KILLED
proof of input_FWnJWm6MK6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 13 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 50 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 321 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 150 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 247 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 366 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 209 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 20.7 s]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 16.1 s]
        (56) CpxRNTS
    (57) CompletionProof [UPPER BOUND(ID), 0 ms]
    (58) CpxTypedWeightedCompleteTrs
        (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
(61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(62) CdtProblem
    (63) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (90) CdtProblem
    (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
    (96) CdtProblem
    (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms]
    (104) CdtProblem
    (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (106) CdtProblem
    (107) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (140) CdtProblem
    (141) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (144) CdtProblem
    (145) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem


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

(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:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   and(true, y) -> y
   and(false, y) -> false
   size(empty) -> 0
   size(edge(x, y, i)) -> s(size(i))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   reachable(x, y, i) -> reach(x, y, 0, i, i)
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
   if1(true, x, y, c, i, j) -> true
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
   if2(false, x, y, c, i, j) -> false
   if2(true, x, y, c, empty, j) -> false
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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:

   eq(0', 0') -> true
   eq(0', s(x)) -> false
   eq(s(x), 0') -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   and(true, y) -> y
   and(false, y) -> false
   size(empty) -> 0'
   size(edge(x, y, i)) -> s(size(i))
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   reachable(x, y, i) -> reach(x, y, 0', i, i)
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
   if1(true, x, y, c, i, j) -> true
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
   if2(false, x, y, c, i, j) -> false
   if2(true, x, y, c, empty, j) -> false
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   and(true, y) -> y
   and(false, y) -> false
   size(empty) -> 0
   size(edge(x, y, i)) -> s(size(i))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   reachable(x, y, i) -> reach(x, y, 0, i, i)
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
   if1(true, x, y, c, i, j) -> true
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
   if2(false, x, y, c, i, j) -> false
   if2(true, x, y, c, empty, j) -> false
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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

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

(6)
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:

   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   and(true, y) -> y [1]
   and(false, y) -> false [1]
   size(empty) -> 0 [1]
   size(edge(x, y, i)) -> s(size(i)) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   reachable(x, y, i) -> reach(x, y, 0, i, i) [1]
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1]
   if1(true, x, y, c, i, j) -> true [1]
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1]
   if2(false, x, y, c, i, j) -> false [1]
   if2(true, x, y, c, empty, j) -> false [1]
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   and(true, y) -> y [1]
   and(false, y) -> false [1]
   size(empty) -> 0 [1]
   size(edge(x, y, i)) -> s(size(i)) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   reachable(x, y, i) -> reach(x, y, 0, i, i) [1]
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1]
   if1(true, x, y, c, i, j) -> true [1]
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1]
   if2(false, x, y, c, i, j) -> false [1]
   if2(true, x, y, c, empty, j) -> false [1]
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
or :: true:false -> true:false -> true:false
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0:s
empty :: empty:edge
edge :: 0:s -> 0:s -> empty:edge -> empty:edge
le :: 0:s -> 0:s -> true:false
reachable :: 0:s -> 0:s -> empty:edge -> true:false
reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false

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

(9) 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:

   reachable_3

(c) The following functions are completely defined:

   le_2
   size_1
   eq_2
   if2_6
   and_2
   reach_5
   if1_6
   or_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(10)
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:

   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   and(true, y) -> y [1]
   and(false, y) -> false [1]
   size(empty) -> 0 [1]
   size(edge(x, y, i)) -> s(size(i)) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   reachable(x, y, i) -> reach(x, y, 0, i, i) [1]
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1]
   if1(true, x, y, c, i, j) -> true [1]
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1]
   if2(false, x, y, c, i, j) -> false [1]
   if2(true, x, y, c, empty, j) -> false [1]
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
or :: true:false -> true:false -> true:false
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0:s
empty :: empty:edge
edge :: 0:s -> 0:s -> empty:edge -> empty:edge
le :: 0:s -> 0:s -> true:false
reachable :: 0:s -> 0:s -> empty:edge -> true:false
reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false

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

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

(12)
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:

