305.75/291.59 WORST_CASE(Omega(n^2), ?) 305.75/291.60 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 305.75/291.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.75/291.60 305.75/291.60 305.75/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.75/291.60 305.75/291.60 (0) CpxTRS 305.75/291.60 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 305.75/291.60 (2) CpxTRS 305.75/291.60 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 305.75/291.60 (4) typed CpxTrs 305.75/291.60 (5) OrderProof [LOWER BOUND(ID), 0 ms] 305.75/291.60 (6) typed CpxTrs 305.75/291.60 (7) RewriteLemmaProof [LOWER BOUND(ID), 307 ms] 305.75/291.60 (8) BEST 305.75/291.60 (9) proven lower bound 305.75/291.60 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 305.75/291.60 (11) BOUNDS(n^1, INF) 305.75/291.60 (12) typed CpxTrs 305.75/291.60 (13) RewriteLemmaProof [LOWER BOUND(ID), 42 ms] 305.75/291.60 (14) typed CpxTrs 305.75/291.60 (15) RewriteLemmaProof [LOWER BOUND(ID), 19 ms] 305.75/291.60 (16) typed CpxTrs 305.75/291.60 (17) RewriteLemmaProof [LOWER BOUND(ID), 50 ms] 305.75/291.60 (18) proven lower bound 305.75/291.60 (19) LowerBoundPropagationProof [FINISHED, 0 ms] 305.75/291.60 (20) BOUNDS(n^2, INF) 305.75/291.60 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (0) 305.75/291.60 Obligation: 305.75/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.75/291.60 305.75/291.60 305.75/291.60 The TRS R consists of the following rules: 305.75/291.60 305.75/291.60 eq(0, 0) -> true 305.75/291.60 eq(0, s(Y)) -> false 305.75/291.60 eq(s(X), 0) -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0, Y) -> true 305.75/291.60 le(s(X), 0) -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0, nil)) -> 0 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 S is empty. 305.75/291.60 Rewrite Strategy: FULL 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 305.75/291.60 Renamed function symbols to avoid clashes with predefined symbol. 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (2) 305.75/291.60 Obligation: 305.75/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 305.75/291.60 305.75/291.60 305.75/291.60 The TRS R consists of the following rules: 305.75/291.60 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 S is empty. 305.75/291.60 Rewrite Strategy: FULL 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 305.75/291.60 Infered types. 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (4) 305.75/291.60 Obligation: 305.75/291.60 TRS: 305.75/291.60 Rules: 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 Types: 305.75/291.60 eq :: 0':s -> 0':s -> true:false 305.75/291.60 0' :: 0':s 305.75/291.60 true :: true:false 305.75/291.60 s :: 0':s -> 0':s 305.75/291.60 false :: true:false 305.75/291.60 le :: 0':s -> 0':s -> true:false 305.75/291.60 min :: nil:cons -> 0':s 305.75/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.75/291.60 nil :: nil:cons 305.75/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.75/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 selsort :: nil:cons -> nil:cons 305.75/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.75/291.60 hole_true:false1_0 :: true:false 305.75/291.60 hole_0':s2_0 :: 0':s 305.75/291.60 hole_nil:cons3_0 :: nil:cons 305.75/291.60 gen_0':s4_0 :: Nat -> 0':s 305.75/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (5) OrderProof (LOWER BOUND(ID)) 305.75/291.60 Heuristically decided to analyse the following defined symbols: 305.75/291.60 eq, le, min, replace, selsort 305.75/291.60 305.75/291.60 They will be analysed ascendingly in the following order: 305.75/291.60 eq < replace 305.75/291.60 eq < selsort 305.75/291.60 le < min 305.75/291.60 min < selsort 305.75/291.60 replace < selsort 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (6) 305.75/291.60 Obligation: 305.75/291.60 TRS: 305.75/291.60 Rules: 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 