_BREADTH#1 [0]
   BREADTH#3(v0, v1) -> null_BREADTH#3 [0]
   BREADTH#5(v0) -> null_BREADTH#5 [0]
   COPYOVER#1(v0) -> null_COPYOVER#1 [0]
   COPYOVER#2(v0, v1) -> null_COPYOVER#2 [0]
   DEQUEUE#1(v0) -> null_DEQUEUE#1 [0]
   DEQUEUE#2(v0, v1) -> null_DEQUEUE#2 [0]
   DEQUEUE#3(v0) -> null_DEQUEUE#3 [0]
   DEQUEUE#4(v0) -> null_DEQUEUE#4 [0]
   BREADTH#2(v0, v1) -> null_BREADTH#2 [0]
   ENQUEUES#1(v0, v1) -> null_ENQUEUES#1 [0]

And the following fresh constants:    null_STARTBREADTH#2, null_STARTBREADTH#1, null_dequeue, null_dequeue#1, null_dequeue#2, null_dequeue#3, null_dequeue#4, null_copyover, null_copyover#1, null_copyover#2, null_enqueues, null_enqueues#1, null_enqueue, null_enqueue#1, null_breadth#4, null_children, null_children#1, null_children#3, null_children#2, null_empty, null_BREADTH#1, null_BREADTH#3, null_BREADTH#5, null_COPYOVER#1, null_COPYOVER#2, null_DEQUEUE#1, null_DEQUEUE#2, null_DEQUEUE#3, null_DEQUEUE#4, null_BREADTH#2, null_ENQUEUES#1, const, const1, const2, const3

----------------------------------------

(78)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1)) [1]
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0)) [1]
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2)) [1]
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1)) [1]
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1))) [1]
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1)) [1]
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2))) [1]
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1))) [1]
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0)) [1]
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0)) [1]
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil)) [1]
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1)) [1]
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1)) [1]
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2)) [1]
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2))) [1]
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1)) [0]
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))) [0]
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1)) [0]
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0) [0]
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil)) [0]
   dequeue#2(nil, z0) -> dequeue#3(z0) [0]
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil)) [0]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [0]
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1) [0]
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1)) [0]
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1) [0]
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2)) [0]
   copyover#2(nil, z0) -> tuple#2(nil, z0) [0]
   enqueues(z0, z1) -> enqueues#1(z0, z1) [0]
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2)) [0]
   enqueues#1(nil, z0) -> z0 [0]
   enqueue(z0, z1) -> enqueue#1(z1, z0) [0]
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1) [0]
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3) [0]
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)) [0]
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1) [0]
   children#1(nil, z0, z1) -> children#2(z1, z0) [0]
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil)) [0]
   children#3(nil, z0, z1, z2) -> nil [0]
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil) [0]
   children#2(nil, z0) -> nil [0]
   empty(z0) -> tuple#2(nil, nil) [0]
   STARTBREADTH#2(v0) -> null_STARTBREADTH#2 [0]
   STARTBREADTH#1(v0) -> null_STARTBREADTH#1 [0]
   dequeue(v0, v1) -> null_dequeue [0]
   dequeue#1(v0) -> null_dequeue#1 [0]
   dequeue#2(v0, v1) -> null_dequeue#2 [0]
   dequeue#3(v0) -> null_dequeue#3 [0]
   dequeue#4(v0) -> null_dequeue#4 [0]
   copyover(v0, v1) -> null_copyover [0]
   copyover#1(v0) -> null_copyover#1 [0]
   copyover#2(v0, v1) -> null_copyover#2 [0]
   enqueues(v0, v1) -> null_enqueues [0]
   enqueues#1(v0, v1) -> null_enqueues#1 [0]
   enqueue(v0, v1) -> null_enqueue [0]
   enqueue#1(v0, v1) -> null_enqueue#1 [0]
   breadth#4(v0) -> null_breadth#4 [0]
   children(v0, v1, v2, v3) -> null_children [0]
   children#1(v0, v1, v2) -> null_children#1 [0]
   children#3(v0, v1, v2, v3) -> null_children#3 [0]
   children#2(v0, v1) -> null_children#2 [0]
   empty(v0) -> null_empty [0]
   BREADTH#1(v0) -> null_BREADTH#1 [0]
   BREADTH#3(v0, v1) -> null_BREADTH#3 [0]
   BREADTH#5(v0) -> null_BREADTH#5 [0]
   COPYOVER#1(v0) -> null_COPYOVER#1 [0]
   COPYOVER#2(v0, v1) -> null_COPYOVER#2 [0]
   DEQUEUE#1(v0) -> null_DEQUEUE#1 [0]
   DEQUEUE#2(v0, v1) -> null_DEQUEUE#2 [0]
   DEQUEUE#3(v0) -> null_DEQUEUE#3 [0]
   DEQUEUE#4(v0) -> null_DEQUEUE#4 [0]
   BREADTH#2(v0, v1) -> null_BREADTH#2 [0]
   ENQUEUES#1(v0, v1) -> null_ENQUEUES#1 [0]

The TRS has the following type information:
BREADTH :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c
c :: c1:null_BREADTH#1 -> c18 -> c
BREADTH#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c1:null_BREADTH#1
dequeue :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
DEQUEUE :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c18
tuple#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
c1 :: c2:null_BREADTH#2 -> c1:null_BREADTH#1
BREADTH#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c2:null_BREADTH#2
BREADTH#3 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c4:null_BREADTH#3
c4 :: c6:null_BREADTH#5 -> c28 -> c4:null_BREADTH#3
BREADTH#5 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c6:null_BREADTH#5
enqueues :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
ENQUEUES :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c28
c6 :: c -> c6:null_BREADTH#5
COPYOVER :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c14
c14 :: c15:null_COPYOVER#1 -> c14
COPYOVER#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c15:null_COPYOVER#1
c15 :: c16:null_COPYOVER#2 -> c15:null_COPYOVER#1
COPYOVER#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c16:null_COPYOVER#2
:: :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
c16 :: c14 -> c16:null_COPYOVER#2
c18 :: c19:null_DEQUEUE#1 -> c18
DEQUEUE#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c19:null_DEQUEUE#1
c19 :: c21:null_DEQUEUE#2 -> c19:null_DEQUEUE#1
DEQUEUE#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c21:null_DEQUEUE#2
nil :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
c21 :: c22:null_DEQUEUE#3 -> c21:null_DEQUEUE#2
DEQUEUE#3 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c22:null_DEQUEUE#3
c22 :: c24:null_DEQUEUE#4 -> c14 -> c22:null_DEQUEUE#3
DEQUEUE#4 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c24:null_DEQUEUE#4
copyover :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
c24 :: c18 -> c24:null_DEQUEUE#4
c28 :: c29:null_ENQUEUES#1 -> c28
ENQUEUES#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c29:null_ENQUEUES#1
c2 :: c4:null_BREADTH#3 -> c2:null_BREADTH#2
breadth#4 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
c29 :: c28 -> c29:null_ENQUEUES#1
enqueue :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
STARTBREADTH#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c34:null_STARTBREADTH#2
c34 :: c -> c34:null_STARTBREADTH#2
STARTBREADTH#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> c32:null_STARTBREADTH#1
c32 :: c34:null_STARTBREADTH#2 -> c32:null_STARTBREADTH#1
tuple#4 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
empty :: #unit -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
#unit :: #unit
dequeue#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
dequeue#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
dequeue#3 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
dequeue#4 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
copyover#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
copyover#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
enqueues#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
enqueue#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
children :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
children#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
children#3 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
children#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty -> tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_STARTBREADTH#2 :: c34:null_STARTBREADTH#2
null_STARTBREADTH#1 :: c32:null_STARTBREADTH#1
null_dequeue :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_dequeue#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_dequeue#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_dequeue#3 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_dequeue#4 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_copyover :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_copyover#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_copyover#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_enqueues :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_enqueues#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_enqueue :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_enqueue#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_breadth#4 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_children :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_children#1 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_children#3 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_children#2 :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_empty :: tuple#2::::nil:tuple#4:null_dequeue:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_copyover:null_copyover#1:null_copyover#2:null_enqueues:null_enqueues#1:null_enqueue:null_enqueue#1:null_breadth#4:null_children:null_children#1:null_children#3:null_children#2:null_empty
null_BREADTH#1 :: c1:null_BREADTH#1
null_BREADTH#3 :: c4:null_BREADTH#3
null_BREADTH#5 :: c6:null_BREADTH#5
null_COPYOVER#1 :: c15:null_COPYOVER#1
null_COPYOVER#2 :: c16:null_COPYOVER#2
null_DEQUEUE#1 :: c19:null_DEQUEUE#1
null_DEQUEUE#2 :: c21:null_DEQUEUE#2
null_DEQUEUE#3 :: c22:null_DEQUEUE#3
null_DEQUEUE#4 :: c24:null_DEQUEUE#4
null_BREADTH#2 :: c2:null_BREADTH#2
null_ENQUEUES#1 :: c29:null_ENQUEUES#1
const :: c
const1 :: c18
const2 :: c28
const3 :: c14

Rewrite Strategy: INNERMOST
----------------------------------------

(79) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   #unit => 0
   null_STARTBREADTH#2 => 0
   null_STARTBREADTH#1 => 0
   null_dequeue => 0
   null_dequeue#1 => 0
   null_dequeue#2 => 0
   null_dequeue#3 => 0
   null_dequeue#4 => 0
   null_copyover => 0
   null_copyover#1 => 0
   null_copyover#2 => 0
   null_enqueues => 0
   null_enqueues#1 => 0
   null_enqueue => 0
   null_enqueue#1 => 0
   null_breadth#4 => 0
   null_children => 0
   null_children#1 => 0
   null_children#3 => 0
   null_children#2 => 0
   null_empty => 0
   null_BREADTH#1 => 0
   null_BREADTH#3 => 0
   null_BREADTH#5 => 0
   null_COPYOVER#1 => 0
   null_COPYOVER#2 => 0
   null_DEQUEUE#1 => 0
   null_DEQUEUE#2 => 0
   null_DEQUEUE#3 => 0
   null_DEQUEUE#4 => 0
   null_BREADTH#2 => 0
   null_ENQUEUES#1 => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0

