MAYBE proof of input_V6KujRPLPY.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 2 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 364 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 303 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 702 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 215 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 56 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) 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: from(X) -> cons(X, from(s(X))) first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) sel(0, cons(X, Z)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) 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: from(X) -> cons(X, from(s(X))) first(0', Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) sel(0', cons(X, Z)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) 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: from(X) -> cons(X, from(s(X))) first(0, Z) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) sel(0, cons(X, Z)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) 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: from(X) -> cons(X, from(s(X))) [1] first(0, Z) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] sel(0, cons(X, Z)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [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: from(X) -> cons(X, from(s(X))) [1] first(0, Z) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] sel(0, cons(X, Z)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 first :: s:0 -> cons:nil -> cons:nil 0 :: s:0 nil :: cons:nil sel :: s:0 -> cons:nil -> s:0 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: from_1 first_2 sel_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X, from(s(X))) [1] first(0, Z) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] sel(0, cons(X, Z)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 first :: s:0 -> cons:nil -> cons:nil 0 :: s:0 nil :: cons:nil sel :: s:0 -> cons:nil -> s:0 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: from(X) -> cons(X, from(s(X))) [1] first(0, Z) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] sel(0, cons(X, Z)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 first :: s:0 -> cons:nil -> cons:nil 0 :: s:0 nil :: cons:nil sel :: s:0 -> cons:nil -> s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z ---------------------------------------- (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: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { sel } { first } { from } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {sel}, {first}, {from} ---------------------------------------- (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: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {sel}, {first}, {from} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sel 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: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {sel}, {first}, {from} Previous analysis results are: sel: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sel after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {first}, {from} Previous analysis results are: sel: runtime: O(n^1) [1 + 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: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {first}, {from} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: first after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {first}, {from} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] first: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: first after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s' :|: s' >= 0, s' <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s' :|: s' >= 0, s' <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] from: runtime: ?, size: O(1) [0] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s' :|: s' >= 0, s' <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] from: runtime: INF, size: O(1) [0] ---------------------------------------- (37) 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: first(v0, v1) -> null_first [0] sel(v0, v1) -> null_sel [0] And the following fresh constants: null_first, null_sel ---------------------------------------- (38) 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: from(X) -> cons(X, from(s(X))) [1] first(0, Z) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] sel(0, cons(X, Z)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] first(v0, v1) -> null_first [0] sel(v0, v1) -> null_sel [0] The TRS has the following type information: from :: s:0:null_sel -> cons:nil:null_first cons :: s:0:null_sel -> cons:nil:null_first -> cons:nil:null_first s :: s:0:null_sel -> s:0:null_sel first :: s:0:null_sel -> cons:nil:null_first -> cons:nil:null_first 0 :: s:0:null_sel nil :: cons:nil:null_first sel :: s:0:null_sel -> cons:nil:null_first -> s:0:null_sel null_first :: cons:nil:null_first null_sel :: s:0:null_sel Rewrite Strategy: INNERMOST ---------------------------------------- (39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 nil => 0 null_first => 0 null_sel => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0 first(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X sel(z, z') -{ 1 }-> X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, from(s(z0))) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, z2) Tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(0, z0) -> c1 FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(0, cons(z0, z1)) -> c3 SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) S tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(0, z0) -> c1 FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(0, cons(z0, z1)) -> c3 SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) K tuples:none Defined Rule Symbols: from_1, first_2, sel_2 Defined Pair Symbols: FROM_1, FIRST_2, SEL_2 Compound Symbols: c_1, c1, c2_1, c3, c4_1 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FIRST(0, z0) -> c1 SEL(0, cons(z0, z1)) -> c3 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, from(s(z0))) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, z2) Tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) S tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) K tuples:none Defined Rule Symbols: from_1, first_2, sel_2 Defined Pair Symbols: FROM_1, FIRST_2, SEL_2 Compound Symbols: c_1, c2_1, c4_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: from(z0) -> cons(z0, from(s(z0))) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, z2) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) S tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FROM_1, FIRST_2, SEL_2 Compound Symbols: c_1, c2_1, c4_1 ---------------------------------------- (47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) We considered the (Usable) Rules:none And the Tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) The order we found is given by the following interpretation: Polynomial interpretation : POL(FIRST(x_1, x_2)) = x_1 + x_2 POL(FROM(x_1)) = x_1 POL(SEL(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = x_1 ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c(FROM(s(z0))) FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) S tuples: FROM(z0) -> c(FROM(s(z0))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, FIRST_2, SEL_2 Compound Symbols: c_1, c2_1, c4_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIRST(s(z0), cons(z1, z2)) -> c2(FIRST(z0, z2)) by FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c(FROM(s(z0))) SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) S tuples: FROM(z0) -> c(FROM(s(z0))) K tuples: SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, SEL_2, FIRST_2 Compound Symbols: c_1, c4_1, c2_1 ---------------------------------------- (51) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(z0) -> c(FROM(s(z0))) by FROM(s(x0)) -> c(FROM(s(s(x0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) FROM(s(x0)) -> c(FROM(s(s(x0)))) S tuples: FROM(s(x0)) -> c(FROM(s(s(x0)))) K tuples: SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: SEL_2, FIRST_2, FROM_1 Compound Symbols: c4_1, c2_1, c_1 ---------------------------------------- (53) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(s(x0)) -> c(FROM(s(s(x0)))) by FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: SEL_2, FIRST_2, FROM_1 Compound Symbols: c4_1, c2_1, c_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, z2)) -> c4(SEL(z0, z2)) by SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: FIRST_2, FROM_1, SEL_2 Compound Symbols: c2_1, c_1, c4_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c2(FIRST(s(y0), cons(y1, y2))) by FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c2(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c2(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c2(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, SEL_2, FIRST_2 Compound Symbols: c_1, c4_1, c2_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c4(SEL(s(y0), cons(y1, y2))) by SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c4(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c2(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c4(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c2(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c4(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, FIRST_2, SEL_2 Compound Symbols: c_1, c2_1, c4_1