/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 210 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 19.1 s] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 1315 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 2180 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 83 ms] (22) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(#) -> # +(#, x) -> x +(x, #) -> x +(0(x), 0(y)) -> 0(+(x, y)) +(0(x), 1(y)) -> 1(+(x, y)) +(1(x), 0(y)) -> 1(+(x, y)) +(0(x), j(y)) -> j(+(x, y)) +(j(x), 0(y)) -> j(+(x, y)) +(1(x), 1(y)) -> j(+(+(x, y), 1(#))) +(j(x), j(y)) -> 1(+(+(x, y), j(#))) +(1(x), j(y)) -> 0(+(x, y)) +(j(x), 1(y)) -> 0(+(x, y)) +(+(x, y), z) -> +(x, +(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +(x, opp(y)) *(#, x) -> # *(0(x), y) -> 0(*(x, y)) *(1(x), y) -> +(0(*(x, y)), y) *(j(x), y) -> -(0(*(x, y)), y) *(*(x, y), z) -> *(x, *(y, z)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(#) -> # +(#, x) -> x +(x, #) -> x +(0(x), 0(y)) -> 0(+(x, y)) +(0(x), 1(y)) -> 1(+(x, y)) +(1(x), 0(y)) -> 1(+(x, y)) +(0(x), j(y)) -> j(+(x, y)) +(j(x), 0(y)) -> j(+(x, y)) +(1(x), 1(y)) -> j(+(+(x, y), 1(#))) +(j(x), j(y)) -> 1(+(+(x, y), j(#))) +(1(x), j(y)) -> 0(+(x, y)) +(j(x), 1(y)) -> 0(+(x, y)) +(+(x, y), z) -> +(x, +(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +(x, opp(y)) *(#, x) -> # *(0(x), y) -> 0(*(x, y)) *(1(x), y) -> +(0(*(x, y)), y) *(j(x), y) -> -(0(*(x, y)), y) *(*(x, y), z) -> *(x, *(y, z)) The (relative) TRS S consists of the following rules: encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(#) -> # +(#, x) -> x +(x, #) -> x +(0(x), 0(y)) -> 0(+(x, y)) +(0(x), 1(y)) -> 1(+(x, y)) +(1(x), 0(y)) -> 1(+(x, y)) +(0(x), j(y)) -> j(+(x, y)) +(j(x), 0(y)) -> j(+(x, y)) +(1(x), 1(y)) -> j(+(+(x, y), 1(#))) +(j(x), j(y)) -> 1(+(+(x, y), j(#))) +(1(x), j(y)) -> 0(+(x, y)) +(j(x), 1(y)) -> 0(+(x, y)) +(+(x, y), z) -> +(x, +(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +(x, opp(y)) *(#, x) -> # *(0(x), y) -> 0(*(x, y)) *(1(x), y) -> +(0(*(x, y)), y) *(j(x), y) -> -(0(*(x, y)), y) *(*(x, y), z) -> *(x, *(y, z)) The (relative) TRS S consists of the following rules: encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) The (relative) TRS S consists of the following rules: encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: 0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* # :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* +' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* 1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* - :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* *' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encArg :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_# :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* hole_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*1_3 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3 :: Nat -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: +', opp, *', encArg They will be analysed ascendingly in the following order: +' < *' +' < encArg opp < encArg *' < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: 0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* # :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* +' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* 1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* - :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* *' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encArg :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_# :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* hole_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*1_3 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3 :: Nat -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* Generator Equations: gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0) <=> # gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(x, 1)) <=> 1(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(x)) The following defined symbols remain to be analysed: +', opp, *', encArg They will be analysed ascendingly in the following order: +' < *' +' < encArg opp < encArg *' < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) Induction Base: +'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, 0)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, 0))) Induction Step: +'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, +(n4_3, 1))), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1) j(+'(+'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3))), 1(#))) ->_IH j(+'(*3_3, 1(#))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: 0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* # :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* +' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* 1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* - :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* *' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encArg :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_# :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* hole_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*1_3 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3 :: Nat -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* Generator Equations: gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0) <=> # gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(x, 1)) <=> 1(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(x)) The following defined symbols remain to be analysed: +', opp, *', encArg They will be analysed ascendingly in the following order: +' < *' +' < encArg opp < encArg *' < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: 0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* # :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* +' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* 1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* - :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* *' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encArg :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_# :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* hole_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*1_3 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3 :: Nat -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* Lemmas: +'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) Generator Equations: gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0) <=> # gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(x, 1)) <=> 1(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(x)) The following defined symbols remain to be analysed: opp, *', encArg They will be analysed ascendingly in the following order: opp < encArg *' < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: opp(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n6452019_3))) -> *3_3, rt in Omega(n6452019_3) Induction Base: opp(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, 0))) Induction Step: opp(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, +(n6452019_3, 1)))) ->_R^Omega(1) j(opp(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n6452019_3)))) ->_IH j(*3_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: 0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* # :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* +' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* 1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* - :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* *' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encArg :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_# :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* hole_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*1_3 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3 :: Nat -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* Lemmas: +'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) opp(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n6452019_3))) -> *3_3, rt in Omega(n6452019_3) Generator Equations: gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0) <=> # gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(x, 1)) <=> 1(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(x)) The following defined symbols remain to be analysed: *', encArg They will be analysed ascendingly in the following order: *' < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: *'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(n6455323_3), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) -> gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0), rt in Omega(1 + n6455323_3) Induction Base: *'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) ->_R^Omega(1) # Induction Step: *'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(n6455323_3, 1)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) ->_R^Omega(1) +'(0(*'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(n6455323_3), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0))), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) ->_IH +'(0(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) ->_R^Omega(1) +'(#, gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) ->_R^Omega(1) gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: Innermost TRS: Rules: 0(#) -> # +'(#, x) -> x +'(x, #) -> x +'(0(x), 0(y)) -> 0(+'(x, y)) +'(0(x), 1(y)) -> 1(+'(x, y)) +'(1(x), 0(y)) -> 1(+'(x, y)) +'(0(x), j(y)) -> j(+'(x, y)) +'(j(x), 0(y)) -> j(+'(x, y)) +'(1(x), 1(y)) -> j(+'(+'(x, y), 1(#))) +'(j(x), j(y)) -> 1(+'(+'(x, y), j(#))) +'(1(x), j(y)) -> 0(+'(x, y)) +'(j(x), 1(y)) -> 0(+'(x, y)) +'(+'(x, y), z) -> +'(x, +'(y, z)) opp(#) -> # opp(0(x)) -> 0(opp(x)) opp(1(x)) -> j(opp(x)) opp(j(x)) -> 1(opp(x)) -(x, y) -> +'(x, opp(y)) *'(#, x) -> # *'(0(x), y) -> 0(*'(x, y)) *'(1(x), y) -> +'(0(*'(x, y)), y) *'(j(x), y) -> -(0(*'(x, y)), y) *'(*'(x, y), z) -> *'(x, *'(y, z)) encArg(#) -> # encArg(1(x_1)) -> 1(encArg(x_1)) encArg(j(x_1)) -> j(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_opp(x_1)) -> opp(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_0(x_1) -> 0(encArg(x_1)) encode_# -> # encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_1(x_1) -> 1(encArg(x_1)) encode_j(x_1) -> j(encArg(x_1)) encode_opp(x_1) -> opp(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: 0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* # :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* +' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* 1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* - :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* *' :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encArg :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* cons_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_0 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_# :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_+ :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_1 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_j :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_opp :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_- :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* encode_* :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* hole_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*1_3 :: #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3 :: Nat -> #:1:j:cons_0:cons_+:cons_opp:cons_-:cons_* Lemmas: +'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3)), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) opp(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(1, n6452019_3))) -> *3_3, rt in Omega(n6452019_3) *'(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(n6455323_3), gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) -> gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0), rt in Omega(1 + n6455323_3) Generator Equations: gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0) <=> # gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(x, 1)) <=> 1(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(n7231512_3)) -> gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(n7231512_3), rt in Omega(0) Induction Base: encArg(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(0)) ->_R^Omega(0) # Induction Step: encArg(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(+(n7231512_3, 1))) ->_R^Omega(0) 1(encArg(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(n7231512_3))) ->_IH 1(gen_#:1:j:cons_0:cons_+:cons_opp:cons_-:cons_*2_3(c7231513_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)