----------------------------------------

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   BREADTH(z, z') -{ 1 }-> 1 + BREADTH#1(dequeue(z0, z1)) + DEQUEUE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   BREADTH#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   BREADTH#1(z) -{ 1 }-> 1 + BREADTH#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   BREADTH#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   BREADTH#2(z, z') -{ 1 }-> 1 + BREADTH#3(breadth#4(z0), z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#3(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   BREADTH#3(z, z') -{ 1 }-> 1 + BREADTH#5(enqueues(z1, z2)) + ENQUEUES(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   BREADTH#5(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   BREADTH#5(z) -{ 1 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER(z, z') -{ 1 }-> 1 + COPYOVER#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   COPYOVER#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   COPYOVER#1(z) -{ 1 }-> 1 + COPYOVER#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   COPYOVER#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   COPYOVER#2(z, z') -{ 1 }-> 1 + COPYOVER(z1, 1 + z0 + z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   DEQUEUE(z, z') -{ 1 }-> 1 + DEQUEUE#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   DEQUEUE#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   DEQUEUE#1(z) -{ 1 }-> 1 + DEQUEUE#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   DEQUEUE#2(z, z') -{ 1 }-> 1 + DEQUEUE#3(z0) :|: z0 >= 0, z = 0, z' = z0
   DEQUEUE#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   DEQUEUE#3(z) -{ 1 }-> 1 + DEQUEUE#4(copyover(1 + z0 + z1, 0)) + COPYOVER(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   DEQUEUE#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   DEQUEUE#4(z) -{ 1 }-> 1 + DEQUEUE(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ENQUEUES(z, z') -{ 1 }-> 1 + ENQUEUES#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   ENQUEUES#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ENQUEUES#1(z, z') -{ 1 }-> 1 + ENQUEUES(z1, enqueue(z0, z2)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   STARTBREADTH#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   STARTBREADTH#1(z) -{ 0 }-> 1 + STARTBREADTH#2(enqueue(1 + z0 + z0 + z1 + z1, empty(0))) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   STARTBREADTH#2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   STARTBREADTH#2(z) -{ 0 }-> 1 + BREADTH(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   breadth#4(z) -{ 0 }-> children(z0, z1, z2, z3) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + z2 + z3, z2 >= 0, z3 >= 0
   breadth#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   children(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children(z, z', z'', z5) -{ 0 }-> 1 + (1 + z0 + z1) + children#1(z2, z1, z3) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z5 = z3, z2 >= 0, z3 >= 0
   children#1(z, z', z'') -{ 0 }-> children#3(z3, z2, z0, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   children#1(z, z', z'') -{ 0 }-> children#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 0 }-> 1 + (1 + z0 + z2 + 0 + z1) + 0 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 0, z5 = z2, z' = z0, z2 >= 0, z'' = z1
   children#3(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z5) -{ 0 }-> 1 + (1 + z3 + z2 + 0 + z4) + (1 + (1 + z3 + z0 + z4 + z1) + 0) :|: z1 >= 0, z5 = z4, z' = z2, z0 >= 0, z'' = z3, z4 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   copyover(z, z') -{ 0 }-> copyover#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   copyover(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#1(z) -{ 0 }-> copyover#2(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   copyover#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   copyover#2(z, z') -{ 0 }-> copyover(z1, 1 + z0 + z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#2(z, z') -{ 0 }-> 1 + 0 + z0 :|: z0 >= 0, z = 0, z' = z0
   dequeue(z, z') -{ 0 }-> dequeue#1(1 + z0 + z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   dequeue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#1(z) -{ 0 }-> dequeue#2(z1, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#2(z, z') -{ 0 }-> dequeue#3(z0) :|: z0 >= 0, z = 0, z' = z0
   dequeue#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#2(z, z') -{ 0 }-> 1 + (1 + z2 + z1) + (1 + z0 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   dequeue#3(z) -{ 0 }-> dequeue#4(copyover(1 + z0 + z1, 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#3(z) -{ 0 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 0 }-> dequeue(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   dequeue#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 0 }-> 1 + 0 + 0 :|: z = z0, z0 >= 0
   enqueue(z, z') -{ 0 }-> enqueue#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   enqueue(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues(z, z') -{ 0 }-> enqueues#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   enqueues(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueues#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   enqueues#1(z, z') -{ 0 }-> enqueues(z1, enqueue(z0, z2)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH(z0, z1) -> c(BREADTH#1(dequeue(z0, z1)), DEQUEUE(z0, z1)) by 
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))

----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, enqueues_2, enqueues#1_2, enqueue_2, enqueue#1_2, breadth#4_1, children_4, children#1_3, children#3_4, children#2_2, empty_1

Defined Pair Symbols:   BREADTH#1_1, BREADTH#3_2, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2

Compound Symbols:   c1_1, c4_2, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c22_2, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1, c_2


----------------------------------------

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#3(tuple#2(z0, z1), z2) -> c4(BREADTH#5(enqueues(z1, z2)), ENQUEUES(z1, z2)) by 
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))

----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, enqueues_2, enqueues#1_2, enqueue_2, enqueue#1_2, breadth#4_1, children_4, children#1_3, children#3_4, children#2_2, empty_1

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#3_1, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c22_2, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1, c_2, c4_2


----------------------------------------

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(::(z0, z1), nil)), COPYOVER(::(z0, z1), nil)) by 
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))

----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, enqueues_2, enqueues#1_2, enqueue_2, enqueue#1_2, breadth#4_1, children_4, children#1_3, children#3_4, children#2_2, empty_1

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1, c_2, c4_2, c22_2


----------------------------------------

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#2(::(z0, z1), z2) -> c2(BREADTH#3(breadth#4(z0), z2)) by 
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))

----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
   empty(z0) -> tuple#2(nil, nil)
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   dequeue_2, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, enqueues_2, enqueues#1_2, enqueue_2, enqueue#1_2, breadth#4_1, children_4, children#1_3, children#3_4, children#2_2, empty_1

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c32_1, c_2, c4_2, c22_2, c2_1


----------------------------------------

(89) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   breadth#4(tuple#4(z0, z1, z2, z3)) -> children(z0, z1, z2, z3)

----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c32_1, c_2, c4_2, c22_2, c2_1


----------------------------------------

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENQUEUES#1(::(z0, z1), z2) -> c29(ENQUEUES(z1, enqueue(z0, z2))) by 
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))

----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, ENQUEUES#1_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c32_1, c_2, c4_2, c22_2, c2_1, c29_1


----------------------------------------

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit)))) by 
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))

----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c_2, c4_2, c22_2, c2_1, c29_1, c32_1


----------------------------------------

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH(z0, z1) -> c(BREADTH#1(dequeue#1(tuple#2(z0, z1))), DEQUEUE(z0, z1)) by 
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))

----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c4_2, c22_2, c2_1, c29_1, c32_1, c_2


----------------------------------------

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#3(tuple#2(x0, z0), z1) -> c4(BREADTH#5(enqueues#1(z0, z1)), ENQUEUES(z0, z1)) by 
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, DEQUEUE#3_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c22_2, c2_1, c29_1, c32_1, c_2, c4_2, c4_1


----------------------------------------

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
We considered the (Usable) Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   enqueue(z0, z1) -> enqueue#1(z1, z0)
And the Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(#unit) = [1]
   POL(::(x_1, x_2)) = 0
   POL(BREADTH(x_1, x_2)) = [1]
   POL(BREADTH#1(x_1)) = [1]
   POL(BREADTH#2(x_1, x_2)) = [1]
   POL(BREADTH#3(x_1, x_2)) = x_1
   POL(BREADTH#5(x_1)) = [1]
   POL(COPYOVER(x_1, x_2)) = 0
   POL(COPYOVER#1(x_1)) = 0
   POL(COPYOVER#2(x_1, x_2)) = 0
   POL(DEQUEUE(x_1, x_2)) = 0
   POL(DEQUEUE#1(x_1)) = 0
   POL(DEQUEUE#2(x_1, x_2)) = 0
   POL(DEQUEUE#3(x_1)) = 0
   POL(DEQUEUE#4(x_1)) = 0
   POL(ENQUEUES(x_1, x_2)) = 0
   POL(ENQUEUES#1(x_1, x_2)) = 0
   POL(STARTBREADTH#1(x_1)) = [1] + x_1
   POL(STARTBREADTH#2(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c32(x_1)) = x_1
   POL(c34(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(children(x_1, x_2, x_3, x_4)) = [1]
   POL(children#1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(children#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(children#3(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(copyover(x_1, x_2)) = [1] + x_1 + x_2
   POL(copyover#1(x_1)) = [1]
   POL(copyover#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#1(x_1)) = [1] + x_1
   POL(dequeue#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#3(x_1)) = [1] + x_1
   POL(dequeue#4(x_1)) = [1] + x_1
   POL(empty(x_1)) = [1] + x_1
   POL(enqueue(x_1, x_2)) = [1]
   POL(enqueue#1(x_1, x_2)) = [1]
   POL(enqueues(x_1, x_2)) = [1] + x_1 + x_2
   POL(enqueues#1(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = 0
   POL(tuple#2(x_1, x_2)) = [1]
   POL(tuple#4(x_1, x_2, x_3, x_4)) = x_1 + x_2

----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, DEQUEUE#3_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c22_2, c2_1, c29_1, c32_1, c_2, c4_2, c4_1


----------------------------------------

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#1(tuple#2(::(x0, x1), nil))), COPYOVER(::(x0, x1), nil)) by 
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))

----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, BREADTH#2_2, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c2_1, c29_1, c32_1, c_2, c4_2, c4_1, c22_2


----------------------------------------

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#2(::(tuple#4(z0, z1, z2, z3), x1), x2) -> c2(BREADTH#3(children(z0, z1, z2, z3), x2)) by 
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))

----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue_2, enqueue#1_2, empty_1, dequeue#1_1, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, enqueues#1_2, enqueues_2, children_4, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c32_1, c_2, c4_2, c4_1, c22_2, c2_1


----------------------------------------

(105) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   children(z0, z1, z2, z3) -> tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3))

----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1))))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue#1_2, empty_1, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c32_1, c_2, c4_2, c4_1, c22_2, c2_1


----------------------------------------

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(empty(#unit), tuple#4(x0, x0, x1, x1)))) by 
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))

----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   empty(z0) -> tuple#2(nil, nil)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue#1_2, empty_1, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c32_1, c_2, c4_2, c4_1, c22_2, c2_1


----------------------------------------

(109) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   empty(z0) -> tuple#2(nil, nil)

----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil))))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, STARTBREADTH#1_1, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c32_1, c_2, c4_2, c4_1, c22_2, c2_1


----------------------------------------

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(x0, x0, x1, x1), tuple#2(nil, nil)))) by 
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))

----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, STARTBREADTH#1_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c_2, c4_2, c4_1, c22_2, c2_1, c32_1


----------------------------------------

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH(z0, z1) -> c(BREADTH#1(dequeue#2(z1, z0)), DEQUEUE(z0, z1)) by 
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))

----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c22_2, c2_1, c32_1, c_2, c_1


----------------------------------------

(115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
We considered the (Usable) Rules:
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
And the Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(::(x_1, x_2)) = 0
   POL(BREADTH(x_1, x_2)) = [1]
   POL(BREADTH#1(x_1)) = x_1
   POL(BREADTH#2(x_1, x_2)) = [1]
   POL(BREADTH#3(x_1, x_2)) = [1]
   POL(BREADTH#5(x_1)) = [1]
   POL(COPYOVER(x_1, x_2)) = 0
   POL(COPYOVER#1(x_1)) = 0
   POL(COPYOVER#2(x_1, x_2)) = 0
   POL(DEQUEUE(x_1, x_2)) = 0
   POL(DEQUEUE#1(x_1)) = 0
   POL(DEQUEUE#2(x_1, x_2)) = 0
   POL(DEQUEUE#3(x_1)) = 0
   POL(DEQUEUE#4(x_1)) = 0
   POL(ENQUEUES(x_1, x_2)) = 0
   POL(ENQUEUES#1(x_1, x_2)) = 0
   POL(STARTBREADTH#1(x_1)) = [3]
   POL(STARTBREADTH#2(x_1)) = [2]
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c32(x_1)) = x_1
   POL(c34(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(children#1(x_1, x_2, x_3)) = [3] + [3]x_1 + [3]x_2 + [3]x_3
   POL(children#2(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(children#3(x_1, x_2, x_3, x_4)) = [3] + [3]x_1 + [3]x_2 + [3]x_3 + [3]x_4
   POL(copyover(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(copyover#1(x_1)) = [3] + [3]x_1
   POL(copyover#2(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(dequeue(x_1, x_2)) = [1]
   POL(dequeue#1(x_1)) = [1]
   POL(dequeue#2(x_1, x_2)) = [1]
   POL(dequeue#3(x_1)) = [1]
   POL(dequeue#4(x_1)) = [1]
   POL(enqueue(x_1, x_2)) = 0
   POL(enqueue#1(x_1, x_2)) = [2]
   POL(enqueues(x_1, x_2)) = [3] + [3]x_1
   POL(enqueues#1(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(nil) = 0
   POL(tuple#2(x_1, x_2)) = [1]
   POL(tuple#4(x_1, x_2, x_3, x_4)) = 0

