1127.48/291.52 WORST_CASE(Omega(n^1), ?) 1136.85/293.92 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1136.85/293.92 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1136.85/293.92 1136.85/293.92 1136.85/293.92 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1136.85/293.92 1136.85/293.92 (0) CpxTRS 1136.85/293.92 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1136.85/293.92 (2) CpxTRS 1136.85/293.92 (3) SlicingProof [LOWER BOUND(ID), 0 ms] 1136.85/293.92 (4) CpxTRS 1136.85/293.92 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1136.85/293.92 (6) typed CpxTrs 1136.85/293.92 (7) OrderProof [LOWER BOUND(ID), 0 ms] 1136.85/293.92 (8) typed CpxTrs 1136.85/293.92 (9) RewriteLemmaProof [LOWER BOUND(ID), 333 ms] 1136.85/293.92 (10) BEST 1136.85/293.92 (11) proven lower bound 1136.85/293.92 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 1136.85/293.92 (13) BOUNDS(n^1, INF) 1136.85/293.92 (14) typed CpxTrs 1136.85/293.92 (15) RewriteLemmaProof [LOWER BOUND(ID), 41 ms] 1136.85/293.92 (16) typed CpxTrs 1136.85/293.92 (17) RewriteLemmaProof [LOWER BOUND(ID), 53 ms] 1136.85/293.92 (18) typed CpxTrs 1136.85/293.92 1136.85/293.92 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (0) 1136.85/293.92 Obligation: 1136.85/293.92 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1136.85/293.92 1136.85/293.92 1136.85/293.92 The TRS R consists of the following rules: 1136.85/293.92 1136.85/293.92 eq(0, 0) -> true 1136.85/293.92 eq(0, s(y)) -> false 1136.85/293.92 eq(s(x), 0) -> false 1136.85/293.92 eq(s(x), s(y)) -> eq(x, y) 1136.85/293.92 lt(0, s(y)) -> true 1136.85/293.92 lt(x, 0) -> false 1136.85/293.92 lt(s(x), s(y)) -> lt(x, y) 1136.85/293.92 bin2s(nil) -> 0 1136.85/293.92 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1136.85/293.92 bin2ss(x, nil) -> x 1136.85/293.92 bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) 1136.85/293.92 bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) 1136.85/293.92 half(0) -> 0 1136.85/293.92 half(s(0)) -> 0 1136.85/293.92 half(s(s(x))) -> s(half(x)) 1136.85/293.92 log(0) -> 0 1136.85/293.92 log(s(0)) -> 0 1136.85/293.92 log(s(s(x))) -> s(log(half(s(s(x))))) 1136.85/293.92 more(nil) -> nil 1136.85/293.92 more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) 1136.85/293.92 s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) 1136.85/293.92 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1136.85/293.92 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1136.85/293.92 if1(false, x, y, lists) -> s2bin2(x, lists) 1136.85/293.92 s2bin2(x, nil) -> bug_list_not 1136.85/293.92 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1136.85/293.92 if2(true, x, xs, ys) -> xs 1136.85/293.92 if2(false, x, xs, ys) -> s2bin2(x, ys) 1136.85/293.92 1136.85/293.92 S is empty. 1136.85/293.92 Rewrite Strategy: INNERMOST 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1136.85/293.92 Renamed function symbols to avoid clashes with predefined symbol. 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (2) 1136.85/293.92 Obligation: 1136.85/293.92 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1136.85/293.92 1136.85/293.92 1136.85/293.92 The TRS R consists of the following rules: 1136.85/293.92 1136.85/293.92 eq(0', 0') -> true 1136.85/293.92 eq(0', s(y)) -> false 1136.85/293.92 eq(s(x), 0') -> false 1136.85/293.92 eq(s(x), s(y)) -> eq(x, y) 1136.85/293.92 lt(0', s(y)) -> true 1136.85/293.92 lt(x, 0') -> false 1136.85/293.92 lt(s(x), s(y)) -> lt(x, y) 1136.85/293.92 bin2s(nil) -> 0' 1136.85/293.92 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1136.85/293.92 bin2ss(x, nil) -> x 1136.85/293.92 bin2ss(x, cons(0', xs)) -> bin2ss(double(x), xs) 1136.85/293.92 bin2ss(x, cons(1', xs)) -> bin2ss(s(double(x)), xs) 1136.85/293.92 half(0') -> 0' 1136.85/293.92 half(s(0')) -> 0' 1136.85/293.92 half(s(s(x))) -> s(half(x)) 1136.85/293.92 log(0') -> 0' 1136.85/293.92 log(s(0')) -> 0' 1136.85/293.92 log(s(s(x))) -> s(log(half(s(s(x))))) 1136.85/293.92 more(nil) -> nil 1136.85/293.92 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1136.85/293.92 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1136.85/293.92 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1136.85/293.92 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1136.85/293.92 if1(false, x, y, lists) -> s2bin2(x, lists) 1136.85/293.92 s2bin2(x, nil) -> bug_list_not 1136.85/293.92 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1136.85/293.92 if2(true, x, xs, ys) -> xs 1136.85/293.92 if2(false, x, xs, ys) -> s2bin2(x, ys) 1136.85/293.92 1136.85/293.92 S is empty. 