   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   and(true, y) -> y [1]
   and(false, y) -> false [1]
   size(empty) -> 0 [1]
   size(edge(x, y, i)) -> s(size(i)) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   reachable(x, y, i) -> reach(x, y, 0, i, i) [1]
   reach(0, 0, c, i, j) -> if1(true, 0, 0, c, i, j) [2]
   reach(0, s(x'), c, i, j) -> if1(false, 0, s(x'), c, i, j) [2]
   reach(s(x''), 0, c, i, j) -> if1(false, s(x''), 0, c, i, j) [2]
   reach(s(x1), s(y'), c, i, j) -> if1(eq(x1, y'), s(x1), s(y'), c, i, j) [2]
   if1(true, x, y, c, i, j) -> true [1]
   if1(false, x, y, c, i, empty) -> if2(le(c, 0), x, y, c, i, empty) [2]
   if1(false, x, y, c, i, edge(x2, y'', i')) -> if2(le(c, s(size(i'))), x, y, c, i, edge(x2, y'', i')) [2]
   if2(false, x, y, c, i, j) -> false [1]
   if2(true, x, y, c, empty, j) -> false [1]
   if2(true, 0, y, c, edge(0, v, empty), j) -> or(false, and(true, if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, 0, y, c, edge(s(x3), v, empty), j) -> or(false, and(false, if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, s(x4), y, c, edge(0, v, empty), j) -> or(false, and(false, if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, s(x5), y, c, edge(s(y1), v, empty), j) -> or(false, and(eq(x5, y1), if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, 0, y, c, edge(0, v, edge(u', v', i'')), j) -> or(or(if2(true, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, s(c), j, j))), and(true, if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, 0, y, c, edge(s(x6), v, edge(u', v', i'')), j) -> or(or(if2(true, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, s(c), j, j))), and(false, if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, s(x7), y, c, edge(0, v, edge(u', v', i'')), j) -> or(or(if2(true, s(x7), y, c, i'', j), and(eq(s(x7), u'), reach(v', y, s(c), j, j))), and(false, if1(eq(v, y), v, y, s(c), j, j))) [4]
   if2(true, s(x8), y, c, edge(s(y2), v, edge(u', v', i'')), j) -> or(or(if2(true, s(x8), y, c, i'', j), and(eq(s(x8), u'), reach(v', y, s(c), j, j))), and(eq(x8, y2), if1(eq(v, y), v, y, s(c), j, j))) [4]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
or :: true:false -> true:false -> true:false
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0:s
empty :: empty:edge
edge :: 0:s -> 0:s -> empty:edge -> empty:edge
le :: 0:s -> 0:s -> true:false
reachable :: 0:s -> 0:s -> empty:edge -> true:false
reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false

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

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

   0 => 0
   true => 1
   false => 0
   empty => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y
   and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 0), x, y, c, i, 0) :|: z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z3 = 0, x >= 0, y >= 0, i >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 1 + size(i')), x, y, c, i, 1 + x2 + y'' + i') :|: z' = x, z'' = y, z1 = c, c >= 0, y >= 0, i >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z2 = i, x >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z3 = j, i >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, j >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x7, y, c, i'', j), and(eq(1 + x7, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: z' = 1 + x7, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x7 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x8, y, c, i'', j), and(eq(1 + x8, u'), reach(v', y, 1 + c, j, j))), and(eq(x8, y2), if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, x8 >= 0, z'' = y, z1 = c, c >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), y >= 0, z3 = j, u' >= 0, y2 >= 0, j >= 0, i'' >= 0, z' = 1 + x8, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(x5, y1), if1(eq(v, y), v, y, 1 + c, j, j))) :|: y1 >= 0, j >= 0, x5 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z' = 1 + x5, z3 = j, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, z2 = 1 + 0 + v + 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, x4 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, z' = 1 + x4, y >= 0, z3 = j, z2 = 1 + 0 + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, z2 = 0, x >= 0, y >= 0, z3 = j
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(x1, y'), 1 + x1, 1 + y', c, i, j) :|: j >= 0, x1 >= 0, c >= 0, z2 = j, i >= 0, z = 1 + x1, y' >= 0, z' = 1 + y', z'' = c, z1 = i
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, c, i, j) :|: j >= 0, c >= 0, z2 = j, i >= 0, z = 0, z'' = c, z1 = i, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + x', c, i, j) :|: j >= 0, z' = 1 + x', c >= 0, z2 = j, x' >= 0, i >= 0, z = 0, z'' = c, z1 = i
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + x'', 0, c, i, j) :|: z = 1 + x'', j >= 0, c >= 0, z2 = j, i >= 0, x'' >= 0, z'' = c, z1 = i, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(x, y, 0, i, i) :|: z'' = i, x >= 0, y >= 0, i >= 0, z = x, z' = y
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0