----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c22_2, c2_1, c32_1, c_2, c_1


----------------------------------------

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#3(tuple#2(x0, ::(z0, z1)), z2) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, z2))), ENQUEUES(::(z0, z1), z2)) by 
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))

----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c22_2, c2_1, c32_1, c_2, c_1


----------------------------------------

(119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
We considered the (Usable) Rules:none
And the Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(::(x_1, x_2)) = 0
   POL(BREADTH(x_1, x_2)) = [1]
   POL(BREADTH#1(x_1)) = [1]
   POL(BREADTH#2(x_1, x_2)) = [1]
   POL(BREADTH#3(x_1, x_2)) = [1]
   POL(BREADTH#5(x_1)) = [1]
   POL(COPYOVER(x_1, x_2)) = 0
   POL(COPYOVER#1(x_1)) = x_1
   POL(COPYOVER#2(x_1, x_2)) = 0
   POL(DEQUEUE(x_1, x_2)) = 0
   POL(DEQUEUE#1(x_1)) = 0
   POL(DEQUEUE#2(x_1, x_2)) = 0
   POL(DEQUEUE#3(x_1)) = 0
   POL(DEQUEUE#4(x_1)) = 0
   POL(ENQUEUES(x_1, x_2)) = 0
   POL(ENQUEUES#1(x_1, x_2)) = 0
   POL(STARTBREADTH#1(x_1)) = [1] + x_1
   POL(STARTBREADTH#2(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c32(x_1)) = x_1
   POL(c34(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(children#1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(children#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(children#3(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(copyover(x_1, x_2)) = [1] + x_1 + x_2
   POL(copyover#1(x_1)) = [1] + x_1
   POL(copyover#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#1(x_1)) = [1] + x_1
   POL(dequeue#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#3(x_1)) = [1] + x_1
   POL(dequeue#4(x_1)) = [1] + x_1
   POL(enqueue(x_1, x_2)) = [1] + x_1 + x_2
   POL(enqueue#1(x_1, x_2)) = x_1
   POL(enqueues(x_1, x_2)) = [1] + x_1
   POL(enqueues#1(x_1, x_2)) = [1] + x_1
   POL(nil) = 0
   POL(tuple#2(x_1, x_2)) = 0
   POL(tuple#4(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4

----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, DEQUEUE#3_1, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c22_2, c2_1, c32_1, c_2, c_1


----------------------------------------

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace DEQUEUE#3(::(x0, x1)) -> c22(DEQUEUE#4(copyover#2(::(x0, x1), nil)), COPYOVER(::(x0, x1), nil)) by 
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))

----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c2_1, c32_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1)))) by 
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(tuple#2(::(tuple#4(x0, x0, x1, x1), nil), nil)))

----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(tuple#2(::(tuple#4(x0, x0, x1, x1), nil), nil)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
K tuples:
   STARTBREADTH#1(::(z0, z1)) -> c32(STARTBREADTH#2(enqueue(tuple#4(z0, z0, z1, z1), empty(#unit))))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c2_1, c32_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(125) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(tuple#2(::(tuple#4(x0, x0, x1, x1), nil), nil)))

----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1))))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
K tuples:
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, STARTBREADTH#1_1, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c2_1, c32_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(enqueue#1(tuple#2(nil, nil), tuple#4(x0, x0, x1, x1)))) by 
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(tuple#2(::(tuple#4(x0, x0, x1, x1), nil), nil)))

----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(tuple#2(::(tuple#4(x0, x0, x1, x1), nil), nil)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
K tuples:
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, STARTBREADTH#2_1, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, STARTBREADTH#1_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c34_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1, c32_1


----------------------------------------

(129) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   STARTBREADTH#1(::(x0, x1)) -> c32(STARTBREADTH#2(tuple#2(::(tuple#4(x0, x0, x1, x1), nil), nil)))
   STARTBREADTH#2(tuple#2(z0, z1)) -> c34(BREADTH(z0, z1))

----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH(z0, nil) -> c(BREADTH#1(dequeue#3(z0)), DEQUEUE(z0, nil)) by 
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))

----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(133) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
We considered the (Usable) Rules:none
And the Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(::(x_1, x_2)) = 0
   POL(BREADTH(x_1, x_2)) = [1]
   POL(BREADTH#1(x_1)) = [1]
   POL(BREADTH#2(x_1, x_2)) = [1]
   POL(BREADTH#3(x_1, x_2)) = x_1
   POL(BREADTH#5(x_1)) = [1]
   POL(COPYOVER(x_1, x_2)) = 0
   POL(COPYOVER#1(x_1)) = 0
   POL(COPYOVER#2(x_1, x_2)) = 0
   POL(DEQUEUE(x_1, x_2)) = 0
   POL(DEQUEUE#1(x_1)) = 0
   POL(DEQUEUE#2(x_1, x_2)) = 0
   POL(DEQUEUE#3(x_1)) = 0
   POL(DEQUEUE#4(x_1)) = 0
   POL(ENQUEUES(x_1, x_2)) = 0
   POL(ENQUEUES#1(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(children#1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(children#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(children#3(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(copyover(x_1, x_2)) = 0
   POL(copyover#1(x_1)) = [1] + x_1
   POL(copyover#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#1(x_1)) = [1] + x_1
   POL(dequeue#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#3(x_1)) = [1] + x_1
   POL(dequeue#4(x_1)) = [1]
   POL(enqueue(x_1, x_2)) = [1] + x_1 + x_2
   POL(enqueue#1(x_1, x_2)) = x_1 + x_2
   POL(enqueues(x_1, x_2)) = [1] + x_1
   POL(enqueues#1(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = 0
   POL(tuple#2(x_1, x_2)) = [1]
   POL(tuple#4(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4

----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
We considered the (Usable) Rules:
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
And the Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(::(x_1, x_2)) = [1]
   POL(BREADTH(x_1, x_2)) = [1]
   POL(BREADTH#1(x_1)) = x_1
   POL(BREADTH#2(x_1, x_2)) = x_1
   POL(BREADTH#3(x_1, x_2)) = [1]
   POL(BREADTH#5(x_1)) = [1]
   POL(COPYOVER(x_1, x_2)) = 0
   POL(COPYOVER#1(x_1)) = 0
   POL(COPYOVER#2(x_1, x_2)) = 0
   POL(DEQUEUE(x_1, x_2)) = 0
   POL(DEQUEUE#1(x_1)) = 0
   POL(DEQUEUE#2(x_1, x_2)) = 0
   POL(DEQUEUE#3(x_1)) = 0
   POL(DEQUEUE#4(x_1)) = 0
   POL(ENQUEUES(x_1, x_2)) = 0
   POL(ENQUEUES#1(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(children#1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(children#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(children#3(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(copyover(x_1, x_2)) = 0
   POL(copyover#1(x_1)) = [1] + x_1
   POL(copyover#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue(x_1, x_2)) = [1]
   POL(dequeue#1(x_1)) = [1]
   POL(dequeue#2(x_1, x_2)) = [1]
   POL(dequeue#3(x_1)) = [1]
   POL(dequeue#4(x_1)) = [1]
   POL(enqueue(x_1, x_2)) = [1] + x_1 + x_2
   POL(enqueue#1(x_1, x_2)) = x_1 + x_2
   POL(enqueues(x_1, x_2)) = [1] + x_1
   POL(enqueues#1(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = 0
   POL(tuple#2(x_1, x_2)) = x_2
   POL(tuple#4(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3

----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue(x1, x3))), ENQUEUES(::(x1, z0), x3)) by 
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))

----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(139) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))

----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1)) by 
   BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue#1(x3, x1))), ENQUEUES(::(x1, z0), x3))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))

----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(143) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))

----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_2, c22_1


----------------------------------------

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil)) by 
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))

----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil)) by BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BREADTH#3(tuple#2(x0, ::(x1, ::(z0, z1))), x3) -> c4(BREADTH#5(enqueues(z1, enqueue(z0, enqueue(x1, x3)))), ENQUEUES(::(x1, ::(z0, z1)), x3)) by BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BREADTH#3(tuple#2(x0, ::(x1, nil)), x3) -> c4(BREADTH#5(enqueue(x1, x3)), ENQUEUES(::(x1, nil), x3)) by BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1)))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BREADTH#3(tuple#2(x0, ::(z2, x2)), tuple#2(z0, z1)) -> c4(BREADTH#5(enqueues(x2, tuple#2(::(z2, z0), z1))), ENQUEUES(::(z2, x2), tuple#2(z0, z1))) by BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace DEQUEUE#3(::(x0, z0)) -> c22(DEQUEUE#4(copyover#1(tuple#2(z0, ::(x0, nil)))), COPYOVER(::(x0, z0), nil)) by DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#1(tuple#2(::(z0, z1), nil)))), DEQUEUE(::(z0, z1), nil)) by BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1)))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER_2, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1

Compound Symbols:   c1_1, c6_1, c14_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2


----------------------------------------

(159) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace COPYOVER(z0, z1) -> c14(COPYOVER#1(tuple#2(z0, z1))) by 
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))

----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
S tuples:
   BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0))
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#1_1, BREADTH#5_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2

Compound Symbols:   c1_1, c6_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1


----------------------------------------

(161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace BREADTH#1(tuple#2(z0, z1)) -> c1(BREADTH#2(z1, z0)) by 
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))

----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(BREADTH#1(tuple#2(tuple#2(nil, nil), nil)), DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1

Compound Symbols:   c6_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1


----------------------------------------

(163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1

Compound Symbols:   c6_1, c15_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1


----------------------------------------

(165) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace COPYOVER#1(tuple#2(z0, z1)) -> c15(COPYOVER#2(z0, z1)) by 
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))

----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2)))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, COPYOVER#2_2, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1

Compound Symbols:   c6_1, c16_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1, c15_1


----------------------------------------

(167) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace COPYOVER#2(::(z0, z1), z2) -> c16(COPYOVER(z1, ::(z0, z2))) by 
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))

----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2

Compound Symbols:   c6_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1, c15_1, c16_1


----------------------------------------

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue(z2, enqueue(z1, z4)))), ENQUEUES(::(z1, ::(z2, z3)), z4)) by BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE_2, DEQUEUE#1_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2

Compound Symbols:   c6_1, c18_1, c19_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1, c15_1, c16_1


----------------------------------------

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEQUEUE#1(tuple#2(z0, z1)) -> c19(DEQUEUE#2(z1, z0)) by 
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))

----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1)))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1

