MAYBE proof of input_pshDkFS0uV.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 930 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 237 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms] (36) CdtProblem (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtInstantiationProof [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 (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0, 0) -> true eq(s(X), s(Y)) -> eq(X, Y) eq(X, Y) -> false inf(X) -> cons(X, inf(s(X))) take(0, X) -> nil take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) length(nil) -> 0 length(cons(X, L)) -> s(length(L)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0', 0') -> true eq(s(X), s(Y)) -> eq(X, Y) eq(X, Y) -> false inf(X) -> cons(X, inf(s(X))) take(0', X) -> nil take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) length(nil) -> 0' length(cons(X, L)) -> s(length(L)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0, 0) -> true eq(s(X), s(Y)) -> eq(X, Y) eq(X, Y) -> false inf(X) -> cons(X, inf(s(X))) take(0, X) -> nil take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) length(nil) -> 0 length(cons(X, L)) -> s(length(L)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(s(X), s(Y)) -> eq(X, Y) [1] eq(X, Y) -> false [1] inf(X) -> cons(X, inf(s(X))) [1] take(0, X) -> nil [1] take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1] length(nil) -> 0 [1] length(cons(X, L)) -> s(length(L)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(s(X), s(Y)) -> eq(X, Y) [1] eq(X, Y) -> false [1] inf(X) -> cons(X, inf(s(X))) [1] take(0, X) -> nil [1] take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1] length(nil) -> 0 [1] length(cons(X, L)) -> s(length(L)) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false inf :: 0:s -> cons:nil cons :: 0:s -> cons:nil -> cons:nil take :: 0:s -> cons:nil -> cons:nil nil :: cons:nil length :: cons:nil -> 0:s 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: eq_2 inf_1 take_2 length_1 (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: eq(0, 0) -> true [1] eq(s(X), s(Y)) -> eq(X, Y) [1] eq(X, Y) -> false [1] inf(X) -> cons(X, inf(s(X))) [1] take(0, X) -> nil [1] take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1] length(nil) -> 0 [1] length(cons(X, L)) -> s(length(L)) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false inf :: 0:s -> cons:nil cons :: 0:s -> cons:nil -> cons:nil take :: 0:s -> cons:nil -> cons:nil nil :: cons:nil length :: cons:nil -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(s(X), s(Y)) -> eq(X, Y) [1] eq(X, Y) -> false [1] inf(X) -> cons(X, inf(s(X))) [1] take(0, X) -> nil [1] take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1] length(nil) -> 0 [1] length(cons(X, L)) -> s(length(L)) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false inf :: 0:s -> cons:nil cons :: 0:s -> cons:nil -> cons:nil take :: 0:s -> cons:nil -> cons:nil nil :: cons:nil length :: cons:nil -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, X >= 0, z = X inf(z) -{ 1 }-> 1 + X + inf(1 + X) :|: X >= 0, z = X length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 take(z, z') -{ 1 }-> 1 + Y + take(X, L) :|: z = 1 + X, Y >= 0, X >= 0, L >= 0, z' = 1 + Y + L ---------------------------------------- (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: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0 inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { inf } { take } { length } { eq } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0 inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L Function symbols to be analyzed: {inf}, {take}, {length}, {eq} ---------------------------------------- (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: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0 inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L Function symbols to be analyzed: {inf}, {take}, {length}, {eq} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: inf after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0 inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L Function symbols to be analyzed: {inf}, {take}, {length}, {eq} Previous analysis results are: inf: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: inf after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0 inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L Function symbols to be analyzed: {inf}, {take}, {length}, {eq} Previous analysis results are: inf: runtime: INF, size: O(1) [0] ---------------------------------------- (25) 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: take(v0, v1) -> null_take [0] length(v0) -> null_length [0] And the following fresh constants: null_take, null_length ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(s(X), s(Y)) -> eq(X, Y) [1] eq(X, Y) -> false [1] inf(X) -> cons(X, inf(s(X))) [1] take(0, X) -> nil [1] take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1] length(nil) -> 0 [1] length(cons(X, L)) -> s(length(L)) [1] take(v0, v1) -> null_take [0] length(v0) -> null_length [0] The TRS has the following type information: eq :: 0:s:null_length -> 0:s:null_length -> true:false 0 :: 0:s:null_length true :: true:false s :: 0:s:null_length -> 0:s:null_length false :: true:false inf :: 0:s:null_length -> cons:nil:null_take cons :: 0:s:null_length -> cons:nil:null_take -> cons:nil:null_take take :: 0:s:null_length -> cons:nil:null_take -> cons:nil:null_take nil :: cons:nil:null_take