1142.99/291.55 WORST_CASE(Omega(n^2), ?) 1158.01/295.34 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1158.01/295.34 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1158.01/295.34 1158.01/295.34 1158.01/295.34 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1158.01/295.34 1158.01/295.34 (0) CpxTRS 1158.01/295.34 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1158.01/295.34 (2) CpxTRS 1158.01/295.34 (3) SlicingProof [LOWER BOUND(ID), 0 ms] 1158.01/295.34 (4) CpxTRS 1158.01/295.34 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1158.01/295.34 (6) typed CpxTrs 1158.01/295.34 (7) OrderProof [LOWER BOUND(ID), 0 ms] 1158.01/295.34 (8) typed CpxTrs 1158.01/295.34 (9) RewriteLemmaProof [LOWER BOUND(ID), 307 ms] 1158.01/295.34 (10) BEST 1158.01/295.34 (11) proven lower bound 1158.01/295.34 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 1158.01/295.34 (13) BOUNDS(n^1, INF) 1158.01/295.34 (14) typed CpxTrs 1158.01/295.34 (15) RewriteLemmaProof [LOWER BOUND(ID), 622 ms] 1158.01/295.34 (16) typed CpxTrs 1158.01/295.34 (17) RewriteLemmaProof [LOWER BOUND(ID), 330 ms] 1158.01/295.34 (18) BEST 1158.01/295.34 (19) proven lower bound 1158.01/295.34 (20) LowerBoundPropagationProof [FINISHED, 0 ms] 1158.01/295.34 (21) BOUNDS(n^2, INF) 1158.01/295.34 (22) typed CpxTrs 1158.01/295.34 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (0) 1158.01/295.34 Obligation: 1158.01/295.34 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1158.01/295.34 1158.01/295.34 1158.01/295.34 The TRS R consists of the following rules: 1158.01/295.34 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0) -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0) -> 0 1158.01/295.34 p(s(0)) -> 0 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, x, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, x, y) -> cons(0, nil) 1158.01/295.34 if1(false, x, y) -> cons(0, int(s(0), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 S is empty. 1158.01/295.34 Rewrite Strategy: INNERMOST 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1158.01/295.34 Renamed function symbols to avoid clashes with predefined symbol. 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (2) 1158.01/295.34 Obligation: 1158.01/295.34 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1158.01/295.34 1158.01/295.34 1158.01/295.34 The TRS R consists of the following rules: 1158.01/295.34 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, x, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, x, y) -> cons(0', nil) 1158.01/295.34 if1(false, x, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 S is empty. 1158.01/295.34 Rewrite Strategy: INNERMOST 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (3) SlicingProof (LOWER BOUND(ID)) 1158.01/295.34 Sliced the following arguments: 1158.01/295.34 if1/1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (4) 1158.01/295.34 Obligation: 1158.01/295.34 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1158.01/295.34 1158.01/295.34 1158.01/295.34 The TRS R consists of the following rules: 1158.01/295.34 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 S is empty. 1158.01/295.34 Rewrite Strategy: INNERMOST 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1158.01/295.34 Infered types. 