Compound Symbols:   c6_1, c18_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1, c15_1, c16_1, c19_1


----------------------------------------

(173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEQUEUE(z0, z1) -> c18(DEQUEUE#1(tuple#2(z0, z1))) by 
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))

----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))), DEQUEUE(z2, ::(z0, z1)))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2

Compound Symbols:   c6_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_2, c_1, c22_1, c22_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1


----------------------------------------

(175) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2

Compound Symbols:   c6_1, c21_1, c24_1, c28_1, c29_1, c4_2, c4_1, c2_1, c_1, c22_1, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1


----------------------------------------

(177) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, nil), z0) -> c4(BREADTH#5(z0), ENQUEUES(nil, z0)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))

----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
K tuples:
   BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#3_2, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2

Compound Symbols:   c6_1, c21_1, c24_1, c28_1, c29_1, c4_1, c2_1, c_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1


----------------------------------------

(179) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, x1), x2) -> c4(ENQUEUES(x1, x2)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))

----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2

Compound Symbols:   c6_1, c21_1, c24_1, c28_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1


----------------------------------------

(181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEQUEUE#2(nil, z0) -> c21(DEQUEUE#3(z0)) by 
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))

----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2

Compound Symbols:   c6_1, c24_1, c28_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1


----------------------------------------

(183) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))

----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2

Compound Symbols:   c6_1, c24_1, c28_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1


----------------------------------------

(185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DEQUEUE#4(tuple#2(z0, z1)) -> c24(DEQUEUE(z0, z1)) by 
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))

----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1

Compound Symbols:   c6_1, c28_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1


----------------------------------------

(187) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
We considered the (Usable) Rules:
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
And the Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(::(x_1, x_2)) = 0
   POL(BREADTH(x_1, x_2)) = 0
   POL(BREADTH#1(x_1)) = 0
   POL(BREADTH#2(x_1, x_2)) = x_1
   POL(BREADTH#3(x_1, x_2)) = 0
   POL(BREADTH#5(x_1)) = 0
   POL(COPYOVER(x_1, x_2)) = 0
   POL(COPYOVER#1(x_1)) = 0
   POL(COPYOVER#2(x_1, x_2)) = 0
   POL(DEQUEUE(x_1, x_2)) = 0
   POL(DEQUEUE#1(x_1)) = 0
   POL(DEQUEUE#2(x_1, x_2)) = 0
   POL(DEQUEUE#3(x_1)) = x_1
   POL(DEQUEUE#4(x_1)) = x_1
   POL(ENQUEUES(x_1, x_2)) = 0
   POL(ENQUEUES#1(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(children#1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(children#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(children#3(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(copyover(x_1, x_2)) = x_2
   POL(copyover#1(x_1)) = x_1
   POL(copyover#2(x_1, x_2)) = x_2
   POL(dequeue(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#1(x_1)) = [1] + x_1
   POL(dequeue#2(x_1, x_2)) = [1] + x_1 + x_2
   POL(dequeue#3(x_1)) = [1] + x_1
   POL(dequeue#4(x_1)) = [1]
   POL(enqueue(x_1, x_2)) = [1] + x_1 + x_2
   POL(enqueue#1(x_1, x_2)) = x_2
   POL(enqueues(x_1, x_2)) = [1] + x_1 + x_2
   POL(enqueues#1(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(tuple#2(x_1, x_2)) = x_2
   POL(tuple#4(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4

----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, ENQUEUES_2, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1

Compound Symbols:   c6_1, c28_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1


----------------------------------------

(189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENQUEUES(z0, z1) -> c28(ENQUEUES#1(z0, z1)) by 
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))

----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5), ENQUEUES(nil, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2

Compound Symbols:   c6_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1, c28_1


----------------------------------------

(191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2

Compound Symbols:   c6_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1, c28_1


----------------------------------------

(193) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))

----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2

Compound Symbols:   c6_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1, c28_1


----------------------------------------

(195) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))

----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0)))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, ENQUEUES#1_2, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2

Compound Symbols:   c6_1, c29_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1, c28_1


----------------------------------------

(197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENQUEUES#1(::(z0, x1), z1) -> c29(ENQUEUES(x1, enqueue#1(z1, z0))) by 
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))

----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(x0, ::(z0, x2)), z1) -> c4(BREADTH#5(enqueues#1(x2, enqueue#1(z1, z0))), ENQUEUES(::(z0, x2), z1))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2

Compound Symbols:   c6_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1, c28_1, c29_1


----------------------------------------

(199) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, ::(x1, z0)), x3) -> c4(BREADTH#5(enqueues#1(z0, enqueue#1(x3, x1))), ENQUEUES(::(x1, z0), x3)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))

----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, BREADTH#2_2, BREADTH_2, BREADTH#3_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2

Compound Symbols:   c6_1, c2_1, c_1, c4_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c21_1, c24_1, c28_1, c29_1


----------------------------------------

(201) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(x0, ::(x1, x2)), x3) -> c4(ENQUEUES(::(x1, x2), x3)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))

----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, BREADTH#3_2, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2

Compound Symbols:   c6_1, c2_1, c_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c4_1, c21_1, c24_1, c28_1, c29_1


----------------------------------------

(203) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(z0, ::(z1, nil)), z2) -> c4(BREADTH#5(enqueue#1(z2, z1)), ENQUEUES(::(z1, nil), z2)) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, nil)), x5) -> c4(BREADTH#5(enqueue#1(x5, z1)), ENQUEUES(::(z1, nil), x5))

----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, nil)), x5) -> c4(BREADTH#5(enqueue#1(x5, z1)), ENQUEUES(::(z1, nil), x5))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, nil)), x5) -> c4(BREADTH#5(enqueue#1(x5, z1)), ENQUEUES(::(z1, nil), x5))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, BREADTH#3_2, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2

Compound Symbols:   c6_1, c2_1, c_1, c22_1, c4_2, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c19_1, c18_1, c4_1, c21_1, c24_1, c28_1, c29_1


----------------------------------------

(205) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace BREADTH#3(tuple#2(z0, ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4))) by 
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))

----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   enqueue#1(tuple#2(z0, z1), z2) -> tuple#2(::(z2, z0), z1)
   enqueue(z0, z1) -> enqueue#1(z1, z0)
   dequeue#2(::(z0, z1), z2) -> tuple#2(tuple#2(z2, z1), ::(z0, nil))
   dequeue#2(nil, z0) -> dequeue#3(z0)
   dequeue#3(::(z0, z1)) -> dequeue#4(copyover(::(z0, z1), nil))
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil)
   dequeue#4(tuple#2(z0, z1)) -> dequeue(z0, z1)
   copyover(z0, z1) -> copyover#1(tuple#2(z0, z1))
   copyover#1(tuple#2(z0, z1)) -> copyover#2(z0, z1)
   copyover#2(::(z0, z1), z2) -> copyover(z1, ::(z0, z2))
   copyover#2(nil, z0) -> tuple#2(nil, z0)
   dequeue(z0, z1) -> dequeue#1(tuple#2(z0, z1))
   dequeue#1(tuple#2(z0, z1)) -> dequeue#2(z1, z0)
   enqueues(z0, z1) -> enqueues#1(z0, z1)
   enqueues#1(::(z0, z1), z2) -> enqueues(z1, enqueue(z0, z2))
   enqueues#1(nil, z0) -> z0
   children#1(::(z0, z1), z2, z3) -> children#3(z3, z2, z0, z1)
   children#1(nil, z0, z1) -> children#2(z1, z0)
   children#3(::(z0, z1), z2, z3, z4) -> ::(tuple#4(z3, z2, nil, z4), ::(tuple#4(z3, z0, z4, z1), nil))
   children#3(nil, z0, z1, z2) -> nil
   children#2(::(z0, z1), z2) -> ::(tuple#4(z0, z2, nil, z1), nil)
   children#2(nil, z0) -> nil
Tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, nil)), x5) -> c4(BREADTH#5(enqueue#1(x5, z1)), ENQUEUES(::(z1, nil), x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
S tuples:
   BREADTH#5(tuple#2(z0, z1)) -> c6(BREADTH(z0, z1))
   BREADTH#2(::(tuple#4(z0, z1, z2, z3), x4), x5) -> c2(BREADTH#3(tuple#2(tuple#2(z0, z1), children#1(z2, z1, z3)), x5))
   DEQUEUE#3(::(x0, x1)) -> c22(COPYOVER(::(x0, x1), nil))
   DEQUEUE#3(::(z0, z1)) -> c22(DEQUEUE#4(copyover#2(z1, ::(z0, nil))), COPYOVER(::(z0, z1), nil))
   BREADTH(::(z0, z1), nil) -> c(BREADTH#1(dequeue#4(copyover#2(::(z0, z1), nil))), DEQUEUE(::(z0, z1), nil))
   COPYOVER(x1, ::(x0, x2)) -> c14(COPYOVER#1(tuple#2(x1, ::(x0, x2))))
   COPYOVER(::(x0, x1), nil) -> c14(COPYOVER#1(tuple#2(::(x0, x1), nil)))
   BREADTH#1(tuple#2(z0, ::(tuple#4(y0, y1, y2, y3), y4))) -> c1(BREADTH#2(::(tuple#4(y0, y1, y2, y3), y4), z0))
   COPYOVER#1(tuple#2(x0, ::(x1, x2))) -> c15(COPYOVER#2(x0, ::(x1, x2)))
   COPYOVER#1(tuple#2(::(x0, x1), nil)) -> c15(COPYOVER#2(::(x0, x1), nil))
   COPYOVER#2(::(z0, z1), ::(x1, x2)) -> c16(COPYOVER(z1, ::(z0, ::(x1, x2))))
   COPYOVER#2(::(x0, x1), nil) -> c16(COPYOVER(x1, ::(x0, nil)))
   BREADTH#3(tuple#2(z0, ::(z1, ::(z2, z3))), z4) -> c4(BREADTH#5(enqueues#1(z3, enqueue#1(enqueue(z1, z4), z2))), ENQUEUES(::(z1, ::(z2, z3)), z4))
   DEQUEUE#1(tuple#2(z0, nil)) -> c19(DEQUEUE#2(nil, z0))
   DEQUEUE(z0, nil) -> c18(DEQUEUE#1(tuple#2(z0, nil)))
   BREADTH(z2, ::(z0, z1)) -> c(BREADTH#1(tuple#2(tuple#2(z2, z1), ::(z0, nil))))
   DEQUEUE#2(nil, ::(y0, y1)) -> c21(DEQUEUE#3(::(y0, y1)))
   ENQUEUES(::(y0, y1), z1) -> c28(ENQUEUES#1(::(y0, y1), z1))
   BREADTH#3(tuple#2(tuple#2(x0, x1), nil), x5) -> c4(BREADTH#5(x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(BREADTH#5(enqueues#1(z2, enqueue#1(x5, z1))), ENQUEUES(::(z1, z2), x5))
   ENQUEUES#1(::(z0, ::(y0, y1)), z2) -> c29(ENQUEUES(::(y0, y1), enqueue#1(z2, z0)))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, nil)), x5) -> c4(BREADTH#5(enqueue#1(x5, z1)), ENQUEUES(::(z1, nil), x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), tuple#2(z3, z4)) -> c4(BREADTH#5(enqueues#1(z2, tuple#2(::(z1, z3), z4))), ENQUEUES(::(z1, z2), tuple#2(z3, z4)))
K tuples:
   BREADTH(x0, x1) -> c(DEQUEUE(x0, x1))
   BREADTH(x0, nil) -> c(DEQUEUE(x0, nil))
   BREADTH(nil, nil) -> c(DEQUEUE(nil, nil))
   BREADTH#3(tuple#2(tuple#2(x0, x1), y0), x5) -> c4(ENQUEUES(y0, x5))
   BREADTH#3(tuple#2(tuple#2(x0, x1), ::(z1, z2)), x5) -> c4(ENQUEUES(::(z1, z2), x5))
   DEQUEUE#4(tuple#2(z0, nil)) -> c24(DEQUEUE(z0, nil))
Defined Rule Symbols:   enqueue#1_2, enqueue_2, dequeue#2_2, dequeue#3_1, dequeue#4_1, copyover_2, copyover#1_1, copyover#2_2, dequeue_2, dequeue#1_1, enqueues_2, enqueues#1_2, children#1_3, children#3_4, children#2_2

