WORST_CASE(Omega(n^1),O(n^2)) proof of input_qLqflhqdsx.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 244 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 1024 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 974 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 110 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 608 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 141 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 616 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 102 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1337 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 475 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 309 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 114 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 206 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 103 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 125 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 136 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) FinalProof [FINISHED, 0 ms] (80) BOUNDS(1, n^2) (81) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 42 ms] (82) CdtProblem (83) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CpxRelTRS (85) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CpxRelTRS (87) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (88) typed CpxTrs (89) OrderProof [LOWER BOUND(ID), 22 ms] (90) typed CpxTrs (91) RewriteLemmaProof [LOWER BOUND(ID), 1824 ms] (92) typed CpxTrs (93) RewriteLemmaProof [LOWER BOUND(ID), 1521 ms] (94) typed CpxTrs (95) RewriteLemmaProof [LOWER BOUND(ID), 552 ms] (96) typed CpxTrs (97) RewriteLemmaProof [LOWER BOUND(ID), 645 ms] (98) typed CpxTrs (99) RewriteLemmaProof [LOWER BOUND(ID), 556 ms] (100) typed CpxTrs (101) RewriteLemmaProof [LOWER BOUND(ID), 821 ms] (102) typed CpxTrs (103) RewriteLemmaProof [LOWER BOUND(ID), 66 ms] (104) typed CpxTrs (105) RewriteLemmaProof [LOWER BOUND(ID), 462 ms] (106) BEST (107) proven lower bound (108) LowerBoundPropagationProof [FINISHED, 0 ms] (109) BOUNDS(n^1, INF) (110) typed CpxTrs (111) RewriteLemmaProof [LOWER BOUND(ID), 504 ms] (112) typed CpxTrs (113) RewriteLemmaProof [LOWER BOUND(ID), 489 ms] (114) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 flatten(@t) -> flatten#1(@t) flatten#1(leaf) -> nil flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) flattensort(@t) -> insertionsort(flatten(@t)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] flatten(@t) -> flatten#1(@t) [1] flatten#1(leaf) -> nil [1] flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) [1] flattensort(@t) -> insertionsort(flatten(@t)) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1] insertionsort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] flatten(@t) -> flatten#1(@t) [1] flatten#1(leaf) -> nil [1] flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) [1] flattensort(@t) -> insertionsort(flatten(@t)) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1] insertionsort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] The TRS has the following type information: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil flatten :: leaf:node -> :::nil flatten#1 :: leaf:node -> :::nil leaf :: leaf:node node :: :::nil -> leaf:node -> leaf:node -> leaf:node flattensort :: leaf:node -> :::nil insertionsort :: :::nil -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true insertionsort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: flattensort_1 (c) The following functions are completely defined: flatten_1 append_2 insertionsort_1 #less_2 flatten#1_1 append#1_2 insertionsort#1_1 insert_2 insert#1_2 insert#2_4 #cklt_1 #compare_2 Due to the following rules being added: #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] insert#2(v0, v1, v2, v3) -> nil [0] And the following fresh constants: null_#cklt, null_#compare ---------------------------------------- (8) 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: #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] flatten(@t) -> flatten#1(@t) [1] flatten#1(leaf) -> nil [1] flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten(@t1), flatten(@t2))) [1] flattensort(@t) -> insertionsort(flatten(@t)) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1] insertionsort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] insert#2(v0, v1, v2, v3) -> nil [0] The TRS has the following type