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (6) 1158.01/295.34 Obligation: 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (7) OrderProof (LOWER BOUND(ID)) 1158.01/295.34 Heuristically decided to analyse the following defined symbols: 1158.01/295.34 p, intlist, int, if1 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 p < int 1158.01/295.34 intlist < int 1158.01/295.34 int = if1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (8) 1158.01/295.34 Obligation: 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 1158.01/295.34 Generator Equations: 1158.01/295.34 gen_nil:cons4_0(0) <=> nil 1158.01/295.34 gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x)) 1158.01/295.34 gen_0':s5_0(0) <=> 0' 1158.01/295.34 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1158.01/295.34 1158.01/295.34 1158.01/295.34 The following defined symbols remain to be analysed: 1158.01/295.34 p, intlist, int, if1 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 p < int 1158.01/295.34 intlist < int 1158.01/295.34 int = if1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (9) RewriteLemmaProof (LOWER BOUND(ID)) 1158.01/295.34 Proved the following rewrite lemma: 1158.01/295.34 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 1158.01/295.34 1158.01/295.34 Induction Base: 1158.01/295.34 p(gen_0':s5_0(+(1, 0))) ->_R^Omega(1) 1158.01/295.34 0' 1158.01/295.34 1158.01/295.34 Induction Step: 1158.01/295.34 p(gen_0':s5_0(+(1, +(n7_0, 1)))) ->_R^Omega(1) 1158.01/295.34 s(p(s(gen_0':s5_0(n7_0)))) ->_IH 1158.01/295.34 s(gen_0':s5_0(c8_0)) 1158.01/295.34 1158.01/295.34 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (10) 1158.01/295.34 Complex Obligation (BEST) 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (11) 1158.01/295.34 Obligation: 1158.01/295.34 Proved the lower bound n^1 for the following obligation: 1158.01/295.34 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 1158.01/295.34 Generator Equations: 1158.01/295.34 gen_nil:cons4_0(0) <=> nil 1158.01/295.34 gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x)) 1158.01/295.34 gen_0':s5_0(0) <=> 0' 1158.01/295.34 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1158.01/295.34 1158.01/295.34 1158.01/295.34 The following defined symbols remain to be analysed: 1158.01/295.34 p, intlist, int, if1 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 p < int 1158.01/295.34 intlist < int 1158.01/295.34 int = if1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (12) LowerBoundPropagationProof (FINISHED) 1158.01/295.34 Propagated lower bound. 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (13) 1158.01/295.34 BOUNDS(n^1, INF) 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (14) 1158.01/295.34 Obligation: 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 1158.01/295.34 Lemmas: 1158.01/295.34 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 1158.01/295.34 1158.01/295.34 1158.01/295.34 Generator Equations: 1158.01/295.34 gen_nil:cons4_0(0) <=> nil 1158.01/295.34 gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x)) 1158.01/295.34 gen_0':s5_0(0) <=> 0' 1158.01/295.34 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1158.01/295.34 1158.01/295.34 1158.01/295.34 The following defined symbols remain to be analysed: 1158.01/295.34 intlist, int, if1 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 intlist < int 1158.01/295.34 int = if1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1158.01/295.34 Proved the following rewrite lemma: 1158.01/295.34 intlist(gen_nil:cons4_0(n328_0)) -> *6_0, rt in Omega(n328_0) 1158.01/295.34 1158.01/295.34 Induction Base: 1158.01/295.34 intlist(gen_nil:cons4_0(0)) 1158.01/295.34 1158.01/295.34 Induction Step: 1158.01/295.34 intlist(gen_nil:cons4_0(+(n328_0, 1))) ->_R^Omega(1) 1158.01/295.34 if_intlist(empty(gen_nil:cons4_0(+(n328_0, 1))), gen_nil:cons4_0(+(n328_0, 1))) ->_R^Omega(1) 1158.01/295.34 if_intlist(false, gen_nil:cons4_0(+(1, n328_0))) ->_R^Omega(1) 1158.01/295.34 