WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 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), 493 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 91 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) FinalProof [FINISHED, 0 ms] (30) BOUNDS(1, n^1) (31) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (32) TRS for Loop Detection (33) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (34) BEST (35) proven lower bound (36) LowerBoundPropagationProof [FINISHED, 0 ms] (37) BOUNDS(n^1, INF) (38) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: p(s(x)) -> x fac(0) -> s(0) fac(s(x)) -> times(s(x), fac(p(s(x)))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. The duplicating contexts are: fac(s([])) The defined contexts are: fac([]) p(s([])) [] just represents basic- or constructor-terms in the following defined contexts: fac([]) As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: p(s(x)) -> x fac(0) -> s(0) fac(s(x)) -> times(s(x), fac(p(s(x)))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(p(s(x)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(p(s(x)))) [1] The TRS has the following type information: p :: s:0:times -> s:0:times s :: s:0:times -> s:0:times fac :: s:0:times -> s:0:times 0 :: s:0:times times :: s:0:times -> s:0:times -> s:0:times Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: fac_1 (c) The following functions are completely defined: p_1 Due to the following rules being added: p(v0) -> 0 [0] And the following fresh constants: none ---------------------------------------- (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: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(p(s(x)))) [1] p(v0) -> 0 [0] The TRS has the following type information: p :: s:0:times -> s:0:times s :: s:0:times -> s:0:times fac :: s:0:times -> s:0:times 0 :: s:0:times times :: s:0:times -> s:0:times -> s:0:times Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: p(s(x)) -> x [1] fac(0) -> s(0) [1] fac(s(x)) -> times(s(x), fac(x)) [2] fac(s(x)) -> times(s(x), fac(0)) [1] p(v0) -> 0 [0] The TRS has the following type information: p :: s:0:times -> s:0:times s :: s:0:times -> s:0:times fac :: s:0:times -> s:0:times 0 :: s:0:times times :: s:0:times -> s:0:times -> s:0:times Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + (1 + x) + fac(x) :|: x >= 0, z = 1 + x fac(z) -{ 1 }-> 1 + (1 + x) + fac(0) :|: x >= 0, z = 1 + x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (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: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 1 }-> 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0 fac(z) -{ 2 }-> 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 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: { fac } { p } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 1 }-> 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0 fac(z) -{ 2 }-> 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {fac}, {p} ---------------------------------------- (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: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 1 }-> 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0 fac(z) -{ 2 }-> 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {fac}, {p} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fac after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + z + z^2 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 1 }-> 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0 fac(z) -{ 2 }-> 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {fac}, {p} Previous analysis results are: fac: runtime: ?, size: O(n^2) [2 + z + z^2] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fac after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 1 }-> 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0 fac(z) -{ 2 }-> 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p} Previous analysis results are: fac: runtime: O(n^1) [4 + 2*z], size: O(n^2) [2 + z + z^2] ---------------------------------------- (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: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 4 + 2*z }-> 1 + (1 + (z - 1)) + s :|: s >= 0, s <= 2 + (z - 1) * (z - 1) + (z - 1), z - 1 >= 0 fac(z) -{ 5 }-> 1 + (1 + (z - 1)) + s' :|: s' >= 0, s' <= 2 + 0 * 0 + 0, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p} Previous analysis results are: fac: runtime: O(n^1) [4 + 2*z], size: O(n^2) [2 + z + z^2] ---------------------------------------- (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: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 4 + 2*z }-> 1 + (1 + (z - 1)) + s :|: s >= 0, s <= 2 + (z - 1) * (z - 1) + (z - 1), z - 1 >= 0 fac(z) -{ 5 }-> 1 + (1 + (z - 1)) + s' :|: s' >= 0, s' <= 2 + 0 * 0 + 0, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p} Previous analysis results are: fac: runtime: O(n^1) [4 + 2*z], size: O(n^2) [2 + z + z^2] 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: fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 4 + 2*z }-> 1 + (1 + (z - 1)) + s :|: s >= 0, s <= 2 + (z - 1) * (z - 1) + (z - 1), z - 1 >= 0 fac(z) -{ 5 }-> 1 + (1 + (z - 1)) + s' :|: s' >= 0, s' <= 2 + 0 * 0 + 0, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: fac: runtime: O(n^1) [4 + 2*z], size: O(n^2) [2 + z + z^2] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (29) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (30) BOUNDS(1, n^1) ---------------------------------------- (31) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (32) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: p(s(x)) -> x fac(0) -> s(0) fac(s(x)) -> times(s(x), fac(p(s(x)))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (33) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence fac(s(x)) ->^+ times(s(x), fac(x)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [x / s(x)]. The result substitution is [ ]. ---------------------------------------- (34) Complex Obligation (BEST) ---------------------------------------- (35) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: p(s(x)) -> x fac(0) -> s(0) fac(s(x)) -> times(s(x), fac(p(s(x)))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (36) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (37) BOUNDS(n^1, INF) ---------------------------------------- (38) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: p(s(x)) -> x fac(0) -> s(0) fac(s(x)) -> times(s(x), fac(p(s(x)))) S is empty. Rewrite Strategy: FULL