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

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

   or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y
   or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y
   and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 0), x, y, c, i, 0) :|: z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z3 = 0, x >= 0, y >= 0, i >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 1 + size(i')), x, y, c, i, 1 + x2 + y'' + i') :|: z' = x, z'' = y, z1 = c, c >= 0, y >= 0, i >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z2 = i, x >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z3 = j, i >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, j >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x7, y, c, i'', j), and(eq(1 + x7, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: z' = 1 + x7, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x7 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x8, y, c, i'', j), and(eq(1 + x8, u'), reach(v', y, 1 + c, j, j))), and(eq(x8, y2), if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, x8 >= 0, z'' = y, z1 = c, c >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), y >= 0, z3 = j, u' >= 0, y2 >= 0, j >= 0, i'' >= 0, z' = 1 + x8, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(x5, y1), if1(eq(v, y), v, y, 1 + c, j, j))) :|: y1 >= 0, j >= 0, x5 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z' = 1 + x5, z3 = j, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, z2 = 1 + 0 + v + 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, x4 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, z' = 1 + x4, y >= 0, z3 = j, z2 = 1 + 0 + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, z2 = 0, x >= 0, y >= 0, z3 = j
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(x1, y'), 1 + x1, 1 + y', c, i, j) :|: j >= 0, x1 >= 0, c >= 0, z2 = j, i >= 0, z = 1 + x1, y' >= 0, z' = 1 + y', z'' = c, z1 = i
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, c, i, j) :|: j >= 0, c >= 0, z2 = j, i >= 0, z = 0, z'' = c, z1 = i, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + x', c, i, j) :|: j >= 0, z' = 1 + x', c >= 0, z2 = j, x' >= 0, i >= 0, z = 0, z'' = c, z1 = i
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + x'', 0, c, i, j) :|: z = 1 + x'', j >= 0, c >= 0, z2 = j, i >= 0, x'' >= 0, z'' = c, z1 = i, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(x, y, 0, i, i) :|: z'' = i, x >= 0, y >= 0, i >= 0, z = x, z' = y
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0


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

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0


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

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

   { le }
   { size }
   { eq }
   { and }
   { or }
   { if2, reach, if1 }
   { reachable }

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {le}, {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable}

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

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {le}, {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {le}, {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {eq}, {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: ?, size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z'

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {and}, {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: ?, size: O(n^1) [z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {or}, {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']
  or: runtime: ?, size: O(n^1) [1 + z']

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']
  or: runtime: O(1) [1], size: O(n^1) [1 + z']

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

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']
  or: runtime: O(1) [1], size: O(n^1) [1 + z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: reach
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']
  or: runtime: O(1) [1], size: O(n^1) [1 + z']
  if2: runtime: ?, size: INF
  reach: runtime: ?, size: INF
  if1: runtime: ?, size: INF

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0
   if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0
   reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Function symbols to be analyzed: {if2,reach,if1}, {reachable}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  size: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [1]
  and: runtime: O(1) [1], size: O(n^1) [z']
  or: runtime: O(1) [1], size: O(n^1) [1 + z']
  if2: runtime: INF, size: INF
  reach: runtime: ?, size: INF
  if1: runtime: ?, size: INF

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

(57) 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:
none

And the following fresh constants: none

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

(58)
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:

   eq(0, 0) -> true [1]
   eq(0, s(x)) -> false [1]
   eq(s(x), 0) -> false [1]
   eq(s(x), s(y)) -> eq(x, y) [1]
   or(true, y) -> true [1]
   or(false, y) -> y [1]
   and(true, y) -> y [1]
   and(false, y) -> false [1]
   size(empty) -> 0 [1]
   size(edge(x, y, i)) -> s(size(i)) [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   reachable(x, y, i) -> reach(x, y, 0, i, i) [1]
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1]
   if1(true, x, y, c, i, j) -> true [1]
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1]
   if2(false, x, y, c, i, j) -> false [1]
   if2(true, x, y, c, empty, j) -> false [1]
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
or :: true:false -> true:false -> true:false
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0:s
empty :: empty:edge
edge :: 0:s -> 0:s -> empty:edge -> empty:edge
le :: 0:s -> 0:s -> true:false
reachable :: 0:s -> 0:s -> empty:edge -> true:false
reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false