Types: 305.75/291.60 eq :: 0':s -> 0':s -> true:false 305.75/291.60 0' :: 0':s 305.75/291.60 true :: true:false 305.75/291.60 s :: 0':s -> 0':s 305.75/291.60 false :: true:false 305.75/291.60 le :: 0':s -> 0':s -> true:false 305.75/291.60 min :: nil:cons -> 0':s 305.75/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.75/291.60 nil :: nil:cons 305.75/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.75/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 selsort :: nil:cons -> nil:cons 305.75/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.75/291.60 hole_true:false1_0 :: true:false 305.75/291.60 hole_0':s2_0 :: 0':s 305.75/291.60 hole_nil:cons3_0 :: nil:cons 305.75/291.60 gen_0':s4_0 :: Nat -> 0':s 305.75/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.75/291.60 305.75/291.60 305.75/291.60 Generator Equations: 305.75/291.60 gen_0':s4_0(0) <=> 0' 305.75/291.60 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.75/291.60 gen_nil:cons5_0(0) <=> nil 305.75/291.60 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.75/291.60 305.75/291.60 305.75/291.60 The following defined symbols remain to be analysed: 305.75/291.60 eq, le, min, replace, selsort 305.75/291.60 305.75/291.60 They will be analysed ascendingly in the following order: 305.75/291.60 eq < replace 305.75/291.60 eq < selsort 305.75/291.60 le < min 305.75/291.60 min < selsort 305.75/291.60 replace < selsort 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (7) RewriteLemmaProof (LOWER BOUND(ID)) 305.75/291.60 Proved the following rewrite lemma: 305.75/291.60 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.75/291.60 305.75/291.60 Induction Base: 305.75/291.60 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 305.75/291.60 true 305.75/291.60 305.75/291.60 Induction Step: 305.75/291.60 eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1) 305.75/291.60 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH 305.75/291.60 true 305.75/291.60 305.75/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (8) 305.75/291.60 Complex Obligation (BEST) 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (9) 305.75/291.60 Obligation: 305.75/291.60 Proved the lower bound n^1 for the following obligation: 305.75/291.60 305.75/291.60 TRS: 305.75/291.60 Rules: 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 Types: 305.75/291.60 eq :: 0':s -> 0':s -> true:false 305.75/291.60 0' :: 0':s 305.75/291.60 true :: true:false 305.75/291.60 s :: 0':s -> 0':s 305.75/291.60 false :: true:false 305.75/291.60 le :: 0':s -> 0':s -> true:false 305.75/291.60 min :: nil:cons -> 0':s 305.75/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.75/291.60 nil :: nil:cons 305.75/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.75/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 selsort :: nil:cons -> nil:cons 305.75/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.75/291.60 hole_true:false1_0 :: true:false 305.75/291.60 hole_0':s2_0 :: 0':s 305.75/291.60 hole_nil:cons3_0 :: nil:cons 305.75/291.60 gen_0':s4_0 :: Nat -> 0':s 305.75/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.75/291.60 305.75/291.60 305.75/291.60 Generator Equations: 305.75/291.60 gen_0':s4_0(0) <=> 0' 305.75/291.60 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.75/291.60 gen_nil:cons5_0(0) <=> nil 305.75/291.60 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.75/291.60 305.75/291.60 305.75/291.60 The following defined symbols remain to be analysed: 305.75/291.60 eq, le, min, replace, selsort 305.75/291.60 305.75/291.60 They will be analysed ascendingly in the following order: 305.75/291.60 eq < replace 305.75/291.60 eq < selsort 305.75/291.60 le < min 305.75/291.60 min < selsort 305.75/291.60 replace < selsort 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (10) LowerBoundPropagationProof (FINISHED) 305.75/291.60 Propagated lower bound. 