Defined Pair Symbols:   BREADTH#5_1, BREADTH#2_2, BREADTH_2, DEQUEUE#3_1, COPYOVER_2, BREADTH#1_1, COPYOVER#1_1, COPYOVER#2_2, BREADTH#3_2, DEQUEUE#1_1, DEQUEUE_2, DEQUEUE#2_2, DEQUEUE#4_1, ENQUEUES_2, ENQUEUES#1_2

Compound Symbols:   c6_1, c2_1, c_1, c22_1, c22_2, c_2, c14_1, c1_1, c15_1, c16_1, c4_2, c19_1, c18_1, c4_1, c21_1, c24_1, c28_1, c29_1


----------------------------------------

(207) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(208)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2)) [1]
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue') [1]
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue') [1]
   breadth#2(nil, @queue') -> nil [1]
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue'))) [1]
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6) [1]
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8) [1]
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2)) [1]
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs) [1]
   children#1(nil, @b, @l2) -> children#2(@l2, @b) [1]
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil) [1]
   children#2(nil, @b) -> nil [1]
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil)) [1]
   children#3(nil, @b, @x, @xs) -> nil [1]
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2)) [1]
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq) [1]
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq)) [1]
   copyover#2(nil, @outq) -> tuple#2(nil, @outq) [1]
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2)) [1]
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq) [1]
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil)) [1]
   dequeue#2(nil, @inq) -> dequeue#3(@inq) [1]
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil)) [1]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [1]
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4) [1]
   empty(@x) -> tuple#2(nil, nil) [1]
   enqueue(@x, @queue) -> enqueue#1(@queue, @x) [1]
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq) [1]
   enqueues(@l, @queue) -> enqueues#1(@l, @queue) [1]
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue)) [1]
   enqueues#1(nil, @queue) -> @queue [1]
   startBreadth(@xs) -> startBreadth#1(@xs) [1]
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit))) [1]
   startBreadth#1(nil) -> nil [1]
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(209) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(210)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2)) [1]
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue') [1]
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue') [1]
   breadth#2(nil, @queue') -> nil [1]
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue'))) [1]
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6) [1]
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8) [1]
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2)) [1]
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs) [1]
   children#1(nil, @b, @l2) -> children#2(@l2, @b) [1]
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil) [1]
   children#2(nil, @b) -> nil [1]
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil)) [1]
   children#3(nil, @b, @x, @xs) -> nil [1]
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2)) [1]
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq) [1]
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq)) [1]
   copyover#2(nil, @outq) -> tuple#2(nil, @outq) [1]
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2)) [1]
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq) [1]
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil)) [1]
   dequeue#2(nil, @inq) -> dequeue#3(@inq) [1]
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil)) [1]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [1]
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4) [1]
   empty(@x) -> tuple#2(nil, nil) [1]
   enqueue(@x, @queue) -> enqueue#1(@queue, @x) [1]
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq) [1]
   enqueues(@l, @queue) -> enqueues#1(@l, @queue) [1]
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue)) [1]
   enqueues#1(nil, @queue) -> @queue [1]
   startBreadth(@xs) -> startBreadth#1(@xs) [1]
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit))) [1]
   startBreadth#1(nil) -> nil [1]
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2) [1]

The TRS has the following type information:
breadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
nil :: tuple#2::::nil:tuple#4
breadth#5 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
empty :: #unit -> tuple#2::::nil:tuple#4
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
#unit :: #unit

Rewrite Strategy: INNERMOST
----------------------------------------

(211) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   breadth_2
   breadth#1_1
   breadth#2_2
   breadth#3_2
   breadth#5_1
   startBreadth_1
   startBreadth#1_1
   startBreadth#2_1

(c) The following functions are completely defined:

   dequeue_2
   enqueue_2
   empty_1
   breadth#4_1
   enqueues_2
   copyover_2
   copyover#1_1
   copyover#2_2
   enqueue#1_2
   children_4
   enqueues#1_2
   dequeue#1_1
   dequeue#2_2
   children#1_3
   children#2_2
   children#3_4
   dequeue#3_1
   dequeue#4_1

Due to the following rules being added:

   breadth#4(v0) -> nil [0]
   copyover#1(v0) -> nil [0]
   copyover#2(v0, v1) -> nil [0]
   enqueue#1(v0, v1) -> nil [0]
   enqueues#1(v0, v1) -> nil [0]
   dequeue#1(v0) -> nil [0]
   dequeue#2(v0, v1) -> nil [0]
   children#1(v0, v1, v2) -> nil [0]
   children#2(v0, v1) -> nil [0]
   children#3(v0, v1, v2, v3) -> nil [0]
   dequeue#3(v0) -> nil [0]
   dequeue#4(v0) -> nil [0]

And the following fresh constants: none

----------------------------------------

(212)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2)) [1]
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue') [1]
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue') [1]
   breadth#2(nil, @queue') -> nil [1]
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue'))) [1]
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6) [1]
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8) [1]
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2)) [1]
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs) [1]
   children#1(nil, @b, @l2) -> children#2(@l2, @b) [1]
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil) [1]
   children#2(nil, @b) -> nil [1]
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil)) [1]
   children#3(nil, @b, @x, @xs) -> nil [1]
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2)) [1]
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq) [1]
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq)) [1]
   copyover#2(nil, @outq) -> tuple#2(nil, @outq) [1]
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2)) [1]
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq) [1]
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil)) [1]
   dequeue#2(nil, @inq) -> dequeue#3(@inq) [1]
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil)) [1]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [1]
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4) [1]
   empty(@x) -> tuple#2(nil, nil) [1]
   enqueue(@x, @queue) -> enqueue#1(@queue, @x) [1]
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq) [1]
   enqueues(@l, @queue) -> enqueues#1(@l, @queue) [1]
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue)) [1]
   enqueues#1(nil, @queue) -> @queue [1]
   startBreadth(@xs) -> startBreadth#1(@xs) [1]
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit))) [1]
   startBreadth#1(nil) -> nil [1]
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2) [1]
   breadth#4(v0) -> nil [0]
   copyover#1(v0) -> nil [0]
   copyover#2(v0, v1) -> nil [0]
   enqueue#1(v0, v1) -> nil [0]
   enqueues#1(v0, v1) -> nil [0]
   dequeue#1(v0) -> nil [0]
   dequeue#2(v0, v1) -> nil [0]
   children#1(v0, v1, v2) -> nil [0]
   children#2(v0, v1) -> nil [0]
   children#3(v0, v1, v2, v3) -> nil [0]
   dequeue#3(v0) -> nil [0]
   dequeue#4(v0) -> nil [0]

The TRS has the following type information:
breadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
nil :: tuple#2::::nil:tuple#4
breadth#5 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
empty :: #unit -> tuple#2::::nil:tuple#4
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
#unit :: #unit

Rewrite Strategy: INNERMOST
----------------------------------------

(213) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(214)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue#1(tuple#2(@breadth@1, @breadth@2))) [2]
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue') [1]
   breadth#2(::(tuple#4(@children@3', @children@4', @children@5', @children@6'), @_@9), @queue') -> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), @queue') [2]
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(nil, @queue') [1]
   breadth#2(nil, @queue') -> nil [1]
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues#1(@ys, @queue'))) [2]
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6) [1]
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8) [1]
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2)) [1]
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs) [1]
   children#1(nil, @b, @l2) -> children#2(@l2, @b) [1]
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil) [1]
   children#2(nil, @b) -> nil [1]
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil)) [1]
   children#3(nil, @b, @x, @xs) -> nil [1]
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2)) [1]
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq) [1]
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq)) [1]
   copyover#2(nil, @outq) -> tuple#2(nil, @outq) [1]
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2)) [1]
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq) [1]
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil)) [1]
   dequeue#2(nil, @inq) -> dequeue#3(@inq) [1]
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover#1(tuple#2(::(@x, @xs), nil))) [2]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [1]
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4) [1]
   empty(@x) -> tuple#2(nil, nil) [1]
   enqueue(@x, @queue) -> enqueue#1(@queue, @x) [1]
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq) [1]
   enqueues(@l, @queue) -> enqueues#1(@l, @queue) [1]
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue#1(@queue, @x)) [2]
   enqueues#1(nil, @queue) -> @queue [1]
   startBreadth(@xs) -> startBreadth#1(@xs) [1]
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), tuple#2(nil, nil))) [2]
   startBreadth#1(nil) -> nil [1]
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2) [1]
   breadth#4(v0) -> nil [0]
   copyover#1(v0) -> nil [0]
   copyover#2(v0, v1) -> nil [0]
   enqueue#1(v0, v1) -> nil [0]
   enqueues#1(v0, v1) -> nil [0]
   dequeue#1(v0) -> nil [0]
   dequeue#2(v0, v1) -> nil [0]
   children#1(v0, v1, v2) -> nil [0]
   children#2(v0, v1) -> nil [0]
   children#3(v0, v1, v2, v3) -> nil [0]
   dequeue#3(v0) -> nil [0]
   dequeue#4(v0) -> nil [0]

The TRS has the following type information:
breadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
tuple#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
:: :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
breadth#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
nil :: tuple#2::::nil:tuple#4
breadth#5 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
tuple#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
children#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
copyover#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#3 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
dequeue#4 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
empty :: #unit -> tuple#2::::nil:tuple#4
enqueue :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueue#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
enqueues#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#1 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
startBreadth#2 :: tuple#2::::nil:tuple#4 -> tuple#2::::nil:tuple#4
#unit :: #unit

