306.62/291.49 WORST_CASE(Omega(n^1), ?) 306.62/291.50 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 306.62/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 306.62/291.50 306.62/291.50 306.62/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.62/291.50 306.62/291.50 (0) CpxTRS 306.62/291.50 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 306.62/291.50 (2) CpxTRS 306.62/291.50 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 306.62/291.50 (4) typed CpxTrs 306.62/291.50 (5) OrderProof [LOWER BOUND(ID), 0 ms] 306.62/291.50 (6) typed CpxTrs 306.62/291.50 (7) RewriteLemmaProof [LOWER BOUND(ID), 266 ms] 306.62/291.50 (8) BEST 306.62/291.50 (9) proven lower bound 306.62/291.50 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 306.62/291.50 (11) BOUNDS(n^1, INF) 306.62/291.50 (12) typed CpxTrs 306.62/291.50 (13) RewriteLemmaProof [LOWER BOUND(ID), 36 ms] 306.62/291.50 (14) typed CpxTrs 306.62/291.50 (15) RewriteLemmaProof [LOWER BOUND(ID), 7 ms] 306.62/291.50 (16) typed CpxTrs 306.62/291.50 (17) RewriteLemmaProof [LOWER BOUND(ID), 8 ms] 306.62/291.50 (18) typed CpxTrs 306.62/291.50 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (0) 306.62/291.50 Obligation: 306.62/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.62/291.50 306.62/291.50 306.62/291.50 The TRS R consists of the following rules: 306.62/291.50 306.62/291.50 le(0, y) -> true 306.62/291.50 le(s(x), 0) -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0, 0) -> true 306.62/291.50 eq(0, s(y)) -> false 306.62/291.50 eq(s(x), 0) -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 S is empty. 306.62/291.50 Rewrite Strategy: FULL 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 306.62/291.50 Renamed function symbols to avoid clashes with predefined symbol. 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (2) 306.62/291.50 Obligation: 306.62/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.62/291.50 306.62/291.50 306.62/291.50 The TRS R consists of the following rules: 306.62/291.50 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 S is empty. 306.62/291.50 Rewrite Strategy: FULL 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 306.62/291.50 Infered types. 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (4) 306.62/291.50 Obligation: 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (5) OrderProof (LOWER BOUND(ID)) 306.62/291.50 Heuristically decided to analyse the following defined symbols: 306.62/291.50 le, eq, minsort, min, rm 306.62/291.50 306.62/291.50 They will be analysed ascendingly in the following order: 306.62/291.50 le < min 306.62/291.50 eq < rm 306.62/291.50 min < minsort 306.62/291.50 rm < minsort 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (6) 306.62/291.50 Obligation: 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 306.62/291.50 Generator Equations: 306.62/291.50 gen_0':s4_0(0) <=> 0' 306.62/291.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 306.62/291.50 gen_nil:cons5_0(0) <=> nil 306.62/291.50 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 306.62/291.50 306.62/291.50 306.62/291.50 The following defined symbols remain to be analysed: 306.62/291.50 le, eq, minsort, min, rm 306.62/291.50 306.62/291.50 They will be analysed ascendingly in the following order: 306.62/291.50 le < min 306.62/291.50 eq < rm 306.62/291.50 min < minsort 306.62/291.50 rm < minsort 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (7) RewriteLemmaProof (LOWER BOUND(ID)) 306.62/291.50 Proved the following rewrite lemma: 306.62/291.50 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 306.62/291.50 306.62/291.50 Induction Base: 306.62/291.50 le(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 306.62/291.50 true 306.62/291.50 306.62/291.50 Induction Step: 306.62/291.50 le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1) 306.62/291.50 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH 306.62/291.50 true 306.62/291.50 306.62/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (8) 306.62/291.50 Complex Obligation (BEST) 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (9) 306.62/291.50 Obligation: 306.62/291.50 Proved the lower bound n^1 for the following obligation: 306.62/291.50 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 306.62/291.50 Generator Equations: 306.62/291.50 gen_0':s4_0(0) <=> 0' 306.62/291.