information: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil flatten :: leaf:node -> :::nil flatten#1 :: leaf:node -> :::nil leaf :: leaf:node node :: :::nil -> leaf:node -> leaf:node -> leaf:node flattensort :: leaf:node -> :::nil insertionsort :: :::nil -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true:null_#cklt #true :: #false:#true:null_#cklt insertionsort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s null_#cklt :: #false:#true:null_#cklt null_#compare :: #EQ:#GT:#LT:null_#compare Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: #less(#0, #0) -> #cklt(#EQ) [1] #less(#0, #neg(@y')) -> #cklt(#GT) [1] #less(#0, #pos(@y'')) -> #cklt(#LT) [1] #less(#0, #s(@y1)) -> #cklt(#LT) [1] #less(#neg(@x'), #0) -> #cklt(#LT) [1] #less(#neg(@x''), #neg(@y2)) -> #cklt(#compare(@y2, @x'')) [1] #less(#neg(@x1), #pos(@y3)) -> #cklt(#LT) [1] #less(#pos(@x2), #0) -> #cklt(#GT) [1] #less(#pos(@x3), #neg(@y4)) -> #cklt(#GT) [1] #less(#pos(@x4), #pos(@y5)) -> #cklt(#compare(@x4, @y5)) [1] #less(#s(@x5), #0) -> #cklt(#GT) [1] #less(#s(@x6), #s(@y6)) -> #cklt(#compare(@x6, @y6)) [1] #less(@x, @y) -> #cklt(null_#compare) [1] append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] flatten(@t) -> flatten#1(@t) [1] flatten#1(leaf) -> nil [1] flatten#1(node(@l, @t1, @t2)) -> append(@l, append(flatten#1(@t1), flatten#1(@t2))) [3] flattensort(@t) -> insertionsort(flatten#1(@t)) [2] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) [2] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort#1(@xs)) [2] insertionsort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] insert#2(v0, v1, v2, v3) -> nil [0] The TRS has the following type information: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil flatten :: leaf:node -> :::nil flatten#1 :: leaf:node -> :::nil leaf :: leaf:node node :: :::nil -> leaf:node -> leaf:node -> leaf:node flattensort :: leaf:node -> :::nil insertionsort :: :::nil -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true:null_#cklt #true :: #false:#true:null_#cklt insertionsort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s null_#cklt :: #false:#true:null_#cklt null_#compare :: #EQ:#GT:#LT:null_#compare Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 leaf => 0 #false => 1 #true => 2 #EQ => 1 #GT => 2 #LT => 3 #0 => 0 null_#cklt => 0 null_#compare => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z = 1 + @x', @x' >= 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1 #less(z, z') -{ 1 }-> #cklt(2) :|: @y' >= 0, z' = 1 + @y', z = 0 #less(z, z') -{ 1 }-> #cklt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0 #less(z, z') -{ 1 }-> #cklt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0 #less(z, z') -{ 1 }-> #cklt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(1) :|: z = 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 append(z, z') -{ 1 }-> append#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 append#1(z, z') -{ 1 }-> @l2 :|: z' = @l2, @l2 >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @l2) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(@t) :|: z = @t, @t >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(@t)) :|: z = @t, @t >= 0 insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertionsort(z) -{ 1 }-> insertionsort#1(@l) :|: z = @l, @l >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 2 :|: z = 3 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 append(z, z') -{ 1 }-> append#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 append#1(z, z') -{ 1 }-> @l2 :|: z' = @l2, @l2 >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @l2) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(@t) :|: z = @t, @t >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(@t)) :|: z = @t, @t >= 0 insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertionsort(z) -{ 1 }-> insertionsort#1(@l) :|: z = @l, @l >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { #compare } { append#1, append } { #cklt } { #less } { insert#2, insert, insert#1 } { flatten#1 } { insertionsort#1 } { flatten } { insertionsort } { flattensort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#compare}, {append#1,append}, {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#compare}, {append#1,append}, {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#compare}, {append#1,append}, {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: ?, size: O(1) [3] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {append#1,append}, {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {append#1,append}, {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' Computed SIZE bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {append#1,append}, {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: ?, size: O(n^1) [z + z'] append: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z Computed RUNTIME bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #cklt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#cklt}, {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: ?, size: O(1) [2] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #cklt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #less after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {#less}, {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: ?, size: O(1) [2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #less after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insert#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1 Computed SIZE bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' Computed SIZE bound using CoFloCo for: insert#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: ?, size: O(n^1) [2 + z' + z'' + z1] insert: runtime: ?, size: O(n^1) [1 + z + z'] insert#1: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insert#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 4*z1 Computed RUNTIME bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 4*z' Computed RUNTIME bound using CoFloCo for: insert#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 4*z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: flatten#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flatten#1}, {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: flatten#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 12*z + 2*z^2 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 1 }-> flatten#1(z) :|: z >= 0 flatten#1(z) -{ 3 }-> append(@l, append(flatten#1(@t1), flatten#1(@t2))) :|: @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 2 }-> insertionsort(flatten#1(z)) :|: z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertionsort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insertionsort#1}, {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insertionsort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 4*z^2 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: flatten after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flatten}, {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: flatten after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 12*z + 2*z^2 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertionsort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {insertionsort}, {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort: runtime: ?, size: O(n^1) [z] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: insertionsort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 4*z^2 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 3 + 12*z + 2*z^2 }-> insertionsort(s16) :|: s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 5 + 4*s16^2 + 12*z + 2*z^2 }-> s20 :|: s20 >= 0, s20 <= s16, s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: flattensort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 5 + 4*s16^2 + 12*z + 2*z^2 }-> s20 :|: s20 >= 0, s20 <= s16, s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {flattensort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] flattensort: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: flattensort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 9 + 20*z + 6*z^2 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s3 :|: s3 >= 0, s3 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s4 :|: s4 >= 0, s4 <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 flatten(z) -{ 2 + 12*z + 2*z^2 }-> s11 :|: s11 >= 0, s11 <= z + 1, z >= 0 flatten#1(z) -{ 13 + 2*@l + 12*@t1 + 2*@t1^2 + 12*@t2 + 2*@t2^2 + 2*s12 }-> s15 :|: s12 >= 0, s12 <= @t1 + 1, s13 >= 0, s13 <= @t2 + 1, s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= @l + s14, @l >= 0, @t1 >= 0, @t2 >= 0, z = 1 + @l + @t1 + @t2 flatten#1(z) -{ 1 }-> 0 :|: z = 0 flattensort(z) -{ 5 + 4*s16^2 + 12*z + 2*z^2 }-> s20 :|: s20 >= 0, s20 <= s16, s16 >= 0, s16 <= z + 1, z >= 0 insert(z, z') -{ 2 + 4*z' }-> s8 :|: s8 >= 0, s8 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s10 :|: s10 >= 0, s10 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s9 :|: s9 >= 0, s9 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s17 :|: s17 >= 0, s17 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s18 }-> s19 :|: s18 >= 0, s18 <= @xs, s19 >= 0, s19 <= @x + s18 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] flatten#1: runtime: O(n^2) [1 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] flatten: runtime: O(n^2) [2 + 12*z + 2*z^2], size: O(n^1) [1 + z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] flattensort: runtime: O(n^2) [9 + 20*z + 6*z^2], size: O(n^1) [1 + z] ---------------------------------------- (79) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (80) BOUNDS(1, n^2) ---------------------------------------- (81) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Tuples: #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 S tuples: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 K tuples:none Defined Rule Symbols: #less_2, append_2, append#1_2, flatten_1, flatten#1_1, flattensort_1, insert_2, insert#1_2, insert#2_4, insertionsort_1, insertionsort#1_1, #cklt_1, #compare_2 Defined Pair Symbols: #CKLT_1, #COMPARE_2, #LESS_2, APPEND_2, APPEND#1_2, FLATTEN_1, FLATTEN#1_1, FLATTENSORT_1, INSERT_2, INSERT#1_2, INSERT#2_4, INSERTIONSORT_1, INSERTIONSORT#1_1 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8_1, c9, c10, c11, c12_1, c13, c14_1, c15_2, c16_1, c17_1, c18, c19_1, c20, c21_3, c22_3, c23_2, c24_1, c25_2, c26, c27, c28_1, c29_1, c30_2, c31 ---------------------------------------- (83) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (84) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 The (relative) TRS S consists of the following rules: #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (85) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (86) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 The (relative) TRS S consists of the following rules: #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (87) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (88) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node ---------------------------------------- (89) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: #compare, #COMPARE, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERT#1, INSERTIONSORT#1, insertionsort, append#1, flatten#1, insert, insert#1, insertionsort#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 insertionsort < INSERTIONSORT#1 insertionsort = insertionsort#1 insert = insert#1 insert < insertionsort#1 ---------------------------------------- (90) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: #compare, #COMPARE, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERT#1, INSERTIONSORT#1, insertionsort, append#1, flatten#1, insert, insert#1, insertionsort#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 insertionsort < INSERTIONSORT#1 insertionsort = insertionsort#1 insert = insert#1 insert < insertionsort#1 ---------------------------------------- (91) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) Induction Base: #compare(gen_#0:#neg:#pos:#s19_32(0), gen_#0:#neg:#pos:#s19_32(0)) ->_R^Omega(0) #EQ Induction Step: #compare(gen_#0:#neg:#pos:#s19_32(+(n24_32, 1)), gen_#0:#neg:#pos:#s19_32(+(n24_32, 1))) ->_R^Omega(0) #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) ->_IH #EQ We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (92) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: #COMPARE, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERT#1, INSERTIONSORT#1, insertionsort, append#1, flatten#1, insert, insert#1, insertionsort#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 insertionsort < INSERTIONSORT#1 insertionsort = insertionsort#1 insert = insert#1 insert < insertionsort#1 ---------------------------------------- (93) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) Induction Base: #COMPARE(gen_#0:#neg:#pos:#s19_32(0), gen_#0:#neg:#pos:#s19_32(0)) ->_R^Omega(0) c3 Induction Step: #COMPARE(gen_#0:#neg:#pos:#s19_32(+(n318895_32, 1)), gen_#0:#neg:#pos:#s19_32(+(n318895_32, 1))) ->_R^Omega(0) c8(#COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32))) ->_IH c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(c318896_32)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (94) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: insert#1, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERT#1, INSERTIONSORT#1, insertionsort, append#1, flatten#1, insert, insertionsort#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 insertionsort < INSERTIONSORT#1 insertionsort = insertionsort#1 insert = insert#1 insert < insertionsort#1 ---------------------------------------- (95) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsort#1(gen_:::nil21_32(n638834_32)) -> *23_32, rt in Omega(0) Induction Base: insertionsort#1(gen_:::nil21_32(0)) Induction Step: insertionsort#1(gen_:::nil21_32(+(n638834_32, 1))) ->_R^Omega(0) insert(#0, insertionsort(gen_:::nil21_32(n638834_32))) ->_R^Omega(0) insert(#0, insertionsort#1(gen_:::nil21_32(n638834_32))) ->_IH insert(#0, *23_32) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (96) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n638834_32)) -> *23_32, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: insertionsort, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERT#1, INSERTIONSORT#1, append#1, flatten#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 insertionsort < INSERTIONSORT#1 insertionsort = insertionsort#1 ---------------------------------------- (97) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) Induction Base: insertionsort(gen_:::nil21_32(0)) Induction Step: insertionsort(gen_:::nil21_32(+(n643249_32, 1))) ->_R^Omega(0) insertionsort#1(gen_:::nil21_32(+(n643249_32, 1))) ->_R^Omega(0) insert(#0, insertionsort(gen_:::nil21_32(n643249_32))) ->_IH insert(#0, *23_32) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (98) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n638834_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: insertionsort#1, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERT#1, INSERTIONSORT#1, append#1, flatten#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 insertionsort < INSERTIONSORT#1 insertionsort = insertionsort#1 ---------------------------------------- (99) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) Induction Base: insertionsort#1(gen_:::nil21_32(0)) Induction Step: insertionsort#1(gen_:::nil21_32(+(n653302_32, 1))) ->_R^Omega(0) insert(#0, insertionsort(gen_:::nil21_32(n653302_32))) ->_R^Omega(0) insert(#0, insertionsort#1(gen_:::nil21_32(n653302_32))) ->_IH insert(#0, *23_32) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (100) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: INSERT#1, APPEND, APPEND#1, FLATTEN, FLATTEN#1, append, flatten, INSERTIONSORT, INSERT, INSERTIONSORT#1, append#1, flatten#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 INSERTIONSORT = INSERTIONSORT#1 INSERT = INSERT#1 INSERT < INSERTIONSORT#1 ---------------------------------------- (101) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) Induction Base: append#1(gen_:::nil21_32(0), gen_:::nil21_32(b)) ->_R^Omega(0) gen_:::nil21_32(b) Induction Step: append#1(gen_:::nil21_32(+(n696814_32, 1)), gen_:::nil21_32(b)) ->_R^Omega(0) ::(#0, append(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b))) ->_R^Omega(0) ::(#0, append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b))) ->_IH ::(#0, gen_:::nil21_32(+(b, c696815_32))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (102) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: append, APPEND, APPEND#1, FLATTEN, FLATTEN#1, flatten, flatten#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 append < FLATTEN#1 flatten < FLATTEN#1 append = append#1 append < flatten#1 flatten = flatten#1 ---------------------------------------- (103) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: flatten#1(gen_leaf:node22_32(n699295_32)) -> gen_:::nil21_32(0), rt in Omega(0) Induction Base: flatten#1(gen_leaf:node22_32(0)) ->_R^Omega(0) nil Induction Step: flatten#1(gen_leaf:node22_32(+(n699295_32, 1))) ->_R^Omega(0) append(nil, append(flatten(leaf), flatten(gen_leaf:node22_32(n699295_32)))) ->_R^Omega(0) append(nil, append(flatten#1(leaf), flatten(gen_leaf:node22_32(n699295_32)))) ->_R^Omega(0) append(nil, append(nil, flatten(gen_leaf:node22_32(n699295_32)))) ->_R^Omega(0) append(nil, append(nil, flatten#1(gen_leaf:node22_32(n699295_32)))) ->_IH append(nil, append(nil, gen_:::nil21_32(0))) ->_R^Omega(0) append(nil, append#1(nil, gen_:::nil21_32(0))) ->_L^Omega(0) append(nil, gen_:::nil21_32(+(0, 0))) ->_R^Omega(0) append#1(nil, gen_:::nil21_32(0)) ->_L^Omega(0) gen_:::nil21_32(+(0, 