WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 201 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), 410 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 377 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Types: active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encArg :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp hole_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp1_0 :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0 :: Nat -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, f, chk, mat, tp, encArg They will be analysed ascendingly in the following order: active = f active < chk active < encArg f < chk f < mat f < tp f < encArg mat < chk chk < tp chk < encArg mat < tp mat < encArg tp < encArg ---------------------------------------- (10) Obligation: TRS: Rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Types: active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encArg :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp hole_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp1_0 :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0 :: Nat -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp Generator Equations: gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(0) <=> X gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(x, 1)) <=> mark(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(x)) The following defined symbols remain to be analysed: f, active, chk, mat, tp, encArg They will be analysed ascendingly in the following order: active = f active < chk active < encArg f < chk f < mat f < tp f < encArg mat < chk chk < tp chk < encArg mat < tp mat < encArg tp < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: f(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: f(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(1, 0))) Induction Step: f(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) mark(f(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(1, n4_0)))) ->_IH mark(*3_0) 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: TRS: Rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Types: active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encArg :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp hole_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp1_0 :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0 :: Nat -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp Generator Equations: gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(0) <=> X gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(x, 1)) <=> mark(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(x)) The following defined symbols remain to be analysed: f, active, chk, mat, tp, encArg They will be analysed ascendingly in the following order: active = f active < chk active < encArg f < chk f < mat f < tp f < encArg mat < chk chk < tp chk < encArg mat < tp mat < encArg tp < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: active(f(x)) -> mark(f(f(x))) chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) mat(f(x), f(y)) -> f(mat(x, y)) chk(no(c)) -> active(c) mat(f(x), c) -> no(c) f(active(x)) -> active(f(x)) f(no(x)) -> no(f(x)) f(mark(x)) -> mark(f(x)) tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x))) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(no(x_1)) -> no(encArg(x_1)) encArg(X) -> X encArg(y) -> y encArg(c) -> c encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_chk(x_1)) -> chk(encArg(x_1)) encArg(cons_mat(x_1, x_2)) -> mat(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_tp(x_1)) -> tp(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_chk(x_1) -> chk(encArg(x_1)) encode_no(x_1) -> no(encArg(x_1)) encode_mat(x_1, x_2) -> mat(encArg(x_1), encArg(x_2)) encode_X -> X encode_y -> y encode_c -> c encode_tp(x_1) -> tp(encArg(x_1)) Types: active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encArg :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp cons_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_active :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_f :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mark :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_chk :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_no :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_mat :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_X :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_y :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_c :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp encode_tp :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp hole_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp1_0 :: mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0 :: Nat -> mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp Lemmas: f(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(0) <=> X gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(x, 1)) <=> mark(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(x)) The following defined symbols remain to be analysed: active, chk, mat, tp, encArg They will be analysed ascendingly in the following order: active = f active < chk active < encArg f < chk f < mat f < tp f < encArg mat < chk chk < tp chk < encArg mat < tp mat < encArg tp < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(n879_0)) -> gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(n879_0), rt in Omega(0) Induction Base: encArg(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(0)) ->_R^Omega(0) X Induction Step: encArg(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(+(n879_0, 1))) ->_R^Omega(0) mark(encArg(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(n879_0))) ->_IH mark(gen_mark:no:X:y:c:cons_active:cons_chk:cons_mat:cons_f:cons_tp2_0(c880_0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)