KILLED proof of input_qDPuZGaoeN.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 124 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 224 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 27 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 34 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 629 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 392 ms] (38) CpxRNTS (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) select(cons(ap, xs)) -> ap select(cons(ap, xs)) -> select(xs) addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) select(cons(ap, xs)) -> ap select(cons(ap, xs)) -> select(xs) addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) select(cons(ap, xs)) -> ap select(cons(ap, xs)) -> select(xs) addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) [1] select(cons(ap, xs)) -> ap [1] select(cons(ap, xs)) -> select(xs) [1] addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) [1] select(cons(ap, xs)) -> ap [1] select(cons(ap, xs)) -> select(xs) [1] addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) [1] The TRS has the following type information: f :: node -> f node :: s -> cons -> node s :: s -> s addchild :: node -> node -> node select :: cons -> node cons :: node -> cons -> cons Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: f_1 (c) The following functions are completely defined: addchild_2 select_1 Due to the following rules being added: addchild(v0, v1) -> const1 [0] select(v0) -> const1 [0] And the following fresh constants: const1, const, const2, const3 ---------------------------------------- (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: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) [1] select(cons(ap, xs)) -> ap [1] select(cons(ap, xs)) -> select(xs) [1] addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) [1] addchild(v0, v1) -> const1 [0] select(v0) -> const1 [0] The TRS has the following type information: f :: node:const1 -> f node :: s -> cons -> node:const1 s :: s -> s addchild :: node:const1 -> node:const1 -> node:const1 select :: cons -> node:const1 cons :: node:const1 -> cons -> cons const1 :: node:const1 const :: f const2 :: s const3 :: cons Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(node(s(n), cons(ap', xs'))) -> f(addchild(ap', node(n, cons(ap', xs')))) [2] f(node(s(n), cons(ap'', xs''))) -> f(addchild(select(xs''), node(n, cons(ap'', xs'')))) [2] f(node(s(n), xs)) -> f(addchild(const1, node(n, xs))) [1] select(cons(ap, xs)) -> ap [1] select(cons(ap, xs)) -> select(xs) [1] addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) [1] addchild(v0, v1) -> const1 [0] select(v0) -> const1 [0] The TRS has the following type information: f :: node:const1 -> f node :: s -> cons -> node:const1 s :: s -> s addchild :: node:const1 -> node:const1 -> node:const1 select :: cons -> node:const1 cons :: node:const1 -> cons -> cons const1 :: node:const1 const :: f const2 :: s const3 :: cons Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const1 => 0 const => 0 const2 => 0 const3 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(ap', 1 + n + (1 + ap' + xs'))) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0 f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 1 }-> f(addchild(0, 1 + n + xs)) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys addchild(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { select } { addchild } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {select}, {addchild}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {select}, {addchild}, {f} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: select after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {select}, {addchild}, {f} Previous analysis results are: select: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: select after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 2 }-> f(addchild(select(xs''), 1 + n + (1 + ap'' + xs''))) :|: n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {addchild}, {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 3 + xs'' }-> f(addchild(s', 1 + n + (1 + ap'' + xs''))) :|: s' >= 0, s' <= xs'', n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 2 + xs }-> s :|: s >= 0, s <= xs, z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {addchild}, {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: addchild after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 3 + xs'' }-> f(addchild(s', 1 + n + (1 + ap'' + xs''))) :|: s' >= 0, s' <= xs'', n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 2 + xs }-> s :|: s >= 0, s <= xs, z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {addchild}, {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] addchild: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: addchild after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 3 + xs'' }-> f(addchild(s', 1 + n + (1 + ap'' + xs''))) :|: s' >= 0, s' <= xs'', n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 2 + xs }-> s :|: s >= 0, s <= xs, z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] addchild: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 4 + xs'' }-> f(s'') :|: s'' >= 0, s'' <= s' + (1 + n + (1 + ap'' + xs'')) + 1, s' >= 0, s' <= xs'', n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 2 + xs }-> s :|: s >= 0, s <= xs, z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] addchild: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f 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: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 4 + xs'' }-> f(s'') :|: s'' >= 0, s'' <= s' + (1 + n + (1 + ap'' + xs'')) + 1, s' >= 0, s' <= xs'', n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 2 + xs }-> s :|: s >= 0, s <= xs, z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] addchild: runtime: O(1) [1], size: O(n^1) [1 + z + z'] f: runtime: ?, size: O(1) [0] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 