Rewrite Strategy: INNERMOST
----------------------------------------

(215) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   #unit => 0

----------------------------------------

(216)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + @breadth@1 + @breadth@2)) :|: @breadth@1 >= 0, z' = @breadth@2, @breadth@2 >= 0, z = @breadth@1
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), @queue') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, @queue' >= 0, @_@9 >= 0, z' = @queue'
   breadth#2(z, z') -{ 1 }-> breadth#3(0, @queue') :|: z = 1 + @z + @_@9, @queue' >= 0, @_@9 >= 0, z' = @queue', @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: @queue' >= 0, z = 0, z' = @queue'
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, @queue')) :|: @x >= 0, @queue' >= 0, @ys >= 0, z = 1 + @x + @ys, z' = @queue'
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + @a + @b) + children#1(@l1, @b, @l2) :|: z'' = @l1, @l1 >= 0, @a >= 0, @l2 >= 0, z = @a, z' = @b, @b >= 0, z1 = @l2
   children#1(z, z', z'') -{ 1 }-> children#3(@l2, @b, @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z'' = @l2, @l2 >= 0, @xs >= 0, z' = @b, @b >= 0
   children#1(z, z', z'') -{ 1 }-> children#2(@l2, @b) :|: z'' = @l2, @l2 >= 0, z' = @b, z = 0, @b >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z' = @b, z = 0, @b >= 0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + @b + 0 + @ys) + 0 :|: z = 1 + @y + @ys, z' = @b, @y >= 0, @b >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z = 0, z'' = @x, @b >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @b + 0 + @xs) + (1 + (1 + @x + @y + @xs + @ys) + 0) :|: z = 1 + @y + @ys, @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z'' = @x, @y >= 0, @b >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + @copyover@1 + @copyover@2) :|: @copyover@2 >= 0, z = @copyover@1, z' = @copyover@2, @copyover@1 >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + @outq) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @outq >= 0, z' = @outq
   copyover#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#2(z, z') -{ 1 }-> 1 + 0 + @outq :|: @outq >= 0, z' = @outq, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + @dequeue@1 + @dequeue@2) :|: @dequeue@2 >= 0, z = @dequeue@1, @dequeue@1 >= 0, z' = @dequeue@2
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(@inq) :|: z' = @inq, @inq >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + @inq + @ys) + (1 + @y + 0) :|: z' = @inq, z = 1 + @y + @ys, @inq >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z = @x, @x >= 0
   enqueue(z, z') -{ 1 }-> enqueue#1(@queue, @x) :|: @queue >= 0, z = @x, @x >= 0, z' = @queue
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + @x + @inq) + @outq :|: z = 1 + @inq + @outq, @x >= 0, @inq >= 0, @outq >= 0, z' = @x
   enqueues(z, z') -{ 1 }-> enqueues#1(@l, @queue) :|: @queue >= 0, z = @l, @l >= 0, z' = @queue
   enqueues#1(z, z') -{ 1 }-> @queue :|: @queue >= 0, z' = @queue, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, enqueue#1(@queue, @x)) :|: @queue >= 0, @x >= 0, z = 1 + @x + @xs, z' = @queue, @xs >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   startBreadth(z) -{ 1 }-> startBreadth#1(@xs) :|: z = @xs, @xs >= 0
   startBreadth#1(z) -{ 2 }-> startBreadth#2(enqueue(1 + @x + @x + @xs + @xs, 1 + 0 + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2


----------------------------------------

(217) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   children#2(z, z') -{ 1 }-> 1 + (1 + @y + @b + 0 + @ys) + 0 :|: z = 1 + @y + @ys, z' = @b, @y >= 0, @b >= 0, @ys >= 0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 1 }-> 0 :|: z' = @b, z = 0, @b >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z = 0, z'' = @x, @b >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @b + 0 + @xs) + (1 + (1 + @x + @y + @xs + @ys) + 0) :|: z = 1 + @y + @ys, @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z'' = @x, @y >= 0, @b >= 0, @ys >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + @x + @inq) + @outq :|: z = 1 + @inq + @outq, @x >= 0, @inq >= 0, @outq >= 0, z' = @x
   children#1(z, z', z'') -{ 1 }-> children#2(@l2, @b) :|: z'' = @l2, @l2 >= 0, z' = @b, z = 0, @b >= 0
   children#1(z, z', z'') -{ 1 }-> children#3(@l2, @b, @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z'' = @l2, @l2 >= 0, @xs >= 0, z' = @b, @b >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   enqueue(z, z') -{ 1 }-> enqueue#1(@queue, @x) :|: @queue >= 0, z = @x, @x >= 0, z' = @queue

----------------------------------------

(218)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + @breadth@1 + @breadth@2)) :|: @breadth@1 >= 0, z' = @breadth@2, @breadth@2 >= 0, z = @breadth@1
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), @queue') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, @queue' >= 0, @_@9 >= 0, z' = @queue'
   breadth#2(z, z') -{ 1 }-> breadth#3(0, @queue') :|: z = 1 + @z + @_@9, @queue' >= 0, @_@9 >= 0, z' = @queue', @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: @queue' >= 0, z = 0, z' = @queue'
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, @queue')) :|: @x >= 0, @queue' >= 0, @ys >= 0, z = 1 + @x + @ys, z' = @queue'
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + @a + @b) + children#3(@l2', @b', @x, @xs) :|: z'' = @l1, @l1 >= 0, @a >= 0, @l2 >= 0, z = @a, z' = @b, @b >= 0, z1 = @l2, @x >= 0, @l1 = 1 + @x + @xs, @l2 = @l2', @l2' >= 0, @xs >= 0, @b = @b', @b' >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + @a + @b) + children#2(@l2', @b') :|: z'' = @l1, @l1 >= 0, @a >= 0, @l2 >= 0, z = @a, z' = @b, @b >= 0, z1 = @l2, @l2 = @l2', @l2' >= 0, @b = @b', @l1 = 0, @b' >= 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + @a + @b) + 0 :|: z'' = @l1, @l1 >= 0, @a >= 0, @l2 >= 0, z = @a, z' = @b, @b >= 0, z1 = @l2, v0 >= 0, @l2 = v2, v1 >= 0, @l1 = v0, @b = v1, v2 >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' = @l2, @l2 >= 0, z' = @b, z = 0, @b >= 0, v0 >= 0, v1 >= 0, @l2 = v0, @b = v1
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' = @l2, @l2 >= 0, z' = @b, z = 0, @b >= 0, @b = @b', @l2 = 0, @b' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' = @l2, @l2 >= 0, @xs >= 0, z' = @b, @b >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, @b = @b', @l2 = 0, @x = @x', @b' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' = @l2, @l2 >= 0, @xs >= 0, z' = @b, @b >= 0, @xs = v3, v0 >= 0, @x = v2, v1 >= 0, @l2 = v0, @b = v1, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + @b' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' = @l2, @l2 >= 0, @xs >= 0, z' = @b, @b >= 0, @l2 = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @b = @b', @x = @x', @y >= 0, @b' >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + @b' + 0 + @ys) + 0 :|: z'' = @l2, @l2 >= 0, z' = @b, z = 0, @b >= 0, @l2 = 1 + @y + @ys, @b = @b', @y >= 0, @b' >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z' = @b, z = 0, @b >= 0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + @b + 0 + @ys) + 0 :|: z = 1 + @y + @ys, z' = @b, @y >= 0, @b >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z = 0, z'' = @x, @b >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @b + 0 + @xs) + (1 + (1 + @x + @y + @xs + @ys) + 0) :|: z = 1 + @y + @ys, @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z'' = @x, @y >= 0, @b >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + @copyover@1 + @copyover@2) :|: @copyover@2 >= 0, z = @copyover@1, z' = @copyover@2, @copyover@1 >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + @outq) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @outq >= 0, z' = @outq
   copyover#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#2(z, z') -{ 1 }-> 1 + 0 + @outq :|: @outq >= 0, z' = @outq, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + @dequeue@1 + @dequeue@2) :|: @dequeue@2 >= 0, z = @dequeue@1, @dequeue@1 >= 0, z' = @dequeue@2
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(@inq) :|: z' = @inq, @inq >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + @inq + @ys) + (1 + @y + 0) :|: z' = @inq, z = 1 + @y + @ys, @inq >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z = @x, @x >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: @queue >= 0, z = @x, @x >= 0, z' = @queue, v0 >= 0, v1 >= 0, @queue = v0, @x = v1
   enqueue(z, z') -{ 2 }-> 1 + (1 + @x' + @inq) + @outq :|: @queue >= 0, z = @x, @x >= 0, z' = @queue, @queue = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + @x + @inq) + @outq :|: z = 1 + @inq + @outq, @x >= 0, @inq >= 0, @outq >= 0, z' = @x
   enqueues(z, z') -{ 1 }-> enqueues#1(@l, @queue) :|: @queue >= 0, z = @l, @l >= 0, z' = @queue
   enqueues#1(z, z') -{ 1 }-> @queue :|: @queue >= 0, z' = @queue, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: @queue >= 0, @x >= 0, z = 1 + @x + @xs, z' = @queue, @xs >= 0, v0 >= 0, v1 >= 0, @queue = v0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: @queue >= 0, @x >= 0, z = 1 + @x + @xs, z' = @queue, @xs >= 0, @queue = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   startBreadth(z) -{ 1 }-> startBreadth#1(@xs) :|: z = @xs, @xs >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2


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(219) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(220)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2


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(221) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { children#1 }
   { empty }
   { enqueue }
   { children#3 }
   { enqueue#1 }
   { children#2 }
   { enqueues#1, enqueues }
   { copyover#2, copyover#1, copyover }
   { children }
   { dequeue, dequeue#3, dequeue#1, dequeue#2, dequeue#4 }
   { breadth#3, breadth#5, breadth#1, breadth, breadth#2 }
   { breadth#4 }
   { startBreadth#2 }
   { startBreadth#1 }
   { startBreadth }

----------------------------------------

(222)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#1}, {empty}, {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}

----------------------------------------

(223) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(224)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#1}, {empty}, {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}

----------------------------------------

(225) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: children#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 2*z + z' + z''

----------------------------------------

(226)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#1}, {empty}, {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: ?, size: O(n^1) [1 + 2*z + z' + z'']

----------------------------------------

(227) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: children#1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(228)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {empty}, {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']

----------------------------------------

(229) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(230)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {empty}, {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']

----------------------------------------

(231) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(232)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {empty}, {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: ?, size: O(1) [1]

----------------------------------------

(233) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(234)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(235) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(236)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(237) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: enqueue
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

----------------------------------------

(238)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueue}, {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(239) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: enqueue
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(240)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']

----------------------------------------

(241) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(242)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']

----------------------------------------

(243) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: children#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z + z' + 2*z'' + 2*z1

----------------------------------------

(244)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#3}, {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: ?, size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]

----------------------------------------

(245) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: children#3
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(246)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#3(z1, z', @x, @xs) :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]

----------------------------------------

(247) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(248)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]

----------------------------------------

(249) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: enqueue#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

----------------------------------------

(250)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueue#1}, {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(251) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: enqueue#1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(252)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 3 }-> startBreadth#2(enqueue#1(@queue, @x')) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']

----------------------------------------

(253) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(254)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']

----------------------------------------

(255) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: children#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

----------------------------------------

(256)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {children#2}, {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(257) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: children#2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(258)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 2 }-> 1 + (1 + z + z') + children#2(z1, z') :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']

----------------------------------------

(259) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(260)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s'' :|: s'' >= 0, s'' <= z1 + z' + 1, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']

----------------------------------------

(261) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: enqueues#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

Computed SIZE bound using CoFloCo for: enqueues
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

----------------------------------------

(262)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s'' :|: s'' >= 0, s'' <= z1 + z' + 1, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {enqueues#1,enqueues}, {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  enqueues#1: runtime: ?, size: O(n^1) [z + z']
  enqueues: runtime: ?, size: O(n^1) [z + z']

----------------------------------------

(263) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: enqueues#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 4*z

