KILLED proof of input_BpGv9vjClw.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 183 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 11 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 134 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (48) CdtProblem (49) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms] (70) CdtProblem (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) 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: incr(nil) -> nil incr(cons(X, L)) -> cons(s(X), incr(L)) adx(nil) -> nil adx(cons(X, L)) -> incr(cons(X, adx(L))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(X, L)) -> X tail(cons(X, L)) -> 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: incr(nil) -> nil incr(cons(X, L)) -> cons(s(X), incr(L)) adx(nil) -> nil adx(cons(X, L)) -> incr(cons(X, adx(L))) nats -> adx(zeros) zeros -> cons(0', zeros) head(cons(X, L)) -> X tail(cons(X, L)) -> 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: incr(nil) -> nil incr(cons(X, L)) -> cons(s(X), incr(L)) adx(nil) -> nil adx(cons(X, L)) -> incr(cons(X, adx(L))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(X, L)) -> X tail(cons(X, L)) -> 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: incr(nil) -> nil [1] incr(cons(X, L)) -> cons(s(X), incr(L)) [1] adx(nil) -> nil [1] adx(cons(X, L)) -> incr(cons(X, adx(L))) [1] nats -> adx(zeros) [1] zeros -> cons(0, zeros) [1] head(cons(X, L)) -> X [1] tail(cons(X, L)) -> 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: incr(nil) -> nil [1] incr(cons(X, L)) -> cons(s(X), incr(L)) [1] adx(nil) -> nil [1] adx(cons(X, L)) -> incr(cons(X, adx(L))) [1] nats -> adx(zeros) [1] zeros -> cons(0, zeros) [1] head(cons(X, L)) -> X [1] tail(cons(X, L)) -> L [1] The TRS has the following type information: incr :: nil:cons -> nil:cons nil :: nil:cons cons :: s:0 -> nil:cons -> nil:cons s :: s:0 -> s:0 adx :: nil:cons -> nil:cons nats :: nil:cons zeros :: nil:cons 0 :: s:0 head :: nil:cons -> s:0 tail :: nil:cons -> nil:cons 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: nats head_1 tail_1 (c) The following functions are completely defined: zeros adx_1 incr_1 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: incr(nil) -> nil [1] incr(cons(X, L)) -> cons(s(X), incr(L)) [1] adx(nil) -> nil [1] adx(cons(X, L)) -> incr(cons(X, adx(L))) [1] nats -> adx(zeros) [1] zeros -> cons(0, zeros) [1] head(cons(X, L)) -> X [1] tail(cons(X, L)) -> L [1] The TRS has the following type information: incr :: nil:cons -> nil:cons nil :: nil:cons cons :: s:0 -> nil:cons -> nil:cons s :: s:0 -> s:0 adx :: nil:cons -> nil:cons nats :: nil:cons zeros :: nil:cons 0 :: s:0 head :: nil:cons -> s:0 tail :: nil:cons -> nil:cons 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: incr(nil) -> nil [1] incr(cons(X, L)) -> cons(s(X), incr(L)) [1] adx(nil) -> nil [1] adx(cons(X, nil)) -> incr(cons(X, nil)) [2] adx(cons(X, cons(X', L'))) -> incr(cons(X, incr(cons(X', adx(L'))))) [2] nats -> adx(cons(0, zeros)) [2] zeros -> cons(0, zeros) [1] head(cons(X, L)) -> X [1] tail(cons(X, L)) -> L [1] The TRS has the following type information: incr :: nil:cons -> nil:cons nil :: nil:cons cons :: s:0 -> nil:cons -> nil:cons s :: s:0 -> s:0 adx :: nil:cons -> nil:cons nats :: nil:cons zeros :: nil:cons 0 :: s:0 head :: nil:cons -> s:0 tail :: nil:cons -> nil:cons 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: nil => 0 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 2 }-> incr(1 + X + 0) :|: z = 1 + X + 0, X >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: ---------------------------------------- (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: adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { incr } { head } { zeros } { tail } { adx } { nats } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {incr}, {head}, {zeros}, {tail}, {adx}, {nats} ---------------------------------------- (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: adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {incr}, {head}, {zeros}, {tail}, {adx}, {nats} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: incr 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: adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {incr}, {head}, {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: incr after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {head}, {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0 adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {head}, {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0 adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {head}, {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] head: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0 adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0 adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: zeros after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0 adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] head: runtime: O(1) [1], size: O(n^1) [z] zeros: runtime: ?, size: O(1) [0] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: zeros after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0 adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L') adx(z) -{ 1 }-> 0 :|: z = 0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0 nats -{ 2 }-> adx(1 + 0 + zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 zeros -{ 1 }-> 1 + 0 + zeros :|: Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats} Previous