4 + xs'' }-> f(s'') :|: s'' >= 0, s'' <= s' + (1 + n + (1 + ap'' + xs'')) + 1, s' >= 0, s' <= xs'', n >= 0, ap'' >= 0, xs'' >= 0, z = 1 + (1 + n) + (1 + ap'' + xs'') f(z) -{ 2 }-> f(0) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, v0 >= 0, v1 >= 0, ap' = v0, 1 + n + (1 + ap' + xs') = v1 f(z) -{ 1 }-> f(0) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs, v0 >= 0, v1 >= 0, 0 = v0, 1 + n + xs = v1 f(z) -{ 3 }-> f(1 + y + (1 + (1 + n' + xs) + ys)) :|: n >= 0, xs' >= 0, z = 1 + (1 + n) + (1 + ap' + xs'), ap' >= 0, n' >= 0, xs >= 0, 1 + n + (1 + ap' + xs') = 1 + n' + xs, ys >= 0, y >= 0, ap' = 1 + y + ys select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 2 + xs }-> s :|: s >= 0, s <= xs, z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: select: runtime: O(n^1) [1 + z], size: O(n^1) [z] addchild: runtime: O(1) [1], size: O(n^1) [1 + z + z'] f: runtime: INF, size: O(1) [0] ---------------------------------------- (39) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: f(v0) -> null_f [0] select(v0) -> null_select [0] addchild(v0, v1) -> null_addchild [0] And the following fresh constants: null_f, null_select, null_addchild, const, const1 ---------------------------------------- (40) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs))) [1] select(cons(ap, xs)) -> ap [1] select(cons(ap, xs)) -> select(xs) [1] addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys)) [1] f(v0) -> null_f [0] select(v0) -> null_select [0] addchild(v0, v1) -> null_addchild [0] The TRS has the following type information: f :: node:null_select:null_addchild -> null_f node :: s -> cons -> node:null_select:null_addchild s :: s -> s addchild :: node:null_select:null_addchild -> node:null_select:null_addchild -> node:null_select:null_addchild select :: cons -> node:null_select:null_addchild cons :: node:null_select:null_addchild -> cons -> cons null_f :: null_f null_select :: node:null_select:null_addchild null_addchild :: node:null_select:null_addchild const :: s const1 :: cons Rewrite Strategy: INNERMOST ---------------------------------------- (41) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_f => 0 null_select => 0 null_addchild => 0 const => 0 const1 => 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: addchild(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 addchild(z, z') -{ 1 }-> 1 + y + (1 + (1 + n + xs) + ys) :|: n >= 0, xs >= 0, z' = 1 + n + xs, ys >= 0, y >= 0, z = 1 + y + ys f(z) -{ 1 }-> f(addchild(select(xs), 1 + n + xs)) :|: n >= 0, xs >= 0, z = 1 + (1 + n) + xs f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 select(z) -{ 1 }-> ap :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 1 }-> select(xs) :|: z = 1 + ap + xs, ap >= 0, xs >= 0 select(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (43) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: f(node(s(z0), z1)) -> f(addchild(select(z1), node(z0, z1))) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) Tuples: F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), ADDCHILD(select(z1), node(z0, z1)), SELECT(z1)) SELECT(cons(z0, z1)) -> c1 SELECT(cons(z0, z1)) -> c2(SELECT(z1)) ADDCHILD(node(z0, z1), node(z2, z3)) -> c3 S tuples: F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), ADDCHILD(select(z1), node(z0, z1)), SELECT(z1)) SELECT(cons(z0, z1)) -> c1 SELECT(cons(z0, z1)) -> c2(SELECT(z1)) ADDCHILD(node(z0, z1), node(z2, z3)) -> c3 K tuples:none Defined Rule Symbols: f_1, select_1, addchild_2 Defined Pair Symbols: F_1, SELECT_1, ADDCHILD_2 Compound Symbols: c_3, c1, c2_1, c3 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: SELECT(cons(z0, z1)) -> c1 ADDCHILD(node(z0, z1), node(z2, z3)) -> c3 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: f(node(s(z0), z1)) -> f(addchild(select(z1), node(z0, z1))) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) Tuples: F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), ADDCHILD(select(z1), node(z0, z1)), SELECT(z1)) SELECT(cons(z0, z1)) -> c2(SELECT(z1)) S tuples: F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), ADDCHILD(select(z1), node(z0, z1)), SELECT(z1)) SELECT(cons(z0, z1)) -> c2(SELECT(z1)) K tuples:none Defined Rule Symbols: f_1, select_1, addchild_2 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_3, c2_1 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: f(node(s(z0), z1)) -> f(addchild(select(z1), node(z0, z1))) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), SELECT(z1)) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), SELECT(z1)) K tuples:none Defined Rule Symbols: f_1, select_1, addchild_2 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2 ---------------------------------------- (49) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(node(s(z0), z1)) -> f(addchild(select(z1), node(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), SELECT(z1)) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), SELECT(z1)) K tuples:none Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(node(s(z0), z1)) -> c(F(addchild(select(z1), node(z0, z1))), SELECT(z1)) by F(node(s(x0), cons(z0, z1))) -> c(F(addchild(z0, node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(z0, node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(z0, node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) K tuples:none Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(node(s(x0), cons(z0, z1))) -> c(F(addchild(z0, node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) by F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) K tuples:none Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2, c_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) We considered the (Usable) Rules:none And the Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(F(x_1)) = [1] POL(SELECT(x_1)) = 0 POL(addchild(x_1, x_2)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(node(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 