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (11) 305.75/291.60 BOUNDS(n^1, INF) 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (12) 305.75/291.60 Obligation: 305.75/291.60 TRS: 305.75/291.60 Rules: 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 Types: 305.75/291.60 eq :: 0':s -> 0':s -> true:false 305.75/291.60 0' :: 0':s 305.75/291.60 true :: true:false 305.75/291.60 s :: 0':s -> 0':s 305.75/291.60 false :: true:false 305.75/291.60 le :: 0':s -> 0':s -> true:false 305.75/291.60 min :: nil:cons -> 0':s 305.75/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.75/291.60 nil :: nil:cons 305.75/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.75/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 selsort :: nil:cons -> nil:cons 305.75/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.75/291.60 hole_true:false1_0 :: true:false 305.75/291.60 hole_0':s2_0 :: 0':s 305.75/291.60 hole_nil:cons3_0 :: nil:cons 305.75/291.60 gen_0':s4_0 :: Nat -> 0':s 305.75/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.75/291.60 305.75/291.60 305.75/291.60 Lemmas: 305.75/291.60 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.75/291.60 305.75/291.60 305.75/291.60 Generator Equations: 305.75/291.60 gen_0':s4_0(0) <=> 0' 305.75/291.60 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.75/291.60 gen_nil:cons5_0(0) <=> nil 305.75/291.60 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.75/291.60 305.75/291.60 305.75/291.60 The following defined symbols remain to be analysed: 305.75/291.60 le, min, replace, selsort 305.75/291.60 305.75/291.60 They will be analysed ascendingly in the following order: 305.75/291.60 le < min 305.75/291.60 min < selsort 305.75/291.60 replace < selsort 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (13) RewriteLemmaProof (LOWER BOUND(ID)) 305.75/291.60 Proved the following rewrite lemma: 305.75/291.60 le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_0) 305.75/291.60 305.75/291.60 Induction Base: 305.75/291.60 le(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 305.75/291.60 true 305.75/291.60 305.75/291.60 Induction Step: 305.75/291.60 le(gen_0':s4_0(+(n530_0, 1)), gen_0':s4_0(+(n530_0, 1))) ->_R^Omega(1) 305.75/291.60 le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) ->_IH 305.75/291.60 true 305.75/291.60 305.75/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (14) 305.75/291.60 Obligation: 305.75/291.60 TRS: 305.75/291.60 Rules: 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 Types: 305.75/291.60 eq :: 0':s -> 0':s -> true:false 305.75/291.60 0' :: 0':s 305.75/291.60 true :: true:false 305.75/291.60 s :: 0':s -> 0':s 305.75/291.60 false :: true:false 305.75/291.60 le :: 0':s -> 0':s -> true:false 305.75/291.60 min :: nil:cons -> 0':s 305.75/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.75/291.60 nil :: nil:cons 305.75/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.75/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 selsort :: nil:cons -> nil:cons 305.75/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.75/291.60 hole_true:false1_0 :: true:false 305.75/291.60 hole_0':s2_0 :: 0':s 305.75/291.60 hole_nil:cons3_0 :: nil:cons 305.75/291.60 gen_0':s4_0 :: Nat -> 0':s 305.75/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.75/291.60 305.75/291.60 305.75/291.60 Lemmas: 305.75/291.60 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.75/291.60 le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_0) 305.75/291.60 305.75/291.60 305.75/291.60 Generator Equations: 305.75/291.60 gen_0':s4_0(0) <=> 0' 305.75/291.60 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.75/291.60 gen_nil:cons5_0(0) <=> nil 305.75/291.60 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.75/291.60 305.75/291.60 305.75/291.60 The following defined symbols remain to be analysed: 305.75/291.60 min, replace, selsort 305.75/291.60 305.75/291.60 They will be analysed