analysis results are: incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] head: runtime: O(1) [1], size: O(n^1) [z] zeros: runtime: INF, size: O(1) [0] ---------------------------------------- (37) 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: head(v0) -> null_head [0] tail(v0) -> null_tail [0] incr(v0) -> null_incr [0] adx(v0) -> null_adx [0] And the following fresh constants: null_head, null_tail, null_incr, null_adx ---------------------------------------- (38) 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: incr(nil) -> nil [1] incr(cons(X, L)) -> cons(s(X), incr(L)) [1] adx(nil) -> nil [1] adx(cons(X, L)) -> incr(cons(X, adx(L))) [1] nats -> adx(zeros) [1] zeros -> cons(0, zeros) [1] head(cons(X, L)) -> X [1] tail(cons(X, L)) -> L [1] head(v0) -> null_head [0] tail(v0) -> null_tail [0] incr(v0) -> null_incr [0] adx(v0) -> null_adx [0] The TRS has the following type information: incr :: nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx nil :: nil:cons:null_tail:null_incr:null_adx cons :: s:0:null_head -> nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx s :: s:0:null_head -> s:0:null_head adx :: nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx nats :: nil:cons:null_tail:null_incr:null_adx zeros :: nil:cons:null_tail:null_incr:null_adx 0 :: s:0:null_head head :: nil:cons:null_tail:null_incr:null_adx -> s:0:null_head tail :: nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx null_head :: s:0:null_head null_tail :: nil:cons:null_tail:null_incr:null_adx null_incr :: nil:cons:null_tail:null_incr:null_adx null_adx :: nil:cons:null_tail:null_incr:null_adx Rewrite Strategy: INNERMOST ---------------------------------------- (39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 null_head => 0 null_tail => 0 null_incr => 0 null_adx => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: adx(z) -{ 1 }-> incr(1 + X + adx(L)) :|: z = 1 + X + L, X >= 0, L >= 0 adx(z) -{ 1 }-> 0 :|: z = 0 adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 incr(z) -{ 1 }-> 0 :|: z = 0 incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0 nats -{ 1 }-> adx(zeros) :|: tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0 tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 zeros -{ 1 }-> 1 + 0 + zeros :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: incr(nil) -> nil incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(z0, z1)) -> z0 tail(cons(z0, z1)) -> z1 Tuples: INCR(nil) -> c INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(nil) -> c2 ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) NATS -> c4(ADX(zeros), ZEROS) ZEROS -> c5(ZEROS) HEAD(cons(z0, z1)) -> c6 TAIL(cons(z0, z1)) -> c7 S tuples: INCR(nil) -> c INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(nil) -> c2 ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) NATS -> c4(ADX(zeros), ZEROS) ZEROS -> c5(ZEROS) HEAD(cons(z0, z1)) -> c6 TAIL(cons(z0, z1)) -> c7 K tuples:none Defined Rule Symbols: incr_1, adx_1, nats, zeros, head_1, tail_1 Defined Pair Symbols: INCR_1, ADX_1, NATS, ZEROS, HEAD_1, TAIL_1 Compound Symbols: c, c1_1, c2, c3_2, c4_2, c5_1, c6, c7 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: HEAD(cons(z0, z1)) -> c6 ADX(nil) -> c2 TAIL(cons(z0, z1)) -> c7 INCR(nil) -> c ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: incr(nil) -> nil incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(z0, z1)) -> z0 tail(cons(z0, z1)) -> z1 Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) NATS -> c4(ADX(zeros), ZEROS) ZEROS -> c5(ZEROS) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) NATS -> c4(ADX(zeros), ZEROS) ZEROS -> c5(ZEROS) K tuples:none Defined Rule Symbols: incr_1, adx_1, nats, zeros, head_1, tail_1 Defined Pair Symbols: INCR_1, ADX_1, NATS, ZEROS Compound Symbols: c1_1, c3_2, c4_2, c5_1 ---------------------------------------- (45) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: incr(nil) -> nil incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(z0, z1)) -> z0 tail(cons(z0, z1)) -> z1 Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(zeros)) NATS -> c(ZEROS) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(zeros)) NATS -> c(ZEROS) K tuples:none Defined Rule Symbols: incr_1, adx_1, nats, zeros, head_1, tail_1 Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (47) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: NATS -> c(ZEROS) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: incr(nil) -> nil incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(z0, z1)) -> z0 tail(cons(z0, z1)) -> z1 Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(zeros)) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(zeros)) K tuples:none Defined Rule Symbols: incr_1, adx_1, nats, zeros, head_1, tail_1 Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (49) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: NATS -> c(ADX(zeros)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: incr(nil) -> nil incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) nats -> adx(zeros) zeros -> cons(0, zeros) head(cons(z0, z1)) -> z0 tail(cons(z0, z1)) -> z1 Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(zeros)) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: incr_1, adx_1, nats, zeros, head_1, tail_1 Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: nats -> adx(zeros) head(cons(z0, z1)) -> z0 tail(cons(z0, z1)) -> z1 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(zeros)) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace NATS -> c(ADX(zeros)) by NATS -> c(ADX(cons(0, zeros))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, zeros))) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace NATS -> c(ADX(cons(0, zeros))) by NATS -> c(ADX(cons(0, cons(0, zeros)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, zeros)))) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (57) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace NATS -> c(ADX(cons(0, cons(0, zeros)))) by NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) S tuples: INCR(cons(z0, z1)) -> c1(INCR(z1)) ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: INCR_1, ADX_1, ZEROS, NATS Compound Symbols: c1_1, c3_2, c5_1, c_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INCR(cons(z0, z1)) -> c1(INCR(z1)) by INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) S tuples: ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ADX_1, ZEROS, NATS, INCR_1 Compound Symbols: c3_2, c5_1, c_1, c1_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) by ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil)) ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil)) ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) S tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil)) ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) K tuples: NATS -> c(ADX(zeros)) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, NATS, INCR_1, ADX_1 Compound Symbols: c5_1, c_1, c1_1, c3_2 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) S tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) K tuples:none Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, NATS, INCR_1, ADX_1 Compound Symbols: c5_1, c_1, c1_1, c3_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) by ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))), ADX(cons(x1, nil))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))), ADX(cons(x1, nil))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) S tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))), ADX(cons(x1, nil))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) K tuples:none Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, NATS, INCR_1, ADX_1 Compound Symbols: c5_1, c_1, c1_1, c3_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) S tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) K tuples:none Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, NATS, INCR_1, ADX_1 Compound Symbols: c5_1, c_1, c1_1, c3_2, c3_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) We considered the (Usable) Rules:none And the Tuples: ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADX(x_1)) = x_1 POL(INCR(x_1)) = 0 POL(NATS) = [1] POL(ZEROS) = 0 POL(adx(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(incr(x_1)) = [1] + x_1 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(zeros) = [1] ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) S tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, NATS, INCR_1, ADX_1 Compound Symbols: c5_1, c_1, c1_1, c3_2, c3_1 ---------------------------------------- (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) by NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) S tuples: ZEROS -> c5(ZEROS) INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, INCR_1, ADX_1, NATS Compound Symbols: c5_1, c1_1, c3_2, c3_1, c_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) by INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) S tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, ADX_1, NATS, INCR_1 Compound Symbols: c5_1, c3_2, c3_1, c_1, c1_1 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) by ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) S tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, ADX_1, NATS, INCR_1 Compound Symbols: c5_1, c3_2, c3_1, c_1, c1_1 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) by ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil))))) S tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, ADX_1, NATS, INCR_1 Compound Symbols: c5_1, c3_2, c_1, c1_1, c3_1 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) by ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil))))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) S tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, ADX_1, NATS, INCR_1 Compound Symbols: c5_1, c3_2, c_1, c1_1, c3_1 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil))))) by ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), nil)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), nil)))) S tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) K tuples: ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, ADX_1, NATS, INCR_1 Compound Symbols: c5_1, c3_2, c_1, c1_1, c3_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), nil)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: adx(nil) -> nil adx(cons(z0, z1)) -> incr(cons(z0, adx(z1))) incr(cons(z0, z1)) -> cons(s(z0), incr(z1)) incr(nil) -> nil zeros -> cons(0, zeros) Tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros)))))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) S tuples: ZEROS -> c5(ZEROS) ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2))) INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2)))) ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) K tuples:none Defined Rule Symbols: adx_1, incr_1, zeros Defined Pair Symbols: ZEROS, ADX_1, NATS, INCR_1 Compound Symbols: c5_1, c3_2, c_1, c1_1