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

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

   0 => 0
   true => 1
   false => 0
   empty => 0

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

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

   and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y
   and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> if2(le(c, size(j)), x, y, c, i, j) :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0
   if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z3 = j, i >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> or(if2(1, x, y, c, i, j), and(eq(x, u), reach(v, y, 1 + c, j, j))) :|: v >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, i >= 0, u >= 0, j >= 0, z2 = 1 + u + v + i, x >= 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0
   if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, z2 = 0, x >= 0, y >= 0, z3 = j
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y
   reach(z, z', z'', z1, z2) -{ 1 }-> if1(eq(x, y), x, y, c, i, j) :|: j >= 0, c >= 0, z2 = j, x >= 0, y >= 0, i >= 0, z = x, z' = y, z'' = c, z1 = i
   reachable(z, z', z'') -{ 1 }-> reach(x, y, 0, i, i) :|: z'' = i, x >= 0, y >= 0, i >= 0, z = x, z' = y
   size(z) -{ 1 }-> 0 :|: z = 0
   size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   and(true, z0) -> z0
   and(false, z0) -> false
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2)
   reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4)
   if1(true, z0, z1, z2, z3, z4) -> true
   if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4)
   if2(false, z0, z1, z2, z3, z4) -> false
   if2(true, z0, z1, z2, empty, z3) -> false
   if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)))
Tuples:
   EQ(0, 0) -> c
   EQ(0, s(z0)) -> c1
   EQ(s(z0), 0) -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   OR(true, z0) -> c4
   OR(false, z0) -> c5
   AND(true, z0) -> c6
   AND(false, z0) -> c7
   SIZE(empty) -> c8
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(0, z0) -> c10
   LE(s(z0), 0) -> c11
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACHABLE(z0, z1, z2) -> c13(REACH(z0, z1, 0, z2, z2))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(true, z0, z1, z2, z3, z4) -> c15
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(false, z0, z1, z2, z3, z4) -> c17
   IF2(true, z0, z1, z2, empty, z3) -> c18
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6))
S tuples:
   EQ(0, 0) -> c
   EQ(0, s(z0)) -> c1
   EQ(s(z0), 0) -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   OR(true, z0) -> c4
   OR(false, z0) -> c5
   AND(true, z0) -> c6
   AND(false, z0) -> c7
   SIZE(empty) -> c8
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(0, z0) -> c10
   LE(s(z0), 0) -> c11
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACHABLE(z0, z1, z2) -> c13(REACH(z0, z1, 0, z2, z2))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(true, z0, z1, z2, z3, z4) -> c15
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(false, z0, z1, z2, z3, z4) -> c17
   IF2(true, z0, z1, z2, empty, z3) -> c18
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6))
K tuples:none
Defined Rule Symbols:   eq_2, or_2, and_2, size_1, le_2, reachable_3, reach_5, if1_6, if2_6

Defined Pair Symbols:   EQ_2, OR_2, AND_2, SIZE_1, LE_2, REACHABLE_3, REACH_5, IF1_6, IF2_6

Compound Symbols:   c, c1, c2, c3_1, c4, c5, c6, c7, c8, c9_1, c10, c11, c12_1, c13_1, c14_2, c15, c16_3, c17, c18, c19_2, c20_3, c21_3


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

(63) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   REACHABLE(z0, z1, z2) -> c13(REACH(z0, z1, 0, z2, z2))
Removed 13 trailing nodes:
   LE(0, z0) -> c10
   IF1(true, z0, z1, z2, z3, z4) -> c15
   SIZE(empty) -> c8
   AND(false, z0) -> c7
   OR(true, z0) -> c4
   EQ(0, 0) -> c
   IF2(false, z0, z1, z2, z3, z4) -> c17
   LE(s(z0), 0) -> c11
   AND(true, z0) -> c6
   OR(false, z0) -> c5
   EQ(s(z0), 0) -> c2
   EQ(0, s(z0)) -> c1
   IF2(true, z0, z1, z2, empty, z3) -> c18

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   and(true, z0) -> z0
   and(false, z0) -> false
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2)
   reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4)
   if1(true, z0, z1, z2, z3, z4) -> true
   if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4)
   if2(false, z0, z1, z2, z3, z4) -> false
   if2(true, z0, z1, z2, empty, z3) -> false
   if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6))
K tuples:none
Defined Rule Symbols:   eq_2, or_2, and_2, size_1, le_2, reachable_3, reach_5, if1_6, if2_6

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6

Compound Symbols:   c3_1, c9_1, c12_1, c14_2, c16_3, c19_2, c20_3, c21_3


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

(65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   or(true, z0) -> true
   or(false, z0) -> z0
   and(true, z0) -> z0
   and(false, z0) -> false
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2)
   reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4)
   if1(true, z0, z1, z2, z3, z4) -> true
   if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4)
   if2(false, z0, z1, z2, z3, z4) -> false
   if2(true, z0, z1, z2, empty, z3) -> false
   if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
K tuples:none
Defined Rule Symbols:   eq_2, or_2, and_2, size_1, le_2, reachable_3, reach_5, if1_6, if2_6