Computed RUNTIME bound using CoFloCo for: enqueues
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 8 + 4*z

----------------------------------------

(264)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 2 }-> 1 + @x + breadth#5(enqueues#1(@ys, z')) :|: @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s'' :|: s'' >= 0, s'' <= z1 + z' + 1, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 1 }-> enqueues#1(z, z') :|: z' >= 0, z >= 0
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 2 }-> enqueues(@xs, 0) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 3 }-> enqueues(@xs, 1 + (1 + @x' + @inq) + @outq) :|: z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  enqueues#1: runtime: O(n^1) [7 + 4*z], size: O(n^1) [z + z']
  enqueues: runtime: O(n^1) [8 + 4*z], size: O(n^1) [z + z']

----------------------------------------

(265) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(266)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 9 + 4*@ys }-> 1 + @x + breadth#5(s1) :|: s1 >= 0, s1 <= @ys + z', @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s'' :|: s'' >= 0, s'' <= z1 + z' + 1, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= z + z', z' >= 0, z >= 0
   enqueues#1(z, z') -{ 10 + 4*@xs }-> s3 :|: s3 >= 0, s3 <= @xs + 0, z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 11 + 4*@xs }-> s4 :|: s4 >= 0, s4 <= @xs + (1 + (1 + @x' + @inq) + @outq), z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  enqueues#1: runtime: O(n^1) [7 + 4*z], size: O(n^1) [z + z']
  enqueues: runtime: O(n^1) [8 + 4*z], size: O(n^1) [z + z']

----------------------------------------

(267) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: copyover#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

Computed SIZE bound using CoFloCo for: copyover#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

Computed SIZE bound using CoFloCo for: copyover
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

----------------------------------------

(268)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 9 + 4*@ys }-> 1 + @x + breadth#5(s1) :|: s1 >= 0, s1 <= @ys + z', @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s'' :|: s'' >= 0, s'' <= z1 + z' + 1, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= z + z', z' >= 0, z >= 0
   enqueues#1(z, z') -{ 10 + 4*@xs }-> s3 :|: s3 >= 0, s3 <= @xs + 0, z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 11 + 4*@xs }-> s4 :|: s4 >= 0, s4 <= @xs + (1 + (1 + @x' + @inq) + @outq), z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  enqueues#1: runtime: O(n^1) [7 + 4*z], size: O(n^1) [z + z']
  enqueues: runtime: O(n^1) [8 + 4*z], size: O(n^1) [z + z']
  copyover#2: runtime: ?, size: O(n^1) [1 + z + z']
  copyover#1: runtime: ?, size: O(n^1) [z]
  copyover: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(269) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: copyover#2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(270)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 2 }-> breadth#1(dequeue#1(1 + z + z')) :|: z >= 0, z' >= 0
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#2(z, z') -{ 2 }-> breadth#3(children(@children@3', @children@4', @children@5', @children@6'), z') :|: @children@5' >= 0, z = 1 + (1 + @children@3' + @children@4' + @children@5' + @children@6') + @_@9, @children@3' >= 0, @children@4' >= 0, @children@6' >= 0, z' >= 0, @_@9 >= 0
   breadth#2(z, z') -{ 1 }-> breadth#3(0, z') :|: z = 1 + @z + @_@9, z' >= 0, @_@9 >= 0, @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   breadth#3(z, z') -{ 9 + 4*@ys }-> 1 + @x + breadth#5(s1) :|: s1 >= 0, s1 <= @ys + z', @x >= 0, z' >= 0, @ys >= 0, z = 1 + @x + @ys
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: z >= 0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s :|: s >= 0, s <= z1 + z' + 2 * @x + 2 * @xs + 3, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, @x >= 0, z'' = 1 + @x + @xs, @xs >= 0
   children(z, z', z'', z1) -{ 3 }-> 1 + (1 + z + z') + s'' :|: s'' >= 0, s'' <= z1 + z' + 1, z'' >= 0, z >= 0, z1 >= 0, z' >= 0, z'' = 0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + z + z') + 0 :|: z'' >= 0, z >= 0, z1 >= 0, z' >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   children#1(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 0, z' >= 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 0
   children#1(z, z', z'') -{ 2 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @x' >= 0, @xs = @xs', @xs' >= 0, z'' = 0, @x = @x'
   children#1(z, z', z'') -{ 1 }-> 0 :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, @xs = v3, @x = v2, v2 >= 0, v3 >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @x' + z' + 0 + @xs') + (1 + (1 + @x' + @y + @xs' + @ys) + 0) :|: @x >= 0, z = 1 + @x + @xs, z'' >= 0, @xs >= 0, z' >= 0, z'' = 1 + @y + @ys, @x' >= 0, @xs = @xs', @xs' >= 0, @x = @x', @y >= 0, @ys >= 0
   children#1(z, z', z'') -{ 2 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z'' >= 0, z = 0, z' >= 0, z'' = 1 + @y + @ys, @y >= 0, @ys >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   children#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + z' + 0 + @ys) + 0 :|: z = 1 + @y + @ys, @y >= 0, z' >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z1 >= 0, z = 0, z' >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z'' + z' + 0 + z1) + (1 + (1 + z'' + @y + z1 + @ys) + 0) :|: z = 1 + @y + @ys, z'' >= 0, z1 >= 0, @y >= 0, z' >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + z + z') :|: z' >= 0, z >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: z >= 0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
   copyover#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   copyover#2(z, z') -{ 1 }-> 1 + 0 + z' :|: z' >= 0, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + z + z') :|: z' >= 0, z >= 0
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(z') :|: z' >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + z' + @ys) + (1 + @y + 0) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 2 }-> dequeue#4(copyover#1(1 + (1 + @x + @xs) + 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: z >= 0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: z >= 0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z >= 0
   enqueue(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   enqueue(z, z') -{ 2 }-> 1 + (1 + z + @inq) + @outq :|: z' >= 0, z >= 0, z' = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   enqueue#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + z' + @inq) + @outq :|: z = 1 + @inq + @outq, z' >= 0, @inq >= 0, @outq >= 0
   enqueues(z, z') -{ 8 + 4*z }-> s2 :|: s2 >= 0, s2 <= z + z', z' >= 0, z >= 0
   enqueues#1(z, z') -{ 10 + 4*@xs }-> s3 :|: s3 >= 0, s3 <= @xs + 0, z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v1 >= 0, @x = v1
   enqueues#1(z, z') -{ 11 + 4*@xs }-> s4 :|: s4 >= 0, s4 <= @xs + (1 + (1 + @x' + @inq) + @outq), z' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' = 1 + @inq + @outq, @x' >= 0, @inq >= 0, @outq >= 0, @x = @x'
   enqueues#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   startBreadth(z) -{ 1 }-> startBreadth#1(z) :|: z >= 0
   startBreadth#1(z) -{ 4 }-> startBreadth#2(s') :|: s' >= 0, s' <= @queue + @x' + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @queue >= 0, 1 + @x + @x + @xs + @xs = @x', @x' >= 0, 1 + 0 + 0 = @queue
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2

Function symbols to be analyzed: {copyover#2,copyover#1,copyover}, {children}, {dequeue,dequeue#3,dequeue#1,dequeue#2,dequeue#4}, {breadth#3,breadth#5,breadth#1,breadth,breadth#2}, {breadth#4}, {startBreadth#2}, {startBreadth#1}, {startBreadth}
Previous analysis results are:
  children#1: runtime: O(1) [2], size: O(n^1) [1 + 2*z + z' + z'']
  empty: runtime: O(1) [1], size: O(1) [1]
  enqueue: runtime: O(1) [2], size: O(n^1) [1 + z + z']
  children#3: runtime: O(1) [1], size: O(n^1) [3 + z + z' + 2*z'' + 2*z1]
  enqueue#1: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  children#2: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  enqueues#1: runtime: O(n^1) [7 + 4*z], size: O(n^1) [z + z']
  enqueues: runtime: O(n^1) [8 + 4*z], size: O(n^1) [z + z']
  copyover#2: runtime: INF, size: O(n^1) [1 + z + z']
  copyover#1: runtime: ?, size: O(n^1) [z]
  copyover: runtime: ?, size: O(n^1) [1 + z + z']

----------------------------------------

(271) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   breadth#1(v0) -> null_breadth#1 [0]
   breadth#2(v0, v1) -> null_breadth#2 [0]
   breadth#3(v0, v1) -> null_breadth#3 [0]
   breadth#4(v0) -> null_breadth#4 [0]
   breadth#5(v0) -> null_breadth#5 [0]
   children#1(v0, v1, v2) -> null_children#1 [0]
   children#2(v0, v1) -> null_children#2 [0]
   children#3(v0, v1, v2, v3) -> null_children#3 [0]
   copyover#1(v0) -> null_copyover#1 [0]
   copyover#2(v0, v1) -> null_copyover#2 [0]
   dequeue#1(v0) -> null_dequeue#1 [0]
   dequeue#2(v0, v1) -> null_dequeue#2 [0]
   dequeue#3(v0) -> null_dequeue#3 [0]
   dequeue#4(v0) -> null_dequeue#4 [0]
   enqueue#1(v0, v1) -> null_enqueue#1 [0]
   enqueues#1(v0, v1) -> null_enqueues#1 [0]
   startBreadth#1(v0) -> null_startBreadth#1 [0]
   startBreadth#2(v0) -> null_startBreadth#2 [0]

And the following fresh constants:    null_breadth#1, null_breadth#2, null_breadth#3, null_breadth#4, null_breadth#5, null_children#1, null_children#2, null_children#3, null_copyover#1, null_copyover#2, null_dequeue#1, null_dequeue#2, null_dequeue#3, null_dequeue#4, null_enqueue#1, null_enqueues#1, null_startBreadth#1, null_startBreadth#2