POL(select(x_1)) = [1] + x_1 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) K tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2, c_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(node(s(x0), cons(z0, z1))) -> c(F(addchild(select(z1), node(x0, cons(z0, z1)))), SELECT(cons(z0, z1))) by F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) K tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2, c_1 ---------------------------------------- (59) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) S tuples: SELECT(cons(z0, z1)) -> c2(SELECT(z1)) F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) K tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: SELECT_1, F_1 Compound Symbols: c2_1, c_2, c_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SELECT(cons(z0, z1)) -> c2(SELECT(z1)) by SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) S tuples: F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) K tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c_1, c2_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(z2), cons(node(z0, z1), x2))) -> c(F(node(z0, cons(node(z2, cons(node(z0, z1), x2)), z1))), SELECT(cons(node(z0, z1), x2))) by F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) K tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_1, c_2, c2_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) by F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) K tuples: F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_1, c_2, c2_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(x0), cons(x1, x2))) -> c(SELECT(cons(x1, x2))) by F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) K tuples: F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c2_1, c_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SELECT(cons(z0, cons(y0, y1))) -> c2(SELECT(cons(y0, y1))) by SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) K tuples: F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c_1, c2_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(z0), cons(node(s(y0), z2), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), z2), z3)), z2))), SELECT(cons(node(s(y0), z2), z3))) by F(node(s(z0), cons(node(s(z1), cons(y2, y3)), z3))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(z1), cons(y2, y3)), z3))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) K tuples: F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c_1, c2_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(z0), cons(node(z1, z2), cons(y1, y2)))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(y1, y2))), z2))), SELECT(cons(node(z1, z2), cons(y1, y2)))) by F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) K tuples: F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c_1, c2_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) by F(node(s(z0), cons(z1, cons(z2, cons(y2, y3))))) -> c(SELECT(cons(z1, cons(z2, cons(y2, y3))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) F(node(s(z0), cons(z1, cons(z2, cons(y2, y3))))) -> c(SELECT(cons(z1, cons(z2, cons(y2, y3))))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) K tuples: F(node(s(z0), cons(z1, cons(z2, cons(y2, y3))))) -> c(SELECT(cons(z1, cons(z2, cons(y2, y3))))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c_1, c2_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(node(s(z0), cons(z1, cons(y1, y2)))) -> c(SELECT(cons(z1, cons(y1, y2)))) by F(node(s(z0), cons(z1, cons(z2, cons(y2, y3))))) -> c(SELECT(cons(z1, cons(z2, cons(y2, y3))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: addchild(node(z0, z1), node(z2, z3)) -> node(z0, cons(node(z2, z3), z1)) select(cons(z0, z1)) -> z0 select(cons(z0, z1)) -> select(z1) Tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) F(node(s(z0), cons(z1, cons(z2, cons(y2, y3))))) -> c(SELECT(cons(z1, cons(z2, cons(y2, y3))))) S tuples: F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(z0, node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(x0), cons(x1, cons(z0, z1)))) -> c(F(addchild(select(z1), node(x0, cons(x1, cons(z0, z1))))), SELECT(cons(x1, cons(z0, z1)))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), z3))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), z3)), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), z3))) SELECT(cons(z0, cons(z1, cons(y1, y2)))) -> c2(SELECT(cons(z1, cons(y1, y2)))) F(node(s(s(y1)), cons(node(s(z1), z2), z3))) -> c(F(node(s(z1), cons(node(s(y1), cons(node(s(z1), z2), z3)), z2))), SELECT(cons(node(s(z1), z2), z3))) F(node(s(z0), cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) -> c(F(node(s(z1), cons(node(z0, cons(node(s(z1), z2), cons(y1, cons(y2, y3)))), z2))), SELECT(cons(node(s(z1), z2), cons(y1, cons(y2, y3))))) F(node(s(z0), cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(z0, cons(node(s(y0), cons(y2, y3)), cons(z3, z4))), cons(y2, y3)))), SELECT(cons(node(s(y0), cons(y2, y3)), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), z2), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(y0), z2), cons(z3, z4)))) F(node(s(z0), cons(node(z1, z2), cons(z3, cons(y2, y3))))) -> c(F(node(z1, cons(node(z0, cons(node(z1, z2), cons(z3, cons(y2, y3)))), z2))), SELECT(cons(node(z1, z2), cons(z3, cons(y2, y3))))) F(node(s(s(y1)), cons(node(s(s(y0)), z2), cons(z3, z4)))) -> c(F(node(s(s(y0)), cons(node(s(y1), cons(node(s(s(y0)), z2), cons(z3, z4))), z2))), SELECT(cons(node(s(s(y0)), z2), cons(z3, z4)))) F(node(s(s(y1)), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) -> c(F(node(s(y0), cons(node(s(y1), cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4))), cons(y3, cons(y4, y5))))), SELECT(cons(node(s(y0), cons(y3, cons(y4, y5))), cons(z3, z4)))) K tuples: F(node(s(z0), cons(z1, cons(z2, cons(y2, y3))))) -> c(SELECT(cons(z1, cons(z2, cons(y2, y3))))) Defined Rule Symbols: addchild_2, select_1 Defined Pair Symbols: F_1, SELECT_1 Compound Symbols: c_2, c2_1, c_1