0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (104) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) flatten#1(gen_leaf:node22_32(n699295_32)) -> gen_:::nil21_32(0), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: flatten, APPEND, APPEND#1, FLATTEN, FLATTEN#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 flatten < FLATTEN#1 flatten = flatten#1 ---------------------------------------- (105) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND#1(gen_:::nil21_32(+(1, n702424_32)), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n702424_32) Induction Base: APPEND#1(gen_:::nil21_32(+(1, 0)), gen_:::nil21_32(b)) Induction Step: APPEND#1(gen_:::nil21_32(+(1, +(n702424_32, 1))), gen_:::nil21_32(b)) ->_R^Omega(1) c17(APPEND(gen_:::nil21_32(+(1, n702424_32)), gen_:::nil21_32(b))) ->_R^Omega(1) c17(c16(APPEND#1(gen_:::nil21_32(+(1, n702424_32)), gen_:::nil21_32(b)))) ->_IH c17(c16(*23_32)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (106) Complex Obligation (BEST) ---------------------------------------- (107) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) flatten#1(gen_leaf:node22_32(n699295_32)) -> gen_:::nil21_32(0), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: APPEND#1, APPEND, FLATTEN, FLATTEN#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 ---------------------------------------- (108) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (109) BOUNDS(n^1, INF) ---------------------------------------- (110) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) flatten#1(gen_leaf:node22_32(n699295_32)) -> gen_:::nil21_32(0), rt in Omega(0) APPEND#1(gen_:::nil21_32(+(1, n702424_32)), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n702424_32) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: APPEND, FLATTEN, FLATTEN#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 ---------------------------------------- (111) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND(gen_:::nil21_32(n706215_32), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n706215_32) Induction Base: APPEND(gen_:::nil21_32(0), gen_:::nil21_32(b)) Induction Step: APPEND(gen_:::nil21_32(+(n706215_32, 1)), gen_:::nil21_32(b)) ->_R^Omega(1) c16(APPEND#1(gen_:::nil21_32(+(n706215_32, 1)), gen_:::nil21_32(b))) ->_R^Omega(1) c16(c17(APPEND(gen_:::nil21_32(n706215_32), gen_:::nil21_32(b)))) ->_IH c16(c17(*23_32)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (112) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) flatten#1(gen_leaf:node22_32(n699295_32)) -> gen_:::nil21_32(0), rt in Omega(0) APPEND#1(gen_:::nil21_32(+(1, n702424_32)), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n702424_32) APPEND(gen_:::nil21_32(n706215_32), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n706215_32) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: APPEND#1, FLATTEN, FLATTEN#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < FLATTEN#1 FLATTEN = FLATTEN#1 ---------------------------------------- (113) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND#1(gen_:::nil21_32(+(1, n711179_32)), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n711179_32) Induction Base: APPEND#1(gen_:::nil21_32(+(1, 0)), gen_:::nil21_32(b)) Induction Step: APPEND#1(gen_:::nil21_32(+(1, +(n711179_32, 1))), gen_:::nil21_32(b)) ->_R^Omega(1) c17(APPEND(gen_:::nil21_32(+(1, n711179_32)), gen_:::nil21_32(b))) ->_R^Omega(1) c17(c16(APPEND#1(gen_:::nil21_32(+(1, n711179_32)), gen_:::nil21_32(b)))) ->_IH c17(c16(*23_32)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (114) Obligation: Innermost TRS: Rules: #LESS(z0, z1) -> c15(#CKLT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c16(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c17(APPEND(z1, z2)) APPEND#1(nil, z0) -> c18 FLATTEN(z0) -> c19(FLATTEN#1(z0)) FLATTEN#1(leaf) -> c20 FLATTEN#1(node(z0, z1, z2)) -> c21(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z1)) FLATTEN#1(node(z0, z1, z2)) -> c22(APPEND(z0, append(flatten(z1), flatten(z2))), APPEND(flatten(z1), flatten(z2)), FLATTEN(z2)) FLATTENSORT(z0) -> c23(INSERTIONSORT(flatten(z0)), FLATTEN(z0)) INSERT(z0, z1) -> c24(INSERT#1(z1, z0)) INSERT#1(::(z0, z1), z2) -> c25(INSERT#2(#less(z0, z2), z2, z0, z1), #LESS(z0, z2)) INSERT#1(nil, z0) -> c26 INSERT#2(#false, z0, z1, z2) -> c27 INSERT#2(#true, z0, z1, z2) -> c28(INSERT(z0, z2)) INSERTIONSORT(z0) -> c29(INSERTIONSORT#1(z0)) INSERTIONSORT#1(::(z0, z1)) -> c30(INSERT(z0, insertionsort(z1)), INSERTIONSORT(z1)) INSERTIONSORT#1(nil) -> c31 #CKLT(#EQ) -> c #CKLT(#GT) -> c1 #CKLT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #less(z0, z1) -> #cklt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 flatten(z0) -> flatten#1(z0) flatten#1(leaf) -> nil flatten#1(node(z0, z1, z2)) -> append(z0, append(flatten(z1), flatten(z2))) flattensort(z0) -> insertionsort(flatten(z0)) insert(z0, z1) -> insert#1(z1, z0) insert#1(::(z0, z1), z2) -> insert#2(#less(z0, z2), z2, z0, z1) insert#1(nil, z0) -> ::(z0, nil) insert#2(#false, z0, z1, z2) -> ::(z0, ::(z1, z2)) insert#2(#true, z0, z1, z2) -> ::(z1, insert(z0, z2)) insertionsort(z0) -> insertionsort#1(z0) insertionsort#1(::(z0, z1)) -> insert(z0, insertionsort(z1)) insertionsort#1(nil) -> nil Types: #LESS :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c15 c15 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c15 #CKLT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c16 c16 :: c17:c18 -> c16 APPEND#1 :: :::nil -> :::nil -> c17:c18 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c17 :: c16 -> c17:c18 nil :: :::nil c18 :: c17:c18 FLATTEN :: leaf:node -> c19 c19 :: c20:c21:c22 -> c19 FLATTEN#1 :: leaf:node -> c20:c21:c22 leaf :: leaf:node c20 :: c20:c21:c22 node :: :::nil -> leaf:node -> leaf:node -> leaf:node c21 :: c16 -> c16 -> c19 -> c20:c21:c22 append :: :::nil -> :::nil -> :::nil flatten :: leaf:node -> :::nil c22 :: c16 -> c16 -> c19 -> c20:c21:c22 FLATTENSORT :: leaf:node -> c23 c23 :: c29 -> c19 -> c23 INSERTIONSORT :: :::nil -> c29 INSERT :: #0:#neg:#pos:#s -> :::nil -> c24 c24 :: c25:c26 -> c24 INSERT#1 :: :::nil -> #0:#neg:#pos:#s -> c25:c26 c25 :: c27:c28 -> c15 -> c25:c26 INSERT#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> c27:c28 #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true c26 :: c25:c26 #false :: #false:#true c27 :: c27:c28 #true :: #false:#true c28 :: c24 -> c27:c28 c29 :: c30:c31 -> c29 INSERTIONSORT#1 :: :::nil -> c30:c31 c30 :: c24 -> c29 -> c30:c31 insertionsort :: :::nil -> :::nil c31 :: c30:c31 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 #0 :: #0:#neg:#pos:#s c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #cklt :: #EQ:#GT:#LT -> #false:#true append#1 :: :::nil -> :::nil -> :::nil flatten#1 :: leaf:node -> :::nil flattensort :: leaf:node -> :::nil insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil hole_c151_32 :: c15 hole_#0:#neg:#pos:#s2_32 :: #0:#neg:#pos:#s hole_c:c1:c23_32 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c144_32 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT5_32 :: #EQ:#GT:#LT hole_c166_32 :: c16 hole_:::nil7_32 :: :::nil hole_c17:c188_32 :: c17:c18 hole_c199_32 :: c19 hole_leaf:node10_32 :: leaf:node hole_c20:c21:c2211_32 :: c20:c21:c22 hole_c2312_32 :: c23 hole_c2913_32 :: c29 hole_c2414_32 :: c24 hole_c25:c2615_32 :: c25:c26 hole_c27:c2816_32 :: c27:c28 hole_#false:#true17_32 :: #false:#true hole_c30:c3118_32 :: c30:c31 gen_#0:#neg:#pos:#s19_32 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil21_32 :: Nat -> :::nil gen_leaf:node22_32 :: Nat -> leaf:node Lemmas: #compare(gen_#0:#neg:#pos:#s19_32(n24_32), gen_#0:#neg:#pos:#s19_32(n24_32)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s19_32(n318895_32), gen_#0:#neg:#pos:#s19_32(n318895_32)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(n318895_32), rt in Omega(0) insertionsort#1(gen_:::nil21_32(n653302_32)) -> *23_32, rt in Omega(0) insertionsort(gen_:::nil21_32(n643249_32)) -> *23_32, rt in Omega(0) append#1(gen_:::nil21_32(n696814_32), gen_:::nil21_32(b)) -> gen_:::nil21_32(+(n696814_32, b)), rt in Omega(0) flatten#1(gen_leaf:node22_32(n699295_32)) -> gen_:::nil21_32(0), rt in Omega(0) APPEND#1(gen_:::nil21_32(+(1, n711179_32)), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n711179_32) APPEND(gen_:::nil21_32(n706215_32), gen_:::nil21_32(b)) -> *23_32, rt in Omega(n706215_32) Generator Equations: gen_#0:#neg:#pos:#s19_32(0) <=> #0 gen_#0:#neg:#pos:#s19_32(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s19_32(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1420_32(x)) gen_:::nil21_32(0) <=> nil gen_:::nil21_32(+(x, 1)) <=> ::(#0, gen_:::nil21_32(x)) gen_leaf:node22_32(0) <=> leaf gen_leaf:node22_32(+(x, 1)) <=> node(nil, leaf, gen_leaf:node22_32(x)) The following defined symbols remain to be analysed: FLATTEN#1, FLATTEN They will be analysed ascendingly in the following order: FLATTEN = FLATTEN#1