14.29/4.59 WORST_CASE(Omega(n^1), O(n^1)) 14.29/4.60 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 14.29/4.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.29/4.60 14.29/4.60 14.29/4.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 14.29/4.60 14.29/4.60 (0) CpxTRS 14.29/4.60 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 14.29/4.60 (2) CpxTRS 14.29/4.60 (3) CpxTrsMatchBoundsTAProof [FINISHED, 107 ms] 14.29/4.60 (4) BOUNDS(1, n^1) 14.29/4.60 (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 14.29/4.60 (6) CpxTRS 14.29/4.60 (7) SlicingProof [LOWER BOUND(ID), 0 ms] 14.29/4.60 (8) CpxTRS 14.29/4.60 (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 14.29/4.60 (10) typed CpxTrs 14.29/4.60 (11) OrderProof [LOWER BOUND(ID), 0 ms] 14.29/4.60 (12) typed CpxTrs 14.29/4.60 (13) RewriteLemmaProof [LOWER BOUND(ID), 240 ms] 14.29/4.60 (14) BEST 14.29/4.60 (15) proven lower bound 14.29/4.60 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 14.29/4.60 (17) BOUNDS(n^1, INF) 14.29/4.60 (18) typed CpxTrs 14.29/4.60 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (0) 14.29/4.60 Obligation: 14.29/4.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 14.29/4.60 14.29/4.60 14.29/4.60 The TRS R consists of the following rules: 14.29/4.60 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 S is empty. 14.29/4.60 Rewrite Strategy: INNERMOST 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 14.29/4.60 transformed relative TRS to TRS 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (2) 14.29/4.60 Obligation: 14.29/4.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 14.29/4.60 14.29/4.60 14.29/4.60 The TRS R consists of the following rules: 14.29/4.60 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 S is empty. 14.29/4.60 Rewrite Strategy: INNERMOST 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (3) CpxTrsMatchBoundsTAProof (FINISHED) 14.29/4.60 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 14.29/4.60 14.29/4.60 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 14.29/4.60 final states : [1, 2, 3, 4, 5] 14.29/4.60 transitions: 14.29/4.60 Cons0(0, 0) -> 0 14.29/4.60 Nil0() -> 0 14.29/4.60 True0() -> 0 14.29/4.60 False0() -> 0 14.29/4.60 dec0(0) -> 1 14.29/4.60 isNilNil0(0) -> 2 14.29/4.60 nestdec0(0) -> 3 14.29/4.60 number170(0) -> 4 14.29/4.60 goal0(0) -> 5 14.29/4.60 Nil1() -> 1 14.29/4.60 Cons1(0, 0) -> 6 14.29/4.60 dec1(6) -> 1 14.29/4.60 Nil1() -> 7 14.29/4.60 dec1(7) -> 1 14.29/4.60 True1() -> 2 14.29/4.60 False1() -> 2 14.29/4.60 Nil1() -> 8 14.29/4.60 Nil1() -> 11 14.29/4.60 Cons1(8, 11) -> 10 14.29/4.60 Cons1(8, 10) -> 9 14.29/4.60 Cons1(8, 9) -> 9 14.29/4.60 Cons1(8, 9) -> 3 14.29/4.60 dec1(6) -> 12 14.29/4.60 nestdec1(12) -> 3 14.29/4.60 Cons1(8, 9) -> 4 14.29/4.60 nestdec1(0) -> 5 14.29/4.60 Nil1() -> 12 14.29/4.60 dec1(7) -> 12 14.29/4.60 Cons1(8, 9) -> 5 14.29/4.60 nestdec1(12) -> 5 14.29/4.60 Nil2() -> 13 14.29/4.60 Nil2() -> 16 14.29/4.60 Cons2(13, 16) -> 15 14.29/4.60 Cons2(13, 15) -> 14 14.29/4.60 Cons2(13, 14) -> 14 14.29/4.60 Cons2(13, 14) -> 3 14.29/4.60 Cons2(13, 14) -> 5 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (4) 14.29/4.60 BOUNDS(1, n^1) 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (5) RenamingProof (BOTH BOUNDS(ID, ID)) 14.29/4.60 Renamed function symbols to avoid clashes with predefined symbol. 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (6) 14.29/4.60 Obligation: 14.29/4.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 14.29/4.60 14.29/4.60 14.29/4.60 The TRS R consists of the following rules: 14.29/4.60 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 S is empty. 14.29/4.60 Rewrite Strategy: INNERMOST 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (7) SlicingProof (LOWER BOUND(ID)) 14.29/4.60 Sliced the following arguments: 14.29/4.60 number17/0 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (8) 14.29/4.60 Obligation: 14.29/4.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 14.29/4.60 14.29/4.60 14.29/4.60 The TRS R consists of the following rules: 14.29/4.60 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 S is empty. 14.29/4.60 Rewrite Strategy: INNERMOST 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 14.29/4.60 Infered types. 