ascendingly in the following order: 305.75/291.60 min < selsort 305.75/291.60 replace < selsort 305.75/291.60 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (15) RewriteLemmaProof (LOWER BOUND(ID)) 305.75/291.60 Proved the following rewrite lemma: 305.75/291.60 min(gen_nil:cons5_0(+(1, n847_0))) -> gen_0':s4_0(0), rt in Omega(1 + n847_0) 305.75/291.60 305.75/291.60 Induction Base: 305.75/291.60 min(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1) 305.75/291.60 0' 305.75/291.60 305.75/291.60 Induction Step: 305.75/291.60 min(gen_nil:cons5_0(+(1, +(n847_0, 1)))) ->_R^Omega(1) 305.75/291.60 ifmin(le(0', 0'), cons(0', cons(0', gen_nil:cons5_0(n847_0)))) ->_L^Omega(1) 305.75/291.60 ifmin(true, cons(0', cons(0', gen_nil:cons5_0(n847_0)))) ->_R^Omega(1) 305.75/291.60 min(cons(0', gen_nil:cons5_0(n847_0))) ->_IH 305.75/291.60 gen_0':s4_0(0) 305.75/291.60 305.75/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 305.75/291.60 ---------------------------------------- 305.75/291.60 305.75/291.60 (16) 305.75/291.60 Obligation: 305.75/291.60 TRS: 305.75/291.60 Rules: 305.75/291.60 eq(0', 0') -> true 305.75/291.60 eq(0', s(Y)) -> false 305.75/291.60 eq(s(X), 0') -> false 305.75/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.75/291.60 le(0', Y) -> true 305.75/291.60 le(s(X), 0') -> false 305.75/291.60 le(s(X), s(Y)) -> le(X, Y) 305.75/291.60 min(cons(0', nil)) -> 0' 305.75/291.60 min(cons(s(N), nil)) -> s(N) 305.75/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.75/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.75/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.75/291.60 replace(N, M, nil) -> nil 305.75/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.75/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.75/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.75/291.60 selsort(nil) -> nil 305.75/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.75/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.75/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.75/291.60 305.75/291.60 Types: 305.75/291.60 eq :: 0':s -> 0':s -> true:false 305.75/291.60 0' :: 0':s 305.75/291.60 true :: true:false 305.75/291.60 s :: 0':s -> 0':s 305.75/291.60 false :: true:false 305.75/291.60 le :: 0':s -> 0':s -> true:false 305.75/291.60 min :: nil:cons -> 0':s 305.75/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.75/291.60 nil :: nil:cons 305.75/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.75/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.75/291.60 selsort :: nil:cons -> nil:cons 305.75/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.75/291.60 hole_true:false1_0 :: true:false 305.75/291.60 hole_0':s2_0 :: 0':s 305.75/291.60 hole_nil:cons3_0 :: nil:cons 305.75/291.60 gen_0':s4_0 :: Nat -> 0':s 305.75/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.75/291.60 305.75/291.60 305.75/291.60 Lemmas: 305.75/291.60 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.75/291.60 le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_0) 305.75/291.60 min(gen_nil:cons5_0(+(1, n847_0))) -> gen_0':s4_0(0), rt in Omega(1 + n847_0) 305.85/291.60 305.85/291.60 305.85/291.60 Generator Equations: 305.85/291.60 gen_0':s4_0(0) <=> 0' 305.85/291.60 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.85/291.60 gen_nil:cons5_0(0) <=> nil 305.85/291.60 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.85/291.60 305.85/291.60 305.85/291.60 The following defined symbols remain to be analysed: 305.85/291.60 replace, selsort 305.85/291.60 305.85/291.60 They will be analysed ascendingly in the following order: 305.85/291.60 replace < selsort 305.85/291.60 305.85/291.60 ---------------------------------------- 305.85/291.60 305.85/291.60 (17) RewriteLemmaProof (LOWER BOUND(ID)) 305.85/291.60 Proved the following rewrite lemma: 305.85/291.60 selsort(gen_nil:cons5_0(n1534_0)) -> gen_nil:cons5_0(n1534_0), rt