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 306.62/291.50 gen_nil:cons5_0(0) <=> nil 306.62/291.50 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 306.62/291.50 306.62/291.50 306.62/291.50 The following defined symbols remain to be analysed: 306.62/291.50 le, eq, minsort, min, rm 306.62/291.50 306.62/291.50 They will be analysed ascendingly in the following order: 306.62/291.50 le < min 306.62/291.50 eq < rm 306.62/291.50 min < minsort 306.62/291.50 rm < minsort 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (10) LowerBoundPropagationProof (FINISHED) 306.62/291.50 Propagated lower bound. 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (11) 306.62/291.50 BOUNDS(n^1, INF) 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (12) 306.62/291.50 Obligation: 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 306.62/291.50 Lemmas: 306.62/291.50 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 306.62/291.50 306.62/291.50 306.62/291.50 Generator Equations: 306.62/291.50 gen_0':s4_0(0) <=> 0' 306.62/291.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 306.62/291.50 gen_nil:cons5_0(0) <=> nil 306.62/291.50 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 306.62/291.50 306.62/291.50 306.62/291.50 The following defined symbols remain to be analysed: 306.62/291.50 eq, minsort, min, rm 306.62/291.50 306.62/291.50 They will be analysed ascendingly in the following order: 306.62/291.50 eq < rm 306.62/291.50 min < minsort 306.62/291.50 rm < minsort 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (13) RewriteLemmaProof (LOWER BOUND(ID)) 306.62/291.50 Proved the following rewrite lemma: 306.62/291.50 eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0) 306.62/291.50 306.62/291.50 Induction Base: 306.62/291.50 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 306.62/291.50 true 306.62/291.50 306.62/291.50 Induction Step: 306.62/291.50 eq(gen_0':s4_0(+(n306_0, 1)), gen_0':s4_0(+(n306_0, 1))) ->_R^Omega(1) 306.62/291.50 eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) ->_IH 306.62/291.50 true 306.62/291.50 306.62/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (14) 306.62/291.50 Obligation: 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 306.62/291.50 Lemmas: 306.62/291.50 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 306.62/291.50 eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0) 306.62/291.50 306.62/291.50 306.62/291.50 Generator Equations: 306.62/291.50 gen_0':s4_0(0) <=> 0' 306.62/291.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 306.62/291.50 gen_nil:cons5_0(0) <=> nil 306.62/291.50 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 306.62/291.50 306.62/291.50 306.62/291.50 The following defined symbols remain to be analysed: 306.62/291.50 min, minsort, rm 306.62/291.50 306.62/291.50 They will be analysed ascendingly in the following order: 306.62/291.50 min < minsort 306.62/291.50 rm < minsort 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (15) RewriteLemmaProof (LOWER BOUND(ID)) 306.62/291.50 Proved the following rewrite lemma: 306.62/291.50 min(gen_nil:cons5_0(+(1, n823_0))) -> gen_0':s4_0(0), rt in Omega(1 + n823_0) 306.62/291.50 306.62/291.50 Induction Base: 306.62/291.50 min(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1) 306.62/291.50 0' 306.62/291.50 306.62/291.50 Induction Step: 306.62/291.50 min(gen_nil:cons5_0(+(1, +(n823_0, 1)))) ->_R^Omega(1) 306.62/291.50 if1(le(0', 0'), 0', 0', gen_nil:cons5_0(n823_0)) ->_L^Omega(1) 306.62/291.50 if1(true, 0', 0', gen_nil:cons5_0(n823_0)) ->_R^Omega(1) 306.62/291.50 min(cons(0', gen_nil:cons5_0(n823_0))) ->_IH 306.62/291.50 gen_0':s4_0(0) 306.62/291.50 306.62/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (16) 306.62/291.50 Obligation: 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 306.62/291.50 Lemmas: 306.62/291.50 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 306.62/291.50 eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0) 306.62/291.50 min(gen_nil:cons5_0(+(1, n823_0))) -> gen_0':s4_0(0), rt in Omega(1 + n823_0) 306.62/291.50 306.62/291.50 306.62/291.50 Generator Equations: 306.62/291.50 gen_0':s4_0(0) <=> 0' 306.62/291.