WORST_CASE(?,O(n^2)) proof of input_spUaNFaaBc.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) InliningProof [UPPER BOUND(ID), 360 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 572 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 262 ms] (34) CpxRNTS (35) FinalProof [FINISHED, 0 ms] (36) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(p(p(x)))) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(p(p(x)))) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(p(p(x)))) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] The TRS has the following type information: cond :: true:false -> 0:s -> cond true :: true:false odd :: 0:s -> true:false p :: 0:s -> 0:s 0 :: 0:s false :: true:false s :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (5) 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: cond_2 (c) The following functions are completely defined: odd_1 p_1 Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (6) 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: cond(true, x) -> cond(odd(x), p(p(p(x)))) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] The TRS has the following type information: cond :: true:false -> 0:s -> cond true :: true:false odd :: 0:s -> true:false p :: 0:s -> 0:s 0 :: 0:s false :: true:false s :: 0:s -> 0:s const :: cond Rewrite Strategy: INNERMOST ---------------------------------------- (7) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: cond(true, 0) -> cond(false, p(p(0))) [3] cond(true, s(0)) -> cond(true, p(p(0))) [3] cond(true, s(s(x'))) -> cond(odd(x'), p(p(s(x')))) [3] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] The TRS has the following type information: cond :: true:false -> 0:s -> cond true :: true:false odd :: 0:s -> true:false p :: 0:s -> 0:s 0 :: 0:s false :: true:false s :: 0:s -> 0:s const :: cond Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: true => 1 0 => 0 false => 0 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 }-> cond(odd(x'), p(p(1 + x'))) :|: z' = 1 + (1 + x'), z = 1, x' >= 0 cond(z, z') -{ 3 }-> cond(1, p(p(0))) :|: z = 1, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(0, p(p(0))) :|: z = 1, z' = 0 odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x) odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (11) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 }-> cond(odd(x'), x'') :|: z' = 1 + (1 + x'), z = 1, x' >= 0, x >= 0, 1 + x' = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 }-> cond(odd(x'), 0) :|: z' = 1 + (1 + x'), z = 1, x' >= 0, x >= 0, 1 + x' = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x) odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { odd } { p } { cond } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {odd}, {p}, {cond} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {odd}, {p}, {cond} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: odd after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {odd}, {p}, {cond} Previous analysis results are: odd: runtime: ?, size: O(1) [1] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: odd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 }-> cond(odd(z' - 2), x'') :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 }-> cond(odd(z' - 2), 0) :|: z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {cond} Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {cond} Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {cond} Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {cond} Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {cond} Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: cond after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {cond} Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: ?, size: O(1) [0] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cond after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 22 + 7*z' + z'^2 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 5 + z' }-> cond(s', x'') :|: s' >= 0, s' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x'' >= 0, x = 1 + x'' cond(z, z') -{ 5 + z' }-> cond(s'', 0) :|: s'' >= 0, s'' <= 1, z = 1, z' - 2 >= 0, x >= 0, 1 + (z' - 2) = 1 + x, x = 0 cond(z, z') -{ 5 }-> cond(1, 0) :|: z = 1, z' = 1 + 0, 0 = 0 cond(z, z') -{ 5 }-> cond(0, 0) :|: z = 1, z' = 0, 0 = 0 odd(z) -{ 1 + z }-> s :|: s >= 0, s <= 1, z - 2 >= 0 odd(z) -{ 1 }-> 1 :|: z = 1 + 0 odd(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: odd: runtime: O(n^1) [2 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [22 + 7*z' + z'^2], size: O(1) [0] ---------------------------------------- (35) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (36) BOUNDS(1, n^2)