----------------------------------------

(272)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   breadth(@breadth@1, @breadth@2) -> breadth#1(dequeue(@breadth@1, @breadth@2)) [1]
   breadth#1(tuple#2(@queue', @elem)) -> breadth#2(@elem, @queue') [1]
   breadth#2(::(@z, @_@9), @queue') -> breadth#3(breadth#4(@z), @queue') [1]
   breadth#2(nil, @queue') -> nil [1]
   breadth#3(tuple#2(@x, @ys), @queue') -> ::(@x, breadth#5(enqueues(@ys, @queue'))) [1]
   breadth#4(tuple#4(@children@3, @children@4, @children@5, @children@6)) -> children(@children@3, @children@4, @children@5, @children@6) [1]
   breadth#5(tuple#2(@breadth@7, @breadth@8)) -> breadth(@breadth@7, @breadth@8) [1]
   children(@a, @b, @l1, @l2) -> tuple#2(tuple#2(@a, @b), children#1(@l1, @b, @l2)) [1]
   children#1(::(@x, @xs), @b, @l2) -> children#3(@l2, @b, @x, @xs) [1]
   children#1(nil, @b, @l2) -> children#2(@l2, @b) [1]
   children#2(::(@y, @ys), @b) -> ::(tuple#4(@y, @b, nil, @ys), nil) [1]
   children#2(nil, @b) -> nil [1]
   children#3(::(@y, @ys), @b, @x, @xs) -> ::(tuple#4(@x, @b, nil, @xs), ::(tuple#4(@x, @y, @xs, @ys), nil)) [1]
   children#3(nil, @b, @x, @xs) -> nil [1]
   copyover(@copyover@1, @copyover@2) -> copyover#1(tuple#2(@copyover@1, @copyover@2)) [1]
   copyover#1(tuple#2(@inq, @outq)) -> copyover#2(@inq, @outq) [1]
   copyover#2(::(@x, @xs), @outq) -> copyover(@xs, ::(@x, @outq)) [1]
   copyover#2(nil, @outq) -> tuple#2(nil, @outq) [1]
   dequeue(@dequeue@1, @dequeue@2) -> dequeue#1(tuple#2(@dequeue@1, @dequeue@2)) [1]
   dequeue#1(tuple#2(@inq, @outq)) -> dequeue#2(@outq, @inq) [1]
   dequeue#2(::(@y, @ys), @inq) -> tuple#2(tuple#2(@inq, @ys), ::(@y, nil)) [1]
   dequeue#2(nil, @inq) -> dequeue#3(@inq) [1]
   dequeue#3(::(@x, @xs)) -> dequeue#4(copyover(::(@x, @xs), nil)) [1]
   dequeue#3(nil) -> tuple#2(tuple#2(nil, nil), nil) [1]
   dequeue#4(tuple#2(@dequeue@3, @dequeue@4)) -> dequeue(@dequeue@3, @dequeue@4) [1]
   empty(@x) -> tuple#2(nil, nil) [1]
   enqueue(@x, @queue) -> enqueue#1(@queue, @x) [1]
   enqueue#1(tuple#2(@inq, @outq), @x) -> tuple#2(::(@x, @inq), @outq) [1]
   enqueues(@l, @queue) -> enqueues#1(@l, @queue) [1]
   enqueues#1(::(@x, @xs), @queue) -> enqueues(@xs, enqueue(@x, @queue)) [1]
   enqueues#1(nil, @queue) -> @queue [1]
   startBreadth(@xs) -> startBreadth#1(@xs) [1]
   startBreadth#1(::(@x, @xs)) -> startBreadth#2(enqueue(tuple#4(@x, @x, @xs, @xs), empty(#unit))) [1]
   startBreadth#1(nil) -> nil [1]
   startBreadth#2(tuple#2(@breadth@1, @breadth@2)) -> breadth(@breadth@1, @breadth@2) [1]
   breadth#1(v0) -> null_breadth#1 [0]
   breadth#2(v0, v1) -> null_breadth#2 [0]
   breadth#3(v0, v1) -> null_breadth#3 [0]
   breadth#4(v0) -> null_breadth#4 [0]
   breadth#5(v0) -> null_breadth#5 [0]
   children#1(v0, v1, v2) -> null_children#1 [0]
   children#2(v0, v1) -> null_children#2 [0]
   children#3(v0, v1, v2, v3) -> null_children#3 [0]
   copyover#1(v0) -> null_copyover#1 [0]
   copyover#2(v0, v1) -> null_copyover#2 [0]
   dequeue#1(v0) -> null_dequeue#1 [0]
   dequeue#2(v0, v1) -> null_dequeue#2 [0]
   dequeue#3(v0) -> null_dequeue#3 [0]
   dequeue#4(v0) -> null_dequeue#4 [0]
   enqueue#1(v0, v1) -> null_enqueue#1 [0]
   enqueues#1(v0, v1) -> null_enqueues#1 [0]
   startBreadth#1(v0) -> null_startBreadth#1 [0]
   startBreadth#2(v0) -> null_startBreadth#2 [0]

The TRS has the following type information:
breadth :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
breadth#1 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
dequeue :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
tuple#2 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
breadth#2 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
:: :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
breadth#3 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
breadth#4 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
nil :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
breadth#5 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
enqueues :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
tuple#4 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
children :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
children#1 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
children#3 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
children#2 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
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startBreadth#1 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2 -> tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
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null_enqueue#1 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
null_enqueues#1 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
null_startBreadth#1 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2
null_startBreadth#2 :: tuple#2::::nil:tuple#4:null_breadth#1:null_breadth#2:null_breadth#3:null_breadth#4:null_breadth#5:null_children#1:null_children#2:null_children#3:null_copyover#1:null_copyover#2:null_dequeue#1:null_dequeue#2:null_dequeue#3:null_dequeue#4:null_enqueue#1:null_enqueues#1:null_startBreadth#1:null_startBreadth#2

Rewrite Strategy: INNERMOST
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(273) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   #unit => 0
   null_breadth#1 => 0
   null_breadth#2 => 0
   null_breadth#3 => 0
   null_breadth#4 => 0
   null_breadth#5 => 0
   null_children#1 => 0
   null_children#2 => 0
   null_children#3 => 0
   null_copyover#1 => 0
   null_copyover#2 => 0
   null_dequeue#1 => 0
   null_dequeue#2 => 0
   null_dequeue#3 => 0
   null_dequeue#4 => 0
   null_enqueue#1 => 0
   null_enqueues#1 => 0
   null_startBreadth#1 => 0
   null_startBreadth#2 => 0

----------------------------------------

(274)
Obligation:
Complexity RNTS consisting of the following rules:

   breadth(z, z') -{ 1 }-> breadth#1(dequeue(@breadth@1, @breadth@2)) :|: @breadth@1 >= 0, z' = @breadth@2, @breadth@2 >= 0, z = @breadth@1
   breadth#1(z) -{ 1 }-> breadth#2(@elem, @queue') :|: z = 1 + @queue' + @elem, @elem >= 0, @queue' >= 0
   breadth#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   breadth#2(z, z') -{ 1 }-> breadth#3(breadth#4(@z), @queue') :|: z = 1 + @z + @_@9, @queue' >= 0, @_@9 >= 0, z' = @queue', @z >= 0
   breadth#2(z, z') -{ 1 }-> 0 :|: @queue' >= 0, z = 0, z' = @queue'
   breadth#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   breadth#3(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   breadth#3(z, z') -{ 1 }-> 1 + @x + breadth#5(enqueues(@ys, @queue')) :|: @x >= 0, @queue' >= 0, @ys >= 0, z = 1 + @x + @ys, z' = @queue'
   breadth#4(z) -{ 1 }-> children(@children@3, @children@4, @children@5, @children@6) :|: @children@4 >= 0, @children@5 >= 0, @children@6 >= 0, @children@3 >= 0, z = 1 + @children@3 + @children@4 + @children@5 + @children@6
   breadth#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   breadth#5(z) -{ 1 }-> breadth(@breadth@7, @breadth@8) :|: @breadth@8 >= 0, z = 1 + @breadth@7 + @breadth@8, @breadth@7 >= 0
   breadth#5(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   children(z, z', z'', z1) -{ 1 }-> 1 + (1 + @a + @b) + children#1(@l1, @b, @l2) :|: z'' = @l1, @l1 >= 0, @a >= 0, @l2 >= 0, z = @a, z' = @b, @b >= 0, z1 = @l2
   children#1(z, z', z'') -{ 1 }-> children#3(@l2, @b, @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z'' = @l2, @l2 >= 0, @xs >= 0, z' = @b, @b >= 0
   children#1(z, z', z'') -{ 1 }-> children#2(@l2, @b) :|: z'' = @l2, @l2 >= 0, z' = @b, z = 0, @b >= 0
   children#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   children#2(z, z') -{ 1 }-> 0 :|: z' = @b, z = 0, @b >= 0
   children#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   children#2(z, z') -{ 1 }-> 1 + (1 + @y + @b + 0 + @ys) + 0 :|: z = 1 + @y + @ys, z' = @b, @y >= 0, @b >= 0, @ys >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 0 :|: @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z = 0, z'' = @x, @b >= 0
   children#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   children#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @b + 0 + @xs) + (1 + (1 + @x + @y + @xs + @ys) + 0) :|: z = 1 + @y + @ys, @x >= 0, z1 = @xs, @xs >= 0, z' = @b, z'' = @x, @y >= 0, @b >= 0, @ys >= 0
   copyover(z, z') -{ 1 }-> copyover#1(1 + @copyover@1 + @copyover@2) :|: @copyover@2 >= 0, z = @copyover@1, z' = @copyover@2, @copyover@1 >= 0
   copyover#1(z) -{ 1 }-> copyover#2(@inq, @outq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   copyover#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   copyover#2(z, z') -{ 1 }-> copyover(@xs, 1 + @x + @outq) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, @outq >= 0, z' = @outq
   copyover#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   copyover#2(z, z') -{ 1 }-> 1 + 0 + @outq :|: @outq >= 0, z' = @outq, z = 0
   dequeue(z, z') -{ 1 }-> dequeue#1(1 + @dequeue@1 + @dequeue@2) :|: @dequeue@2 >= 0, z = @dequeue@1, @dequeue@1 >= 0, z' = @dequeue@2
   dequeue#1(z) -{ 1 }-> dequeue#2(@outq, @inq) :|: z = 1 + @inq + @outq, @inq >= 0, @outq >= 0
   dequeue#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#2(z, z') -{ 1 }-> dequeue#3(@inq) :|: z' = @inq, @inq >= 0, z = 0
   dequeue#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   dequeue#2(z, z') -{ 1 }-> 1 + (1 + @inq + @ys) + (1 + @y + 0) :|: z' = @inq, z = 1 + @y + @ys, @inq >= 0, @y >= 0, @ys >= 0
   dequeue#3(z) -{ 1 }-> dequeue#4(copyover(1 + @x + @xs, 0)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   dequeue#3(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dequeue#3(z) -{ 1 }-> 1 + (1 + 0 + 0) + 0 :|: z = 0
   dequeue#4(z) -{ 1 }-> dequeue(@dequeue@3, @dequeue@4) :|: @dequeue@3 >= 0, @dequeue@4 >= 0, z = 1 + @dequeue@3 + @dequeue@4
   dequeue#4(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   empty(z) -{ 1 }-> 1 + 0 + 0 :|: z = @x, @x >= 0
   enqueue(z, z') -{ 1 }-> enqueue#1(@queue, @x) :|: @queue >= 0, z = @x, @x >= 0, z' = @queue
   enqueue#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   enqueue#1(z, z') -{ 1 }-> 1 + (1 + @x + @inq) + @outq :|: z = 1 + @inq + @outq, @x >= 0, @inq >= 0, @outq >= 0, z' = @x
   enqueues(z, z') -{ 1 }-> enqueues#1(@l, @queue) :|: @queue >= 0, z = @l, @l >= 0, z' = @queue
   enqueues#1(z, z') -{ 1 }-> @queue :|: @queue >= 0, z' = @queue, z = 0
   enqueues#1(z, z') -{ 1 }-> enqueues(@xs, enqueue(@x, @queue)) :|: @queue >= 0, @x >= 0, z = 1 + @x + @xs, z' = @queue, @xs >= 0
   enqueues#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   startBreadth(z) -{ 1 }-> startBreadth#1(@xs) :|: z = @xs, @xs >= 0
   startBreadth#1(z) -{ 1 }-> startBreadth#2(enqueue(1 + @x + @x + @xs + @xs, empty(0))) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
   startBreadth#1(z) -{ 1 }-> 0 :|: z = 0
   startBreadth#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   startBreadth#2(z) -{ 1 }-> breadth(@breadth@1, @breadth@2) :|: @breadth@1 >= 0, @breadth@2 >= 0, z = 1 + @breadth@1 + @breadth@2
   startBreadth#2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.