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6

Compound Symbols:   c3_1, c9_1, c12_1, c14_2, c16_3, c19_1, c20_1, c21_1


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

(67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   or(true, z0) -> true
   or(false, z0) -> z0
   and(true, z0) -> z0
   and(false, z0) -> false
   reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2)
   reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4)
   if1(true, z0, z1, z2, z3, z4) -> true
   if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4)
   if2(false, z0, z1, z2, z3, z4) -> false
   if2(true, z0, z1, z2, empty, z3) -> false
   if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6

Compound Symbols:   c3_1, c9_1, c12_1, c14_2, c16_3, c19_1, c20_1, c21_1


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
We considered the (Usable) Rules:
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(0, 0) -> true
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_3
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = [1] + x_3
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2
   POL(SIZE(x_1)) = 0
   POL(c12(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(empty) = [1]
   POL(eq(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1
   POL(size(x_1)) = [1]
   POL(true) = [1]

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6

Compound Symbols:   c3_1, c9_1, c12_1, c14_2, c16_3, c19_1, c20_1, c21_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) by 
   REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0)))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0)))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0)))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF1_6, IF2_6, REACH_5

Compound Symbols:   c3_1, c9_1, c12_1, c16_3, c19_1, c20_1, c21_1, c14_2


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

(73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0)))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0)))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF1_6, IF2_6, REACH_5

Compound Symbols:   c3_1, c9_1, c12_1, c16_3, c19_1, c20_1, c21_1, c14_2


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

(75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF1_6, IF2_6, REACH_5

Compound Symbols:   c3_1, c9_1, c12_1, c16_3, c19_1, c20_1, c21_1, c14_2, c14_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) by 
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), LE(0, size(x4)), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)), SIZE(empty))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), LE(0, size(x4)), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)), SIZE(empty))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), LE(0, size(x4)), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)), SIZE(empty))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3


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

(79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
We considered the (Usable) Rules:none
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_4
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(SIZE(x_1)) = 0
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = x_1 + x_2
   POL(empty) = 0
   POL(eq(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(size(x_1)) = 0
   POL(true) = [1]

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1)))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) by 
   REACH(s(0), s(0), x2, x3, x4) -> c14(IF1(true, s(0), s(0), x2, x3, x4), EQ(s(0), s(0)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(IF1(true, s(0), s(0), x2, x3, x4), EQ(s(0), s(0)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(IF1(true, s(0), s(0), x2, x3, x4), EQ(s(0), s(0)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c16_2, c14_2


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c16_2, c14_2


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
We considered the (Usable) Rules:
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(0, 0) -> true
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_4
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = [1] + x_2 + x_4
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(SIZE(x_1)) = 0
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(empty) = 0
   POL(eq(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(size(x_1)) = 0
   POL(true) = [1]

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c16_2, c14_2


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) by 
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), LE(0, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), LE(0, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), LE(0, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3


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

(93) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1


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

(95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
We considered the (Usable) Rules:none
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_4
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [2]x_3
   POL(SIZE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = [2]
   POL(empty) = 0
   POL(eq(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(false) = 0
   POL(le(x_1, x_2)) = [3]
   POL(s(x_1)) = 0
   POL(size(x_1)) = [2] + [2]x_1
   POL(true) = 0

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1


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

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
We considered the (Usable) Rules:
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(0, 0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   le(0, z0) -> true
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(SIZE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = x_2
   POL(empty) = 0
   POL(eq(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(size(x_1)) = 0
   POL(true) = [1]

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) by 
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty), LE(0, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(IF2(false, x0, x1, s(z0), x3, empty), LE(s(z0), size(empty)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty), LE(0, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(IF2(false, x0, x1, s(z0), x3, empty), LE(s(z0), size(empty)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty), LE(0, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(IF2(false, x0, x1, s(z0), x3, empty), LE(s(z0), size(empty)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1


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

(101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1


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

(103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
We considered the (Usable) Rules:none
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_2 + x_6
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = x_2 + x_6
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5
   POL(SIZE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = x_1 + x_2
   POL(empty) = [1]
   POL(eq(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(size(x_1)) = 0
   POL(true) = [1]

