/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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), 516 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (22) CpxRNTS (23) FinalProof [FINISHED, 0 ms] (24) BOUNDS(1, n^1) (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTRS (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (28) typed CpxTrs (29) OrderProof [LOWER BOUND(ID), 0 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 668 ms] (32) proven lower bound (33) LowerBoundPropagationProof [FINISHED, 0 ms] (34) BOUNDS(n^1, INF) ---------------------------------------- (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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS does not nest defined symbols, 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) 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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- 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: D_1 (c) The following functions are completely defined: none Due to the following rules being added: none 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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- 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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- 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: t => 3 1 => 1 constant => 2 0 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 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: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { D } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {D} ---------------------------------------- (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: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {D} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: D after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 2*z + z^2 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {D} Previous analysis results are: D: runtime: ?, size: O(n^2) [1 + 2*z + z^2] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: D after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: Previous analysis results are: D: runtime: O(n^1) [3*z], size: O(n^2) [1 + 2*z + z^2] ---------------------------------------- (23) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (24) BOUNDS(1, n^1) ---------------------------------------- (25) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (26) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (27) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (28) Obligation: TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) Types: D :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- t :: t:1':constant:0':+':*':- 1' :: t:1':constant:0':+':*':- constant :: t:1':constant:0':+':*':- 0' :: t:1':constant:0':+':*':- +' :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- *' :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- - :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':- gen_t:1':constant:0':+':*':-2_0 :: Nat -> t:1':constant:0':+':*':- ---------------------------------------- (29) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: D ---------------------------------------- (30) Obligation: TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) Types: D :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- t :: t:1':constant:0':+':*':- 1' :: t:1':constant:0':+':*':- constant :: t:1':constant:0':+':*':- 0' :: t:1':constant:0':+':*':- +' :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- *' :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- - :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':- gen_t:1':constant:0':+':*':-2_0 :: Nat -> t:1':constant:0':+':*':- Generator Equations: gen_t:1':constant:0':+':*':-2_0(0) <=> t gen_t:1':constant:0':+':*':-2_0(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-2_0(x)) The following defined symbols remain to be analysed: D ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: D(gen_t:1':constant:0':+':*':-2_0(n4_0)) -> *3_0, rt in Omega(n4_0) Induction Base: D(gen_t:1':constant:0':+':*':-2_0(0)) Induction Step: D(gen_t:1':constant:0':+':*':-2_0(+(n4_0, 1))) ->_R^Omega(1) +'(D(t), D(gen_t:1':constant:0':+':*':-2_0(n4_0))) ->_R^Omega(1) +'(1', D(gen_t:1':constant:0':+':*':-2_0(n4_0))) ->_IH +'(1', *3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (32) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) Types: D :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- t :: t:1':constant:0':+':*':- 1' :: t:1':constant:0':+':*':- constant :: t:1':constant:0':+':*':- 0' :: t:1':constant:0':+':*':- +' :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- *' :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- - :: t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- -> t:1':constant:0':+':*':- hole_t:1':constant:0':+':*':-1_0 :: t:1':constant:0':+':*':- gen_t:1':constant:0':+':*':-2_0 :: Nat -> t:1':constant:0':+':*':- Generator Equations: gen_t:1':constant:0':+':*':-2_0(0) <=> t gen_t:1':constant:0':+':*':-2_0(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-2_0(x)) The following defined symbols remain to be analysed: D ---------------------------------------- (33) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (34) BOUNDS(n^1, INF)