1136.85/293.92 Rewrite Strategy: INNERMOST 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (3) SlicingProof (LOWER BOUND(ID)) 1136.85/293.92 Sliced the following arguments: 1136.85/293.92 double/0 1136.85/293.92 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (4) 1136.85/293.92 Obligation: 1136.85/293.92 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1136.85/293.92 1136.85/293.92 1136.85/293.92 The TRS R consists of the following rules: 1136.85/293.92 1136.85/293.92 eq(0', 0') -> true 1136.85/293.92 eq(0', s(y)) -> false 1136.85/293.92 eq(s(x), 0') -> false 1136.85/293.92 eq(s(x), s(y)) -> eq(x, y) 1136.85/293.92 lt(0', s(y)) -> true 1136.85/293.92 lt(x, 0') -> false 1136.85/293.92 lt(s(x), s(y)) -> lt(x, y) 1136.85/293.92 bin2s(nil) -> 0' 1136.85/293.92 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1136.85/293.92 bin2ss(x, nil) -> x 1136.85/293.92 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1136.85/293.92 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1136.85/293.92 half(0') -> 0' 1136.85/293.92 half(s(0')) -> 0' 1136.85/293.92 half(s(s(x))) -> s(half(x)) 1136.85/293.92 log(0') -> 0' 1136.85/293.92 log(s(0')) -> 0' 1136.85/293.92 log(s(s(x))) -> s(log(half(s(s(x))))) 1136.85/293.92 more(nil) -> nil 1136.85/293.92 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1136.85/293.92 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1136.85/293.92 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1136.85/293.92 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1136.85/293.92 if1(false, x, y, lists) -> s2bin2(x, lists) 1136.85/293.92 s2bin2(x, nil) -> bug_list_not 1136.85/293.92 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1136.85/293.92 if2(true, x, xs, ys) -> xs 1136.85/293.92 if2(false, x, xs, ys) -> s2bin2(x, ys) 1136.85/293.92 1136.85/293.92 S is empty. 1136.85/293.92 Rewrite Strategy: INNERMOST 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1136.85/293.92 Infered types. 1136.85/293.92 ---------------------------------------- 1136.85/293.92 1136.85/293.92 (6) 1136.85/293.92 Obligation: 1136.85/293.92 Innermost TRS: 1136.85/293.92 Rules: 1136.85/293.92 eq(0', 0') -> true 1136.85/293.92 eq(0', s(y)) -> false 1136.85/293.92 eq(s(x), 0') -> false 1136.85/293.92 eq(s(x), s(y)) -> eq(x, y) 1136.85/293.92 lt(0', s(y)) -> true 1136.85/293.92 lt(x, 0') -> false 1136.85/293.92 lt(s(x), s(y)) -> lt(x, y) 1137.17/293.93 bin2s(nil) -> 0' 1137.17/293.93 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1137.17/293.93 bin2ss(x, nil) -> x 1137.17/293.93 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1137.17/293.93 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1137.17/293.93 half(0') -> 0' 1137.17/293.93 half(s(0')) -> 0' 1137.17/293.93 half(s(s(x))) -> s(half(x)) 1137.17/293.93 log(0') -> 0' 1137.17/293.93 log(s(0')) -> 0' 1137.17/293.93 log(s(s(x))) -> s(log(half(s(s(x))))) 1137.17/293.93 more(nil) -> nil 1137.17/293.93 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1137.17/293.93 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1137.17/293.93 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1137.17/293.93 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1137.17/293.93 if1(false, x, y, lists) -> s2bin2(x, lists) 1137.17/293.93 s2bin2(x, nil) -> bug_list_not 1137.17/293.93 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1137.17/293.93 if2(true, x, xs, ys) -> xs 1137.17/293.93 if2(false, x, xs, ys) -> s2bin2(x, ys) 1137.17/293.93 1137.17/293.93 Types: 1137.17/293.93 eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 0' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 true :: true:false 1137.17/293.93 s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 false :: true:false 1137.17/293.93 lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 nil :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 double :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bug_list_not :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 