in Omega(1 + n1534_0 + n1534_0^2) 305.85/291.60 305.85/291.60 Induction Base: 305.85/291.60 selsort(gen_nil:cons5_0(0)) ->_R^Omega(1) 305.85/291.60 nil 305.85/291.60 305.85/291.60 Induction Step: 305.85/291.60 selsort(gen_nil:cons5_0(+(n1534_0, 1))) ->_R^Omega(1) 305.85/291.60 ifselsort(eq(0', min(cons(0', gen_nil:cons5_0(n1534_0)))), cons(0', gen_nil:cons5_0(n1534_0))) ->_L^Omega(1 + n1534_0) 305.85/291.60 ifselsort(eq(0', gen_0':s4_0(0)), cons(0', gen_nil:cons5_0(n1534_0))) ->_L^Omega(1) 305.85/291.60 ifselsort(true, cons(0', gen_nil:cons5_0(n1534_0))) ->_R^Omega(1) 305.85/291.60 cons(0', selsort(gen_nil:cons5_0(n1534_0))) ->_IH 305.85/291.60 cons(0', gen_nil:cons5_0(c1535_0)) 305.85/291.60 305.85/291.60 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 305.85/291.60 ---------------------------------------- 305.85/291.60 305.85/291.60 (18) 305.85/291.60 Obligation: 305.85/291.60 Proved the lower bound n^2 for the following obligation: 305.85/291.60 305.85/291.60 TRS: 305.85/291.60 Rules: 305.85/291.60 eq(0', 0') -> true 305.85/291.60 eq(0', s(Y)) -> false 305.85/291.60 eq(s(X), 0') -> false 305.85/291.60 eq(s(X), s(Y)) -> eq(X, Y) 305.85/291.60 le(0', Y) -> true 305.85/291.60 le(s(X), 0') -> false 305.85/291.60 le(s(X), s(Y)) -> le(X, Y) 305.85/291.60 min(cons(0', nil)) -> 0' 305.85/291.60 min(cons(s(N), nil)) -> s(N) 305.85/291.60 min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) 305.85/291.60 ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) 305.85/291.60 ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) 305.85/291.60 replace(N, M, nil) -> nil 305.85/291.60 replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) 305.85/291.60 ifrepl(true, N, M, cons(K, L)) -> cons(M, L) 305.85/291.60 ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) 305.85/291.60 selsort(nil) -> nil 305.85/291.60 selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) 305.85/291.60 ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) 305.85/291.60 ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 305.85/291.60 305.85/291.60 Types: 305.85/291.60 eq :: 0':s -> 0':s -> true:false 305.85/291.60 0' :: 0':s 305.85/291.60 true :: true:false 305.85/291.60 s :: 0':s -> 0':s 305.85/291.60 false :: true:false 305.85/291.60 le :: 0':s -> 0':s -> true:false 305.85/291.60 min :: nil:cons -> 0':s 305.85/291.60 cons :: 0':s -> nil:cons -> nil:cons 305.85/291.60 nil :: nil:cons 305.85/291.60 ifmin :: true:false -> nil:cons -> 0':s 305.85/291.60 replace :: 0':s -> 0':s -> nil:cons -> nil:cons 305.85/291.60 ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 305.85/291.60 selsort :: nil:cons -> nil:cons 305.85/291.60 ifselsort :: true:false -> nil:cons -> nil:cons 305.85/291.60 hole_true:false1_0 :: true:false 305.85/291.60 hole_0':s2_0 :: 0':s 305.85/291.60 hole_nil:cons3_0 :: nil:cons 305.85/291.60 gen_0':s4_0 :: Nat -> 0':s 305.85/291.60 gen_nil:cons5_0 :: Nat -> nil:cons 305.85/291.60 305.85/291.60 305.85/291.60 Lemmas: 305.85/291.60 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 305.85/291.60 le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_0) 305.85/291.60 min(gen_nil:cons5_0(+(1, n847_0))) -> gen_0':s4_0(0), rt in Omega(1 + n847_0) 305.85/291.60 305.85/291.60 305.85/291.60 Generator Equations: 305.85/291.60 gen_0':s4_0(0) <=> 0' 305.85/291.60 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 305.85/291.60 gen_nil:cons5_0(0) <=> nil 305.85/291.60 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 305.85/291.60 305.85/291.60 305.85/291.60 The following defined symbols remain to be analysed: 305.85/291.60 selsort 305.85/291.60 ---------------------------------------- 305.85/291.60 305.85/291.60 (19) LowerBoundPropagationProof (FINISHED) 305.85/291.60 Propagated lower bound. 305.85/291.60 ---------------------------------------- 305.85/291.60 305.85/291.60 (20) 305.85/291.60 BOUNDS(n^2, INF) 305.85/291.62 EOF