cons(s(head(gen_nil:cons4_0(+(1, n328_0)))), intlist(tail(gen_nil:cons4_0(+(1, n328_0))))) ->_R^Omega(1) 1158.01/295.34 cons(s(0'), intlist(tail(gen_nil:cons4_0(+(1, n328_0))))) ->_R^Omega(1) 1158.01/295.34 cons(s(0'), intlist(gen_nil:cons4_0(n328_0))) ->_IH 1158.01/295.34 cons(s(0'), *6_0) 1158.01/295.34 1158.01/295.34 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (16) 1158.01/295.34 Obligation: 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 1158.01/295.34 Lemmas: 1158.01/295.34 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 1158.01/295.34 intlist(gen_nil:cons4_0(n328_0)) -> *6_0, rt in Omega(n328_0) 1158.01/295.34 1158.01/295.34 1158.01/295.34 Generator Equations: 1158.01/295.34 gen_nil:cons4_0(0) <=> nil 1158.01/295.34 gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x)) 1158.01/295.34 gen_0':s5_0(0) <=> 0' 1158.01/295.34 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1158.01/295.34 1158.01/295.34 1158.01/295.34 The following defined symbols remain to be analysed: 1158.01/295.34 if1, int 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 int = if1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (17) RewriteLemmaProof (LOWER BOUND(ID)) 1158.01/295.34 Proved the following rewrite lemma: 1158.01/295.34 int(gen_0':s5_0(+(1, n5556_0)), gen_0':s5_0(n5556_0)) -> gen_nil:cons4_0(0), rt in Omega(1 + n5556_0 + n5556_0^2) 1158.01/295.34 1158.01/295.34 Induction Base: 1158.01/295.34 int(gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) ->_R^Omega(1) 1158.01/295.34 if_int(zero(gen_0':s5_0(+(1, 0))), zero(gen_0':s5_0(0)), gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) ->_R^Omega(1) 1158.01/295.34 if_int(false, zero(gen_0':s5_0(0)), gen_0':s5_0(1), gen_0':s5_0(0)) ->_R^Omega(1) 1158.01/295.34 if_int(false, true, gen_0':s5_0(1), gen_0':s5_0(0)) ->_R^Omega(1) 1158.01/295.34 if2(true, gen_0':s5_0(1), gen_0':s5_0(0)) ->_R^Omega(1) 1158.01/295.34 nil 1158.01/295.34 1158.01/295.34 Induction Step: 1158.01/295.34 int(gen_0':s5_0(+(1, +(n5556_0, 1))), gen_0':s5_0(+(n5556_0, 1))) ->_R^Omega(1) 1158.01/295.34 if_int(zero(gen_0':s5_0(+(1, +(n5556_0, 1)))), zero(gen_0':s5_0(+(n5556_0, 1))), gen_0':s5_0(+(1, +(n5556_0, 1))), gen_0':s5_0(+(n5556_0, 1))) ->_R^Omega(1) 1158.01/295.34 if_int(false, zero(gen_0':s5_0(+(1, n5556_0))), gen_0':s5_0(+(2, n5556_0)), gen_0':s5_0(+(1, n5556_0))) ->_R^Omega(1) 1158.01/295.34 if_int(false, false, gen_0':s5_0(+(2, n5556_0)), gen_0':s5_0(+(1, n5556_0))) ->_R^Omega(1) 1158.01/295.34 if2(false, gen_0':s5_0(+(2, n5556_0)), gen_0':s5_0(+(1, n5556_0))) ->_R^Omega(1) 1158.01/295.34 intlist(int(p(gen_0':s5_0(+(2, n5556_0))), p(gen_0':s5_0(+(1, n5556_0))))) ->_L^Omega(2 + n5556_0) 1158.01/295.34 intlist(int(gen_0':s5_0(+(1, n5556_0)), p(gen_0':s5_0(+(1, n5556_0))))) ->_L^Omega(1 + n5556_0) 1158.01/295.34 intlist(int(gen_0':s5_0(+(1, n5556_0)), gen_0':s5_0(n5556_0))) ->_IH 1158.01/295.34 intlist(gen_nil:cons4_0(0)) ->_R^Omega(1) 1158.01/295.34 if_intlist(empty(gen_nil:cons4_0(0)), gen_nil:cons4_0(0)) ->_R^Omega(1) 1158.01/295.34 if_intlist(true, gen_nil:cons4_0(0)) ->_R^Omega(1) 1158.01/295.34 nil 1158.01/295.34 1158.01/295.34 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (18) 1158.01/295.34 Complex Obligation (BEST) 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (19) 1158.01/295.34 Obligation: 1158.01/295.34 Proved the lower bound n^2 for the following obligation: 1158.01/295.34 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 1158.01/295.34 Lemmas: 1158.01/295.34 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 1158.01/295.34 intlist(gen_nil:cons4_0(n328_0)) -> *6_0, rt in