hole_true:false1_0 :: true:false 1137.17/293.93 hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (7) OrderProof (LOWER BOUND(ID)) 1137.17/293.93 Heuristically decided to analyse the following defined symbols: 1137.17/293.93 eq, lt, bin2ss, half, log, s2bin1, s2bin2 1137.17/293.93 1137.17/293.93 They will be analysed ascendingly in the following order: 1137.17/293.93 eq < s2bin2 1137.17/293.93 lt < s2bin1 1137.17/293.93 half < log 1137.17/293.93 log < s2bin1 1137.17/293.93 s2bin2 < s2bin1 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (8) 1137.17/293.93 Obligation: 1137.17/293.93 Innermost TRS: 1137.17/293.93 Rules: 1137.17/293.93 eq(0', 0') -> true 1137.17/293.93 eq(0', s(y)) -> false 1137.17/293.93 eq(s(x), 0') -> false 1137.17/293.93 eq(s(x), s(y)) -> eq(x, y) 1137.17/293.93 lt(0', s(y)) -> true 1137.17/293.93 lt(x, 0') -> false 1137.17/293.93 lt(s(x), s(y)) -> lt(x, y) 1137.17/293.93 bin2s(nil) -> 0' 1137.17/293.93 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1137.17/293.93 bin2ss(x, nil) -> x 1137.17/293.93 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1137.17/293.93 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1137.17/293.93 half(0') -> 0' 1137.17/293.93 half(s(0')) -> 0' 1137.17/293.93 half(s(s(x))) -> s(half(x)) 1137.17/293.93 log(0') -> 0' 1137.17/293.93 log(s(0')) -> 0' 1137.17/293.93 log(s(s(x))) -> s(log(half(s(s(x))))) 1137.17/293.93 more(nil) -> nil 1137.17/293.93 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1137.17/293.93 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1137.17/293.93 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1137.17/293.93 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1137.17/293.93 if1(false, x, y, lists) -> s2bin2(x, lists) 1137.17/293.93 s2bin2(x, nil) -> bug_list_not 1137.17/293.93 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1137.17/293.93 if2(true, x, xs, ys) -> xs 1137.17/293.93 if2(false, x, xs, ys) -> s2bin2(x, ys) 1137.17/293.93 1137.17/293.93 Types: 1137.17/293.93 eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 0' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 true :: true:false 1137.17/293.93 s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 false :: true:false 1137.17/293.93 lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 nil :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 double :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bug_list_not :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 hole_true:false1_0 :: true:false 1137.17/293.93 hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1137.17/293.93 1137.17/293.93 Generator Equations: 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) 1137.17/293.93 1137.17/293.93 1137.17/293.93 The following defined symbols remain to be analysed: 1137.17/293.93 eq, lt, bin2ss, half, log, s2bin1, s2bin2 1137.17/293.93 1137.17/293.93 They will be analysed ascendingly in the following order: 1137.17/293.93 eq < s2bin2 1137.17/293.93 lt < s2bin1 1137.17/293.93 half < log 1137.17/293.93 log < s2bin1 1137.17/293.93 s2bin2 < s2bin1 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (9) RewriteLemmaProof (LOWER BOUND(ID)) 1137.17/293.93 Proved the following rewrite lemma: 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 1137.17/293.93 1137.17/293.93 Induction Base: 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(0), gen_0':s:nil:cons:double:1':bug_list_not3_0(0)) ->_R^Omega(1) 1137.17/293.93 true 1137.17/293.93 1137.17/293.93 Induction Step: 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n5_0, 1)), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n5_0, 1))) ->_R^Omega(1) 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) ->_IH 1137.17/293.93 true 1137.17/293.93 1137.17/293.93 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (10) 1137.17/293.93 Complex Obligation (BEST) 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (11) 1137.17/293.93 Obligation: 1137.17/293.93 Proved the lower bound n^1 for the following obligation: 1137.17/293.93 1137.17/293.93 Innermost TRS: 1137.17/293.93 Rules: 1137.17/293.93 eq(0', 0') -> true 1137.17/293.93 eq(0', s(y)) -> false 1137.17/293.93 eq(s(x), 0') -> false 1137.17/293.93 eq(s(x), s(y)) -> eq(x, y) 1137.17/293.93 lt(0', s(y)) -> true 