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 306.62/291.50 gen_nil:cons5_0(0) <=> nil 306.62/291.50 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 306.62/291.50 306.62/291.50 306.62/291.50 The following defined symbols remain to be analysed: 306.62/291.50 rm, minsort 306.62/291.50 306.62/291.50 They will be analysed ascendingly in the following order: 306.62/291.50 rm < minsort 306.62/291.50 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (17) RewriteLemmaProof (LOWER BOUND(ID)) 306.62/291.50 Proved the following rewrite lemma: 306.62/291.50 rm(gen_0':s4_0(0), gen_nil:cons5_0(n1254_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1254_0) 306.62/291.50 306.62/291.50 Induction Base: 306.62/291.50 rm(gen_0':s4_0(0), gen_nil:cons5_0(0)) ->_R^Omega(1) 306.62/291.50 nil 306.62/291.50 306.62/291.50 Induction Step: 306.62/291.50 rm(gen_0':s4_0(0), gen_nil:cons5_0(+(n1254_0, 1))) ->_R^Omega(1) 306.62/291.50 if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1254_0)) ->_L^Omega(1) 306.62/291.50 if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1254_0)) ->_R^Omega(1) 306.62/291.50 rm(gen_0':s4_0(0), gen_nil:cons5_0(n1254_0)) ->_IH 306.62/291.50 gen_nil:cons5_0(0) 306.62/291.50 306.62/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 306.62/291.50 ---------------------------------------- 306.62/291.50 306.62/291.50 (18) 306.62/291.50 Obligation: 306.62/291.50 TRS: 306.62/291.50 Rules: 306.62/291.50 le(0', y) -> true 306.62/291.50 le(s(x), 0') -> false 306.62/291.50 le(s(x), s(y)) -> le(x, y) 306.62/291.50 eq(0', 0') -> true 306.62/291.50 eq(0', s(y)) -> false 306.62/291.50 eq(s(x), 0') -> false 306.62/291.50 eq(s(x), s(y)) -> eq(x, y) 306.62/291.50 minsort(nil) -> nil 306.62/291.50 minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) 306.62/291.50 min(nil) -> 0' 306.62/291.50 min(cons(x, nil)) -> x 306.62/291.50 min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) 306.62/291.50 if1(true, x, y, xs) -> min(cons(x, xs)) 306.62/291.50 if1(false, x, y, xs) -> min(cons(y, xs)) 306.62/291.50 rm(x, nil) -> nil 306.62/291.50 rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) 306.62/291.50 if2(true, x, y, xs) -> rm(x, xs) 306.62/291.50 if2(false, x, y, xs) -> cons(y, rm(x, xs)) 306.62/291.50 306.62/291.50 Types: 306.62/291.50 le :: 0':s -> 0':s -> true:false 306.62/291.50 0' :: 0':s 306.62/291.50 true :: true:false 306.62/291.50 s :: 0':s -> 0':s 306.62/291.50 false :: true:false 306.62/291.50 eq :: 0':s -> 0':s -> true:false 306.62/291.50 minsort :: nil:cons -> nil:cons 306.62/291.50 nil :: nil:cons 306.62/291.50 cons :: 0':s -> nil:cons -> nil:cons 306.62/291.50 min :: nil:cons -> 0':s 306.62/291.50 rm :: 0':s -> nil:cons -> nil:cons 306.62/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s 306.62/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons 306.62/291.50 hole_true:false1_0 :: true:false 306.62/291.50 hole_0':s2_0 :: 0':s 306.62/291.50 hole_nil:cons3_0 :: nil:cons 306.62/291.50 gen_0':s4_0 :: Nat -> 0':s 306.62/291.50 gen_nil:cons5_0 :: Nat -> nil:cons 306.62/291.50 306.62/291.50 306.62/291.50 Lemmas: 306.62/291.50 le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 306.62/291.50 eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0) 306.62/291.50 min(gen_nil:cons5_0(+(1, n823_0))) -> gen_0':s4_0(0), rt in Omega(1 + n823_0) 306.62/291.50 rm(gen_0':s4_0(0), gen_nil:cons5_0(n1254_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1254_0) 306.62/291.50 306.62/291.50 306.62/291.50 Generator Equations: 306.62/291.50 gen_0':s4_0(0) <=> 0' 306.62/291.50 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 306.62/291.50 gen_nil:cons5_0(0) <=> nil 306.62/291.50 gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x)) 306.62/291.50 306.62/291.50 306.62/291.50 The following defined symbols remain to be analysed: 306.62/291.50 minsort 306.76/291.55 EOF