Omega(n328_0) 1158.01/295.34 1158.01/295.34 1158.01/295.34 Generator Equations: 1158.01/295.34 gen_nil:cons4_0(0) <=> nil 1158.01/295.34 gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x)) 1158.01/295.34 gen_0':s5_0(0) <=> 0' 1158.01/295.34 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1158.01/295.34 1158.01/295.34 1158.01/295.34 The following defined symbols remain to be analysed: 1158.01/295.34 int 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 int = if1 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (20) LowerBoundPropagationProof (FINISHED) 1158.01/295.34 Propagated lower bound. 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (21) 1158.01/295.34 BOUNDS(n^2, INF) 1158.01/295.34 1158.01/295.34 ---------------------------------------- 1158.01/295.34 1158.01/295.34 (22) 1158.01/295.34 Obligation: 1158.01/295.34 Innermost TRS: 1158.01/295.34 Rules: 1158.01/295.34 empty(nil) -> true 1158.01/295.34 empty(cons(x, y)) -> false 1158.01/295.34 tail(nil) -> nil 1158.01/295.34 tail(cons(x, y)) -> y 1158.01/295.34 head(cons(x, y)) -> x 1158.01/295.34 zero(0') -> true 1158.01/295.34 zero(s(x)) -> false 1158.01/295.34 p(0') -> 0' 1158.01/295.34 p(s(0')) -> 0' 1158.01/295.34 p(s(s(x))) -> s(p(s(x))) 1158.01/295.34 intlist(x) -> if_intlist(empty(x), x) 1158.01/295.34 if_intlist(true, x) -> nil 1158.01/295.34 if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) 1158.01/295.34 int(x, y) -> if_int(zero(x), zero(y), x, y) 1158.01/295.34 if_int(true, b, x, y) -> if1(b, y) 1158.01/295.34 if_int(false, b, x, y) -> if2(b, x, y) 1158.01/295.34 if1(true, y) -> cons(0', nil) 1158.01/295.34 if1(false, y) -> cons(0', int(s(0'), y)) 1158.01/295.34 if2(true, x, y) -> nil 1158.01/295.34 if2(false, x, y) -> intlist(int(p(x), p(y))) 1158.01/295.34 1158.01/295.34 Types: 1158.01/295.34 empty :: nil:cons -> true:false 1158.01/295.34 nil :: nil:cons 1158.01/295.34 true :: true:false 1158.01/295.34 cons :: 0':s -> nil:cons -> nil:cons 1158.01/295.34 false :: true:false 1158.01/295.34 tail :: nil:cons -> nil:cons 1158.01/295.34 head :: nil:cons -> 0':s 1158.01/295.34 zero :: 0':s -> true:false 1158.01/295.34 0' :: 0':s 1158.01/295.34 s :: 0':s -> 0':s 1158.01/295.34 p :: 0':s -> 0':s 1158.01/295.34 intlist :: nil:cons -> nil:cons 1158.01/295.34 if_intlist :: true:false -> nil:cons -> nil:cons 1158.01/295.34 int :: 0':s -> 0':s -> nil:cons 1158.01/295.34 if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 if1 :: true:false -> 0':s -> nil:cons 1158.01/295.34 if2 :: true:false -> 0':s -> 0':s -> nil:cons 1158.01/295.34 hole_true:false1_0 :: true:false 1158.01/295.34 hole_nil:cons2_0 :: nil:cons 1158.01/295.34 hole_0':s3_0 :: 0':s 1158.01/295.34 gen_nil:cons4_0 :: Nat -> nil:cons 1158.01/295.34 gen_0':s5_0 :: Nat -> 0':s 1158.01/295.34 1158.01/295.34 1158.01/295.34 Lemmas: 1158.01/295.34 p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) 1158.01/295.34 intlist(gen_nil:cons4_0(n328_0)) -> *6_0, rt in Omega(n328_0) 1158.01/295.34 int(gen_0':s5_0(+(1, n5556_0)), gen_0':s5_0(n5556_0)) -> gen_nil:cons4_0(0), rt in Omega(1 + n5556_0 + n5556_0^2) 1158.01/295.34 1158.01/295.34 1158.01/295.34 Generator Equations: 1158.01/295.34 gen_nil:cons4_0(0) <=> nil 1158.01/295.34 gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x)) 1158.01/295.34 gen_0':s5_0(0) <=> 0' 1158.01/295.34 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1158.01/295.34 1158.01/295.34 1158.01/295.34 The following defined symbols remain to be analysed: 1158.01/295.34 if1 1158.01/295.34 1158.01/295.34 They will be analysed ascendingly in the following order: 1158.01/295.34 int = if1 1159.91/295.88 EOF