1137.17/293.93 lt(x, 0') -> false 1137.17/293.93 lt(s(x), s(y)) -> lt(x, y) 1137.17/293.93 bin2s(nil) -> 0' 1137.17/293.93 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1137.17/293.93 bin2ss(x, nil) -> x 1137.17/293.93 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1137.17/293.93 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1137.17/293.93 half(0') -> 0' 1137.17/293.93 half(s(0')) -> 0' 1137.17/293.93 half(s(s(x))) -> s(half(x)) 1137.17/293.93 log(0') -> 0' 1137.17/293.93 log(s(0')) -> 0' 1137.17/293.93 log(s(s(x))) -> s(log(half(s(s(x))))) 1137.17/293.93 more(nil) -> nil 1137.17/293.93 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1137.17/293.93 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1137.17/293.93 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1137.17/293.93 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1137.17/293.93 if1(false, x, y, lists) -> s2bin2(x, lists) 1137.17/293.93 s2bin2(x, nil) -> bug_list_not 1137.17/293.93 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1137.17/293.93 if2(true, x, xs, ys) -> xs 1137.17/293.93 if2(false, x, xs, ys) -> s2bin2(x, ys) 1137.17/293.93 1137.17/293.93 Types: 1137.17/293.93 eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 0' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 true :: true:false 1137.17/293.93 s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 false :: true:false 1137.17/293.93 lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 nil :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 double :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bug_list_not :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 hole_true:false1_0 :: true:false 1137.17/293.93 hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1137.17/293.93 1137.17/293.93 Generator Equations: 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) 1137.17/293.93 1137.17/293.93 1137.17/293.93 The following defined symbols remain to be analysed: 1137.17/293.93 eq, lt, bin2ss, half, log, s2bin1, s2bin2 1137.17/293.93 1137.17/293.93 They will be analysed ascendingly in the following order: 1137.17/293.93 eq < s2bin2 1137.17/293.93 lt < s2bin1 1137.17/293.93 half < log 1137.17/293.93 log < s2bin1 1137.17/293.93 s2bin2 < s2bin1 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (12) LowerBoundPropagationProof (FINISHED) 1137.17/293.93 Propagated lower bound. 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (13) 1137.17/293.93 BOUNDS(n^1, INF) 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (14) 1137.17/293.93 Obligation: 1137.17/293.93 Innermost TRS: 1137.17/293.93 Rules: 1137.17/293.93 eq(0', 0') -> true 1137.17/293.93 eq(0', s(y)) -> false 1137.17/293.93 eq(s(x), 0') -> false 1137.17/293.93 eq(s(x), s(y)) -> eq(x, y) 1137.17/293.93 lt(0', s(y)) -> true 1137.17/293.93 lt(x, 0') -> false 1137.17/293.93 lt(s(x), s(y)) -> lt(x, y) 1137.17/293.93 bin2s(nil) -> 0' 1137.17/293.93 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1137.17/293.93 bin2ss(x, nil) -> x 1137.17/293.93 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1137.17/293.93 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1137.17/293.93 half(0') -> 0' 1137.17/293.93 half(s(0')) -> 0' 1137.17/293.93 half(s(s(x))) -> s(half(x)) 1137.17/293.93 log(0') -> 0' 1137.17/293.93 log(s(0')) -> 0' 1137.17/293.93 log(s(s(x))) -> s(log(half(s(s(x))))) 1137.17/293.93 more(nil) -> nil 1137.17/293.93 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1137.17/293.93 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1137.17/293.93 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1137.17/293.93 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1137.17/293.93 if1(false, x, y, lists) -> s2bin2(x, lists) 1137.17/293.93 s2bin2(x, nil) -> bug_list_not 1137.17/293.93 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1137.17/293.93 if2(true, x, xs, ys) -> xs 1137.17/293.93 if2(false, x, xs, ys) -> s2bin2(x, ys) 1137.17/293.93 1137.17/293.93 Types: 1137.17/293.93 eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 0' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 true :: true:false 1137.17/293.93 s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 false :: true:false 1137.17/293.93 lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 