length :: cons:nil:null_take -> 0:s:null_length null_take :: cons:nil:null_take null_length :: 0:s:null_length Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 null_take => 0 null_length => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, X >= 0, z = X inf(z) -{ 1 }-> 1 + X + inf(1 + X) :|: X >= 0, z = X length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0 take(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 take(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 take(z, z') -{ 1 }-> 1 + Y + take(X, L) :|: z = 1 + X, Y >= 0, X >= 0, L >= 0, z' = 1 + Y + L Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(s(z0), s(z1)) -> eq(z0, z1) eq(z0, z1) -> false inf(z0) -> cons(z0, inf(s(z0))) take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: EQ(0, 0) -> c EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) EQ(z0, z1) -> c2 INF(z0) -> c3(INF(s(z0))) TAKE(0, z0) -> c4 TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(nil) -> c6 LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) S tuples: EQ(0, 0) -> c EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) EQ(z0, z1) -> c2 INF(z0) -> c3(INF(s(z0))) TAKE(0, z0) -> c4 TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(nil) -> c6 LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) K tuples:none Defined Rule Symbols: eq_2, inf_1, take_2, length_1 Defined Pair Symbols: EQ_2, INF_1, TAKE_2, LENGTH_1 Compound Symbols: c, c1_1, c2, c3_1, c4, c5_1, c6, c7_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: LENGTH(nil) -> c6 EQ(z0, z1) -> c2 EQ(0, 0) -> c TAKE(0, z0) -> c4 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(s(z0), s(z1)) -> eq(z0, z1) eq(z0, z1) -> false inf(z0) -> cons(z0, inf(s(z0))) take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) INF(z0) -> c3(INF(s(z0))) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) S tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) INF(z0) -> c3(INF(s(z0))) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) K tuples:none Defined Rule Symbols: eq_2, inf_1, take_2, length_1 Defined Pair Symbols: EQ_2, INF_1, TAKE_2, LENGTH_1 Compound Symbols: c1_1, c3_1, c5_1, c7_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: eq(0, 0) -> true eq(s(z0), s(z1)) -> eq(z0, z1) eq(z0, z1) -> false inf(z0) -> cons(z0, inf(s(z0))) take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) INF(z0) -> c3(INF(s(z0))) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) S tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) INF(z0) -> c3(INF(s(z0))) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: EQ_2, INF_1, TAKE_2, LENGTH_1 Compound Symbols: c1_1, c3_1, c5_1, c7_1 ---------------------------------------- (35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) INF(z0) -> c3(INF(s(z0))) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(EQ(x_1, x_2)) = x_1 POL(INF(x_1)) = 0 POL(LENGTH(x_1)) = x_1 POL(TAKE(x_1, x_2)) = x_2 POL(c1(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) INF(z0) -> c3(INF(s(z0))) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) S tuples: INF(z0) -> c3(INF(s(z0))) K tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) Defined Rule Symbols:none Defined Pair Symbols: EQ_2, INF_1, TAKE_2, LENGTH_1 Compound Symbols: c1_1, c3_1, c5_1, c7_1 ---------------------------------------- (37) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace INF(z0) -> c3(INF(s(z0))) by INF(s(x0)) -> c3(INF(s(s(x0)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) INF(s(x0)) -> c3(INF(s(s(x0)))) S tuples: INF(s(x0)) -> c3(INF(s(s(x0)))) K tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) Defined Rule Symbols:none Defined Pair Symbols: EQ_2, TAKE_2, LENGTH_1, INF_1 Compound Symbols: c1_1, c5_1, c7_1, c3_1 ---------------------------------------- (39) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace INF(s(x0)) -> c3(INF(s(s(x0)))) by INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) Defined Rule Symbols:none Defined Pair Symbols: EQ_2, TAKE_2, LENGTH_1, INF_1 Compound Symbols: c1_1, c5_1, c7_1, c3_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) Defined Rule Symbols:none Defined Pair Symbols: TAKE_2, LENGTH_1, INF_1, EQ_2 Compound Symbols: c5_1, c7_1, c3_1, c1_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) by TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, INF_1, EQ_2, TAKE_2 Compound Symbols: c7_1, c3_1, c1_1, c5_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) by LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) Defined Rule Symbols:none Defined Pair Symbols: INF_1, EQ_2, TAKE_2, LENGTH_1 Compound Symbols: c3_1, c1_1, c5_1, c7_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) by EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols:none Defined Pair Symbols: INF_1, TAKE_2, LENGTH_1, EQ_2 Compound Symbols: c3_1, c5_1, c7_1, c1_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) by TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: INF_1, LENGTH_1, EQ_2, TAKE_2 Compound Symbols: c3_1, c7_1, c1_1, c5_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) by LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2)))) S tuples: INF(s(s(x0))) -> c3(INF(s(s(s(x0))))) K tuples: EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1)))) TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2)))) Defined Rule Symbols:none Defined Pair Symbols: INF_1, EQ_2, TAKE_2, LENGTH_1 Compound Symbols: c3_1, c1_1, c5_1, c7_1