/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 0 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 1430 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 24 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 44 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 665 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 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)) +(1(x), 1(y)) -> 0(+(+(x, y), 1(#))) +(+(x, y), z) -> +(x, +(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) Types: 0 :: #:1 -> #:1 # :: #:1 +' :: #:1 -> #:1 -> #:1 1 :: #:1 -> #:1 - :: #:1 -> #:1 -> #:1 not :: true:false -> true:false true :: true:false false :: true:false if :: true:false -> #:1 -> #:1 -> #:1 ge :: #:1 -> #:1 -> true:false log :: #:1 -> #:1 log' :: #:1 -> #:1 hole_#:11_2 :: #:1 hole_true:false2_2 :: true:false gen_#:13_2 :: Nat -> #:1 ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: +', -, ge, log' They will be analysed ascendingly in the following order: +' < log' ge < log' ---------------------------------------- (6) Obligation: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) Types: 0 :: #:1 -> #:1 # :: #:1 +' :: #:1 -> #:1 -> #:1 1 :: #:1 -> #:1 - :: #:1 -> #:1 -> #:1 not :: true:false -> true:false true :: true:false false :: true:false if :: true:false -> #:1 -> #:1 -> #:1 ge :: #:1 -> #:1 -> true:false log :: #:1 -> #:1 log' :: #:1 -> #:1 hole_#:11_2 :: #:1 hole_true:false2_2 :: true:false gen_#:13_2 :: Nat -> #:1 Generator Equations: gen_#:13_2(0) <=> # gen_#:13_2(+(x, 1)) <=> 1(gen_#:13_2(x)) The following defined symbols remain to be analysed: +', -, ge, log' They will be analysed ascendingly in the following order: +' < log' ge < log' ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) -> *4_2, rt in Omega(n5_2) Induction Base: +'(gen_#:13_2(0), gen_#:13_2(0)) Induction Step: +'(gen_#:13_2(+(n5_2, 1)), gen_#:13_2(+(n5_2, 1))) ->_R^Omega(1) 0(+'(+'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)), 1(#))) ->_IH 0(+'(*4_2, 1(#))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) Types: 0 :: #:1 -> #:1 # :: #:1 +' :: #:1 -> #:1 -> #:1 1 :: #:1 -> #:1 - :: #:1 -> #:1 -> #:1 not :: true:false -> true:false true :: true:false false :: true:false if :: true:false -> #:1 -> #:1 -> #:1 ge :: #:1 -> #:1 -> true:false log :: #:1 -> #:1 log' :: #:1 -> #:1 hole_#:11_2 :: #:1 hole_true:false2_2 :: true:false gen_#:13_2 :: Nat -> #:1 Generator Equations: gen_#:13_2(0) <=> # gen_#:13_2(+(x, 1)) <=> 1(gen_#:13_2(x)) The following defined symbols remain to be analysed: +', -, ge, log' They will be analysed ascendingly in the following order: +' < log' ge < log' ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) Types: 0 :: #:1 -> #:1 # :: #:1 +' :: #:1 -> #:1 -> #:1 1 :: #:1 -> #:1 - :: #:1 -> #:1 -> #:1 not :: true:false -> true:false true :: true:false false :: true:false if :: true:false -> #:1 -> #:1 -> #:1 ge :: #:1 -> #:1 -> true:false log :: #:1 -> #:1 log' :: #:1 -> #:1 hole_#:11_2 :: #:1 hole_true:false2_2 :: true:false gen_#:13_2 :: Nat -> #:1 Lemmas: +'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) -> *4_2, rt in Omega(n5_2) Generator Equations: gen_#:13_2(0) <=> # gen_#:13_2(+(x, 1)) <=> 1(gen_#:13_2(x)) The following defined symbols remain to be analysed: -, ge, log' They will be analysed ascendingly in the following order: ge < log' ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -(gen_#:13_2(n105167_2), gen_#:13_2(n105167_2)) -> gen_#:13_2(0), rt in Omega(1 + n105167_2) Induction Base: -(gen_#:13_2(0), gen_#:13_2(0)) ->_R^Omega(1) # Induction Step: -(gen_#:13_2(+(n105167_2, 1)), gen_#:13_2(+(n105167_2, 1))) ->_R^Omega(1) 0(-(gen_#:13_2(n105167_2), gen_#:13_2(n105167_2))) ->_IH 0(gen_#:13_2(0)) ->_R^Omega(1) # We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Obligation: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) Types: 0 :: #:1 -> #:1 # :: #:1 +' :: #:1 -> #:1 -> #:1 1 :: #:1 -> #:1 - :: #:1 -> #:1 -> #:1 not :: true:false -> true:false true :: true:false false :: true:false if :: true:false -> #:1 -> #:1 -> #:1 ge :: #:1 -> #:1 -> true:false log :: #:1 -> #:1 log' :: #:1 -> #:1 hole_#:11_2 :: #:1 hole_true:false2_2 :: true:false gen_#:13_2 :: Nat -> #:1 Lemmas: +'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) -> *4_2, rt in Omega(n5_2) -(gen_#:13_2(n105167_2), gen_#:13_2(n105167_2)) -> gen_#:13_2(0), rt in Omega(1 + n105167_2) Generator Equations: gen_#:13_2(0) <=> # gen_#:13_2(+(x, 1)) <=> 1(gen_#:13_2(x)) The following defined symbols remain to be analysed: ge, log' They will be analysed ascendingly in the following order: ge < log' ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: ge(gen_#:13_2(n107134_2), gen_#:13_2(n107134_2)) -> true, rt in Omega(1 + n107134_2) Induction Base: ge(gen_#:13_2(0), gen_#:13_2(0)) ->_R^Omega(1) true Induction Step: ge(gen_#:13_2(+(n107134_2, 1)), gen_#:13_2(+(n107134_2, 1))) ->_R^Omega(1) ge(gen_#:13_2(n107134_2), gen_#:13_2(n107134_2)) ->_IH true We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Obligation: 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)) +'(1(x), 1(y)) -> 0(+'(+'(x, y), 1(#))) +'(+'(x, y), z) -> +'(x, +'(y, z)) -(#, x) -> # -(x, #) -> x -(0(x), 0(y)) -> 0(-(x, y)) -(0(x), 1(y)) -> 1(-(-(x, y), 1(#))) -(1(x), 0(y)) -> 1(-(x, y)) -(1(x), 1(y)) -> 0(-(x, y)) not(true) -> false not(false) -> true if(true, x, y) -> x if(false, x, y) -> y ge(0(x), 0(y)) -> ge(x, y) ge(0(x), 1(y)) -> not(ge(y, x)) ge(1(x), 0(y)) -> ge(x, y) ge(1(x), 1(y)) -> ge(x, y) ge(x, #) -> true ge(#, 0(x)) -> ge(#, x) ge(#, 1(x)) -> false log(x) -> -(log'(x), 1(#)) log'(#) -> # log'(1(x)) -> +'(log'(x), 1(#)) log'(0(x)) -> if(ge(x, 1(#)), +'(log'(x), 1(#)), #) Types: 0 :: #:1 -> #:1 # :: #:1 +' :: #:1 -> #:1 -> #:1 1 :: #:1 -> #:1 - :: #:1 -> #:1 -> #:1 not :: true:false -> true:false true :: true:false false :: true:false if :: true:false -> #:1 -> #:1 -> #:1 ge :: #:1 -> #:1 -> true:false log :: #:1 -> #:1 log' :: #:1 -> #:1 hole_#:11_2 :: #:1 hole_true:false2_2 :: true:false gen_#:13_2 :: Nat -> #:1 Lemmas: +'(gen_#:13_2(n5_2), gen_#:13_2(n5_2)) -> *4_2, rt in Omega(n5_2) -(gen_#:13_2(n105167_2), gen_#:13_2(n105167_2)) -> gen_#:13_2(0), rt in Omega(1 + n105167_2) ge(gen_#:13_2(n107134_2), gen_#:13_2(n107134_2)) -> true, rt in Omega(1 + n107134_2) Generator Equations: gen_#:13_2(0) <=> # gen_#:13_2(+(x, 1)) <=> 1(gen_#:13_2(x)) The following defined symbols remain to be analysed: log' ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: log'(gen_#:13_2(+(1, n110653_2))) -> *4_2, rt in Omega(n110653_2) Induction Base: log'(gen_#:13_2(+(1, 0))) Induction Step: log'(gen_#:13_2(+(1, +(n110653_2, 1)))) ->_R^Omega(1) +'(log'(gen_#:13_2(+(1, n110653_2))), 1(#)) ->_IH +'(*4_2, 1(#)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) BOUNDS(1, INF)