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1


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

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
We considered the (Usable) Rules:none
And the Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(EQ(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_4
   POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(REACH(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(SIZE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(edge(x_1, x_2, x_3)) = x_1 + x_2
   POL(empty) = 0
   POL(eq(x_1, x_2)) = [1]
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(size(x_1)) = 0
   POL(true) = [1]

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1


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

(107) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) by 
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) by IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1


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

(111) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) by 
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, IF1_6, REACH_5

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c16_2, c14_2, c14_1, c16_3, c_1, c16_1


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

(113) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) by 
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2, c_1, c16_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2, c_1, c16_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0))))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(119) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) by 
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(121) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) by IF1(false, z0, z1, z2, z3, empty) -> c16(LE(z2, 0))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, empty) -> c16(LE(z2, 0))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6))))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF1(false, z0, z1, z2, z3, empty) -> c16(LE(z2, 0))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) by IF1(false, z0, z1, s(z2), z3, empty) -> c16(LE(s(z2), 0))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, s(z2), z3, empty) -> c16(LE(s(z2), 0))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF1(false, z0, z1, s(z2), z3, empty) -> c16(LE(s(z2), 0))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(137) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) by 
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(139) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1))))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3


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

(141) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) by 
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c_1, c16_1, c16_3, c14_2


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

(143) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6)))
   IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty))
   IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1)))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c_1, c16_3, c14_2


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

(145) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) by 
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0)))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c_1, c16_3, c14_2


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

(147) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) by 
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6)))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, IF1_6, REACH_5

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c_1, c14_1, c16_3, c14_2


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

(149) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) by 
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c14_2, c_1


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

(151) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) by 
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6

Compound Symbols:   c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3


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

(153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) by 
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
S tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2

Compound Symbols:   c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1


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

(155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0)))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
S tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0)))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2

Compound Symbols:   c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1


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

(157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6))))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
S tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2

Compound Symbols:   c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c_1, c16_3, c14_2, c3_1


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

(159) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) by 
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
S tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2

Compound Symbols:   c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c14_2, c_1, c3_1


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

(161) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) by 
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
S tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2

Compound Symbols:   c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c14_2, c_1, c3_1


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

(163) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) by 
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
S tuples:
   SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2))
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2

Compound Symbols:   c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) by 
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
S tuples:
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   LE_2, IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1

Compound Symbols:   c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1


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

(167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c12(LE(z0, z1))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   LE_2, IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1

Compound Symbols:   c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c16_2


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(z0), s(z1)) -> c12(LE(z0, z1)) by 
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
S tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2

Compound Symbols:   c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c16_2, c12_1


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

(171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
S tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2

Compound Symbols:   c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1


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

(173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) by 
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
S tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
K tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2

Compound Symbols:   c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1


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

(175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) by 
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
S tuples:
   IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6))
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
K tuples:
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2

Compound Symbols:   c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1


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

(177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) by 
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
S tuples:
   REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6))
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
K tuples:
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) by 
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
S tuples:
   REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6))
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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(181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) by 
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c14_1, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) by 
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by 
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by 
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by 
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by 
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6)))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by 
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6

Compound Symbols:   c14_2, c16_3, c3_1, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1


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

(195) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) by 
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
S tuples:
   REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
K tuples:
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   REACH_5, IF1_6, SIZE_1, LE_2, IF2_6, EQ_2

Compound Symbols:   c14_2, c16_3, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c3_1


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

(197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) by 
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))
K tuples:
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF1_6, REACH_5, SIZE_1, LE_2, IF2_6, EQ_2

Compound Symbols:   c16_3, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c3_1, c14_2


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

(199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) by 
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
S tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF1_6, REACH_5, SIZE_1, LE_2, IF2_6, EQ_2

Compound Symbols:   c16_3, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c3_1, c14_2


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

(201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) by 
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   size(empty) -> 0
   size(edge(z0, z1, z2)) -> s(size(z2))
Tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6))))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
S tuples:
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8))))
   SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2)))
   LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1)))
   IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)))
   IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6))
   IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6))
   IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6))
   IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6))
   IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6))
   IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6))
   IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6))
   REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1))))
   REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1)))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))))
   REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)))
K tuples:
   IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1))))
   REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1))))
   IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4))))
   IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4))))
Defined Rule Symbols:   eq_2, le_2, size_1

Defined Pair Symbols:   IF1_6, SIZE_1, LE_2, IF2_6, REACH_5, EQ_2

Compound Symbols:   c16_3, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c14_1, c3_1, c14_2