MAYBE proof of input_tP2JPMBA6L.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) 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), 352 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 100 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 660 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 249 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) 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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) lookup(Nil, Cons(x, xs)) -> x run(e, p) -> intlookup(e, p) intlookup(e, p) -> intlookup(lookup(e, p), p) 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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) lookup(Nil, Cons(x, xs)) -> x run(e, p) -> intlookup(e, p) intlookup(e, p) -> intlookup(lookup(e, p), p) 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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) lookup(Nil, Cons(x, xs)) -> x run(e, p) -> intlookup(e, p) intlookup(e, p) -> intlookup(lookup(e, p), p) 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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1] lookup(Nil, Cons(x, xs)) -> x [1] run(e, p) -> intlookup(e, p) [1] intlookup(e, p) -> intlookup(lookup(e, p), p) [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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1] lookup(Nil, Cons(x, xs)) -> x [1] run(e, p) -> intlookup(e, p) [1] intlookup(e, p) -> intlookup(lookup(e, p), p) [1] The TRS has the following type information: lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil run :: Cons:Nil -> Cons:Nil -> run:intlookup intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup 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: run_2 intlookup_2 (c) The following functions are completely defined: lookup_2 Due to the following rules being added: lookup(v0, v1) -> Nil [0] And the following fresh constants: const ---------------------------------------- (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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1] lookup(Nil, Cons(x, xs)) -> x [1] run(e, p) -> intlookup(e, p) [1] intlookup(e, p) -> intlookup(lookup(e, p), p) [1] lookup(v0, v1) -> Nil [0] The TRS has the following type information: lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil run :: Cons:Nil -> Cons:Nil -> run:intlookup intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup const :: run:intlookup 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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1] lookup(Nil, Cons(x, xs)) -> x [1] run(e, p) -> intlookup(e, p) [1] intlookup(Cons(x'', xs''), Cons(x1, xs1)) -> intlookup(lookup(xs'', xs1), Cons(x1, xs1)) [2] intlookup(Nil, Cons(x2, xs2)) -> intlookup(x2, Cons(x2, xs2)) [2] intlookup(e, p) -> intlookup(Nil, p) [1] lookup(v0, v1) -> Nil [0] The TRS has the following type information: lookup :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil run :: Cons:Nil -> Cons:Nil -> run:intlookup intlookup :: Cons:Nil -> Cons:Nil -> run:intlookup const :: run:intlookup 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: Nil => 0 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 1 }-> intlookup(0, p) :|: z = e, z' = p, p >= 0, e >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 run(z, z') -{ 1 }-> intlookup(e, p) :|: z = e, z' = p, p >= 0, e >= 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: intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lookup } { intlookup } { run } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {lookup}, {intlookup}, {run} ---------------------------------------- (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: intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {lookup}, {intlookup}, {run} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: lookup after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {lookup}, {intlookup}, {run} Previous analysis results are: lookup: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lookup after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 2 }-> intlookup(lookup(xs'', xs1), 1 + x1 + xs1) :|: x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {intlookup}, {run} Previous analysis results are: lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] ---------------------------------------- (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: intlookup(z, z') -{ 4 + xs1 }-> intlookup(s', 1 + x1 + xs1) :|: s' >= 0, s' <= xs1, x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= xs, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {intlookup}, {run} Previous analysis results are: lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: intlookup after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 4 + xs1 }-> intlookup(s', 1 + x1 + xs1) :|: s' >= 0, s' <= xs1, x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= xs, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {intlookup}, {run} Previous analysis results are: lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] intlookup: runtime: ?, size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: intlookup after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 4 + xs1 }-> intlookup(s', 1 + x1 + xs1) :|: s' >= 