WORST_CASE(?,O(n^3)) proof of input_yLzoglcDlh.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). (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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRNTS (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) IntTrsBoundProof [UPPER BOUND(ID), 391 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 420 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 16 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 614 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) FinalProof [FINISHED, 0 ms] (34) BOUNDS(1, n^3) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) mul0(Nil, y) -> Nil add0(Nil, y) -> y goal(xs, ys) -> mul0(xs, ys) 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^3). The TRS R consists of the following rules: mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) [1] add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1] mul0(Nil, y) -> Nil [1] add0(Nil, y) -> y [1] goal(xs, ys) -> mul0(xs, ys) [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: mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) [1] add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1] mul0(Nil, y) -> Nil [1] add0(Nil, y) -> y [1] goal(xs, ys) -> mul0(xs, ys) [1] The TRS has the following type information: mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: S -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil S :: S Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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: goal_2 (c) The following functions are completely defined: mul0_2 add0_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) [1] add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1] mul0(Nil, y) -> Nil [1] add0(Nil, y) -> y [1] goal(xs, ys) -> mul0(xs, ys) [1] The TRS has the following type information: mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: S -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil S :: S Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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: mul0(Cons(x, Cons(x', xs')), y) -> add0(add0(mul0(xs', y), y), y) [2] mul0(Cons(x, Nil), y) -> add0(Nil, y) [2] add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1] mul0(Nil, y) -> Nil [1] add0(Nil, y) -> y [1] goal(xs, ys) -> mul0(xs, ys) [1] The TRS has the following type information: mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: S -> Cons:Nil -> Cons:Nil add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil S :: S Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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: S => 0 Nil => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y goal(z, z') -{ 1 }-> mul0(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', y), y), y) :|: x >= 0, x' >= 0, xs' >= 0, y >= 0, z = 1 + x + (1 + x' + xs'), z' = y mul0(z, z') -{ 2 }-> add0(0, y) :|: x >= 0, y >= 0, z' = y, z = 1 + x + 0 mul0(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y ---------------------------------------- (11) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0 mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { add0 } { mul0 } { goal } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0 mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {add0}, {mul0}, {goal} ---------------------------------------- (15) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0 mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {add0}, {mul0}, {goal} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: add0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0 mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {add0}, {mul0}, {goal} Previous analysis results are: add0: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: add0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0 mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {mul0}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {mul0}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: mul0 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {mul0}, {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] mul0: runtime: ?, size: O(n^2) [2*z*z' + z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: mul0 after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 4 + 4*z + 3*z*z' + 4*z^2*z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0 mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0 mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0 mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0 mul0(z, z') -{ 8 + s'' + s1 + 4*xs' + 3*xs'*z' + 4*xs'^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: goal after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0 mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0 mul0(z, z') -{ 8 + s'' + s1 + 4*xs' + 3*xs'*z' + 4*xs'^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {goal} Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z'] goal: runtime: ?, size: O(n^2) [2*z*z' + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: goal after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 5 + 4*z + 3*z*z' + 4*z^2*z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0 add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0 mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0 mul0(z, z') -{ 8 + s'' + s1 + 4*xs' + 3*xs'*z' + 4*xs'^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs') mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: Previous analysis results are: add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z'] goal: runtime: O(n^3) [5 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (33) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (34) BOUNDS(1, n^3)