nil :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 double :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bug_list_not :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 hole_true:false1_0 :: true:false 1137.17/293.93 hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1137.17/293.93 1137.17/293.93 Lemmas: 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 1137.17/293.93 1137.17/293.93 1137.17/293.93 Generator Equations: 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) 1137.17/293.93 1137.17/293.93 1137.17/293.93 The following defined symbols remain to be analysed: 1137.17/293.93 lt, bin2ss, half, log, s2bin1, s2bin2 1137.17/293.93 1137.17/293.93 They will be analysed ascendingly in the following order: 1137.17/293.93 lt < s2bin1 1137.17/293.93 half < log 1137.17/293.93 log < s2bin1 1137.17/293.93 s2bin2 < s2bin1 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1137.17/293.93 Proved the following rewrite lemma: 1137.17/293.93 lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n698_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n698_0))) -> true, rt in Omega(1 + n698_0) 1137.17/293.93 1137.17/293.93 Induction Base: 1137.17/293.93 lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, 0))) ->_R^Omega(1) 1137.17/293.93 true 1137.17/293.93 1137.17/293.93 Induction Step: 1137.17/293.93 lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n698_0, 1)), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, +(n698_0, 1)))) ->_R^Omega(1) 1137.17/293.93 lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n698_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n698_0))) ->_IH 1137.17/293.93 true 1137.17/293.93 1137.17/293.93 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (16) 1137.17/293.93 Obligation: 1137.17/293.93 Innermost TRS: 1137.17/293.93 Rules: 1137.17/293.93 eq(0', 0') -> true 1137.17/293.93 eq(0', s(y)) -> false 1137.17/293.93 eq(s(x), 0') -> false 1137.17/293.93 eq(s(x), s(y)) -> eq(x, y) 1137.17/293.93 lt(0', s(y)) -> true 1137.17/293.93 lt(x, 0') -> false 1137.17/293.93 lt(s(x), s(y)) -> lt(x, y) 1137.17/293.93 bin2s(nil) -> 0' 1137.17/293.93 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1137.17/293.93 bin2ss(x, nil) -> x 1137.17/293.93 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1137.17/293.93 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1137.17/293.93 half(0') -> 0' 1137.17/293.93 half(s(0')) -> 0' 1137.17/293.93 half(s(s(x))) -> s(half(x)) 1137.17/293.93 log(0') -> 0' 1137.17/293.93 log(s(0')) -> 0' 1137.17/293.93 log(s(s(x))) -> s(log(half(s(s(x))))) 1137.17/293.93 more(nil) -> nil 1137.17/293.93 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1137.17/293.93 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1137.17/293.93 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1137.17/293.93 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1137.17/293.93 if1(false, x, y, lists) -> s2bin2(x, lists) 1137.17/293.93 s2bin2(x, nil) -> bug_list_not 1137.17/293.93 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1137.17/293.93 if2(true, x, xs, ys) -> xs 1137.17/293.93 if2(false, x, xs, ys) -> s2bin2(x, ys) 1137.17/293.93 1137.17/293.93 Types: 1137.17/293.93 eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 0' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 true :: true:false 1137.17/293.93 s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 false :: true:false 1137.17/293.93 lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 nil :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 double :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bug_list_not :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 hole_true:false1_0 :: true:false 1137.17/293.93 hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1137.17/293.93 1137.17/293.93 Lemmas: 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 1137.17/293.93 lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n698_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n698_0))) -> true, rt in Omega(1 + n698_0) 1137.17/293.93 1137.17/293.93 1137.17/293.93 Generator Equations: 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) 1137.17/293.93 1137.17/293.93 1137.17/293.93 The following defined symbols remain to be analysed: 1137.17/293.93 