0, s' <= xs1, x1 >= 0, xs'' >= 0, z = 1 + x'' + xs'', xs1 >= 0, x'' >= 0, z' = 1 + x1 + xs1 intlookup(z, z') -{ 2 }-> intlookup(x2, 1 + x2 + xs2) :|: xs2 >= 0, z' = 1 + x2 + xs2, z = 0, x2 >= 0 intlookup(z, z') -{ 1 }-> intlookup(0, z') :|: z' >= 0, z >= 0 lookup(z, z') -{ 3 + xs }-> s :|: s >= 0, s <= xs, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 run(z, z') -{ 1 }-> intlookup(z, z') :|: z' >= 0, z >= 0 Function symbols to be analyzed: {intlookup}, {run} Previous analysis results are: lookup: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] intlookup: runtime: INF, size: O(1) [0] ---------------------------------------- (31) 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: lookup(v0, v1) -> null_lookup [0] And the following fresh constants: null_lookup, const ---------------------------------------- (32) 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: lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs) [1] lookup(Nil, Cons(x, xs)) -> x [1] run(e, p) -> intlookup(e, p) [1] intlookup(e, p) -> intlookup(lookup(e, p), p) [1] lookup(v0, v1) -> null_lookup [0] The TRS has the following type information: lookup :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> Cons:Nil:null_lookup Cons :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> Cons:Nil:null_lookup Nil :: Cons:Nil:null_lookup run :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> run:intlookup intlookup :: Cons:Nil:null_lookup -> Cons:Nil:null_lookup -> run:intlookup null_lookup :: Cons:Nil:null_lookup const :: run:intlookup Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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 null_lookup => 0 const => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: intlookup(z, z') -{ 1 }-> intlookup(lookup(e, p), p) :|: z = e, z' = p, p >= 0, e >= 0 lookup(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0 lookup(z, z') -{ 1 }-> lookup(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs' lookup(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 run(z, z') -{ 1 }-> intlookup(e, p) :|: z = e, z' = p, p >= 0, e >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3) lookup(Nil, Cons(z0, z1)) -> z0 run(z0, z1) -> intlookup(z0, z1) intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1) Tuples: LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) LOOKUP(Nil, Cons(z0, z1)) -> c1 RUN(z0, z1) -> c2(INTLOOKUP(z0, z1)) INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) S tuples: LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) LOOKUP(Nil, Cons(z0, z1)) -> c1 RUN(z0, z1) -> c2(INTLOOKUP(z0, z1)) INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) K tuples:none Defined Rule Symbols: lookup_2, run_2, intlookup_2 Defined Pair Symbols: LOOKUP_2, RUN_2, INTLOOKUP_2 Compound Symbols: c_1, c1, c2_1, c3_2 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: RUN(z0, z1) -> c2(INTLOOKUP(z0, z1)) Removed 1 trailing nodes: LOOKUP(Nil, Cons(z0, z1)) -> c1 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3) lookup(Nil, Cons(z0, z1)) -> z0 run(z0, z1) -> intlookup(z0, z1) intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1) Tuples: LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) S tuples: LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) K tuples:none Defined Rule Symbols: lookup_2, run_2, intlookup_2 Defined Pair Symbols: LOOKUP_2, INTLOOKUP_2 Compound Symbols: c_1, c3_2 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: run(z0, z1) -> intlookup(z0, z1) intlookup(z0, z1) -> intlookup(lookup(z0, z1), z1) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3) lookup(Nil, Cons(z0, z1)) -> z0 Tuples: LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) S tuples: LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) K tuples:none Defined Rule Symbols: lookup_2 Defined Pair Symbols: LOOKUP_2, INTLOOKUP_2 Compound Symbols: c_1, c3_2 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LOOKUP(Cons(z0, z1), Cons(z2, z3)) -> c(LOOKUP(z1, z3)) by LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3) lookup(Nil, Cons(z0, z1)) -> z0 Tuples: INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3))) S tuples: INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3))) K tuples:none Defined Rule Symbols: lookup_2 Defined Pair Symbols: INTLOOKUP_2, LOOKUP_2 Compound Symbols: c3_2, c_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LOOKUP(Cons(z0, Cons(y0, y1)), Cons(z2, Cons(y2, y3))) -> c(LOOKUP(Cons(y0, y1), Cons(y2, y3))) by LOOKUP(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c(LOOKUP(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: lookup(Cons(z0, z1), Cons(z2, z3)) -> lookup(z1, z3) lookup(Nil, Cons(z0, z1)) -> z0 Tuples: INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) LOOKUP(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c(LOOKUP(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) S tuples: INTLOOKUP(z0, z1) -> c3(INTLOOKUP(lookup(z0, z1), z1), LOOKUP(z0, z1)) LOOKUP(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Cons(z4, Cons(y4, y5)))) -> c(LOOKUP(Cons(z1, Cons(y1, y2)), Cons(z4, Cons(y4, y5)))) K tuples:none Defined Rule Symbols: lookup_2 Defined Pair Symbols: INTLOOKUP_2, LOOKUP_2 Compound Symbols: c3_2, c_1