bin2ss, half, log, s2bin1, s2bin2 1137.17/293.93 1137.17/293.93 They will be analysed ascendingly in the following order: 1137.17/293.93 half < log 1137.17/293.93 log < s2bin1 1137.17/293.93 s2bin2 < s2bin1 1137.17/293.93 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (17) RewriteLemmaProof (LOWER BOUND(ID)) 1137.17/293.93 Proved the following rewrite lemma: 1137.17/293.93 half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1204_0))) -> gen_0':s:nil:cons:double:1':bug_list_not3_0(n1204_0), rt in Omega(1 + n1204_0) 1137.17/293.93 1137.17/293.93 Induction Base: 1137.17/293.93 half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, 0))) ->_R^Omega(1) 1137.17/293.93 0' 1137.17/293.93 1137.17/293.93 Induction Step: 1137.17/293.93 half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, +(n1204_0, 1)))) ->_R^Omega(1) 1137.17/293.93 s(half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1204_0)))) ->_IH 1137.17/293.93 s(gen_0':s:nil:cons:double:1':bug_list_not3_0(c1205_0)) 1137.17/293.93 1137.17/293.93 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1137.17/293.93 ---------------------------------------- 1137.17/293.93 1137.17/293.93 (18) 1137.17/293.93 Obligation: 1137.17/293.93 Innermost TRS: 1137.17/293.93 Rules: 1137.17/293.93 eq(0', 0') -> true 1137.17/293.93 eq(0', s(y)) -> false 1137.17/293.93 eq(s(x), 0') -> false 1137.17/293.93 eq(s(x), s(y)) -> eq(x, y) 1137.17/293.93 lt(0', s(y)) -> true 1137.17/293.93 lt(x, 0') -> false 1137.17/293.93 lt(s(x), s(y)) -> lt(x, y) 1137.17/293.93 bin2s(nil) -> 0' 1137.17/293.93 bin2s(cons(x, xs)) -> bin2ss(x, xs) 1137.17/293.93 bin2ss(x, nil) -> x 1137.17/293.93 bin2ss(x, cons(0', xs)) -> bin2ss(double, xs) 1137.17/293.93 bin2ss(x, cons(1', xs)) -> bin2ss(s(double), xs) 1137.17/293.93 half(0') -> 0' 1137.17/293.93 half(s(0')) -> 0' 1137.17/293.93 half(s(s(x))) -> s(half(x)) 1137.17/293.93 log(0') -> 0' 1137.17/293.93 log(s(0')) -> 0' 1137.17/293.93 log(s(s(x))) -> s(log(half(s(s(x))))) 1137.17/293.93 more(nil) -> nil 1137.17/293.93 more(cons(xs, ys)) -> cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys))) 1137.17/293.93 s2bin(x) -> s2bin1(x, 0', cons(nil, nil)) 1137.17/293.93 s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) 1137.17/293.93 if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) 1137.17/293.93 if1(false, x, y, lists) -> s2bin2(x, lists) 1137.17/293.93 s2bin2(x, nil) -> bug_list_not 1137.17/293.93 s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) 1137.17/293.93 if2(true, x, xs, ys) -> xs 1137.17/293.93 if2(false, x, xs, ys) -> s2bin2(x, ys) 1137.17/293.93 1137.17/293.93 Types: 1137.17/293.93 eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 0' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 true :: true:false 1137.17/293.93 s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 false :: true:false 1137.17/293.93 lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 1137.17/293.93 bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 nil :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 double :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1' :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 bug_list_not :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 hole_true:false1_0 :: true:false 1137.17/293.93 hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not 1137.17/293.93 1137.17/293.93 1137.17/293.93 Lemmas: 1137.17/293.93 eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 1137.17/293.93 lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n698_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n698_0))) -> true, rt in Omega(1 + n698_0) 1137.17/293.93 half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1204_0))) -> gen_0':s:nil:cons:double:1':bug_list_not3_0(n1204_0), rt in Omega(1 + n1204_0) 1137.17/293.93 1137.17/293.93 1137.17/293.93 Generator Equations: 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' 1137.17/293.93 gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) 1137.17/293.93 1137.17/293.93 1137.17/293.93 The following defined symbols remain to be analysed: 1137.17/293.93 log, s2bin1, s2bin2 1137.17/293.93 1137.17/293.93 They will be analysed ascendingly in the following order: 1137.17/293.93 log < s2bin1 1137.17/293.93 s2bin2 < s2bin1 1137.27/294.00 EOF