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (10) 14.29/4.60 Obligation: 14.29/4.60 Innermost TRS: 14.29/4.60 Rules: 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 Types: 14.29/4.60 dec :: Nil:Cons -> Nil:Cons 14.29/4.60 Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons 14.29/4.60 Nil :: Nil:Cons 14.29/4.60 isNilNil :: Nil:Cons -> True:False 14.29/4.60 True :: True:False 14.29/4.60 False :: True:False 14.29/4.60 nestdec :: Nil:Cons -> Nil:Cons 14.29/4.60 number17 :: Nil:Cons 14.29/4.60 goal :: Nil:Cons -> Nil:Cons 14.29/4.60 hole_Nil:Cons1_0 :: Nil:Cons 14.29/4.60 hole_True:False2_0 :: True:False 14.29/4.60 gen_Nil:Cons3_0 :: Nat -> Nil:Cons 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (11) OrderProof (LOWER BOUND(ID)) 14.29/4.60 Heuristically decided to analyse the following defined symbols: 14.29/4.60 dec, nestdec 14.29/4.60 14.29/4.60 They will be analysed ascendingly in the following order: 14.29/4.60 dec < nestdec 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (12) 14.29/4.60 Obligation: 14.29/4.60 Innermost TRS: 14.29/4.60 Rules: 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 Types: 14.29/4.60 dec :: Nil:Cons -> Nil:Cons 14.29/4.60 Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons 14.29/4.60 Nil :: Nil:Cons 14.29/4.60 isNilNil :: Nil:Cons -> True:False 14.29/4.60 True :: True:False 14.29/4.60 False :: True:False 14.29/4.60 nestdec :: Nil:Cons -> Nil:Cons 14.29/4.60 number17 :: Nil:Cons 14.29/4.60 goal :: Nil:Cons -> Nil:Cons 14.29/4.60 hole_Nil:Cons1_0 :: Nil:Cons 14.29/4.60 hole_True:False2_0 :: True:False 14.29/4.60 gen_Nil:Cons3_0 :: Nat -> Nil:Cons 14.29/4.60 14.29/4.60 14.29/4.60 Generator Equations: 14.29/4.60 gen_Nil:Cons3_0(0) <=> Nil 14.29/4.60 gen_Nil:Cons3_0(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons3_0(x)) 14.29/4.60 14.29/4.60 14.29/4.60 The following defined symbols remain to be analysed: 14.29/4.60 dec, nestdec 14.29/4.60 14.29/4.60 They will be analysed ascendingly in the following order: 14.29/4.60 dec < nestdec 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (13) RewriteLemmaProof (LOWER BOUND(ID)) 14.29/4.60 Proved the following rewrite lemma: 14.29/4.60 dec(gen_Nil:Cons3_0(+(1, n5_0))) -> gen_Nil:Cons3_0(0), rt in Omega(1 + n5_0) 14.29/4.60 14.29/4.60 Induction Base: 14.29/4.60 dec(gen_Nil:Cons3_0(+(1, 0))) ->_R^Omega(1) 14.29/4.60 Nil 14.29/4.60 14.29/4.60 Induction Step: 14.29/4.60 dec(gen_Nil:Cons3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1) 14.29/4.60 dec(Cons(Nil, gen_Nil:Cons3_0(n5_0))) ->_IH 14.29/4.60 gen_Nil:Cons3_0(0) 14.29/4.60 14.29/4.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (14) 14.29/4.60 Complex Obligation (BEST) 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (15) 14.29/4.60 Obligation: 14.29/4.60 Proved the lower bound n^1 for the following obligation: 14.29/4.60 14.29/4.60 Innermost TRS: 14.29/4.60 Rules: 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 Types: 14.29/4.60 dec :: Nil:Cons -> Nil:Cons 14.29/4.60 Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons 14.29/4.60 Nil :: Nil:Cons 14.29/4.60 isNilNil :: Nil:Cons -> True:False 14.29/4.60 True :: True:False 14.29/4.60 False :: True:False 14.29/4.60 nestdec :: Nil:Cons -> Nil:Cons 14.29/4.60 number17 :: Nil:Cons 14.29/4.60 goal :: Nil:Cons -> Nil:Cons 14.29/4.60 hole_Nil:Cons1_0 :: Nil:Cons 14.29/4.60 hole_True:False2_0 :: True:False 14.29/4.60 gen_Nil:Cons3_0 :: Nat -> Nil:Cons 14.29/4.60 14.29/4.60 14.29/4.60 Generator Equations: 14.29/4.60 gen_Nil:Cons3_0(0) <=> Nil 14.29/4.60 gen_Nil:Cons3_0(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons3_0(x)) 14.29/4.60 14.29/4.60 14.29/4.60 The following defined symbols remain to be analysed: 14.29/4.60 dec, nestdec 14.29/4.60 14.29/4.60 They will be analysed ascendingly in the following order: 14.29/4.60 dec < nestdec 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (16) LowerBoundPropagationProof (FINISHED) 14.29/4.60 Propagated lower bound. 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (17) 14.29/4.60 BOUNDS(n^1, INF) 14.29/4.60 14.29/4.60 ---------------------------------------- 14.29/4.60 14.29/4.60 (18) 14.29/4.60 Obligation: 14.29/4.60 Innermost TRS: 14.29/4.60 Rules: 14.29/4.60 dec(Cons(Nil, Nil)) -> Nil 14.29/4.60 dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 dec(Cons(Cons(x, xs), Nil)) -> dec(Nil) 14.29/4.60 dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs)) 14.29/4.60 isNilNil(Cons(Nil, Nil)) -> True 14.29/4.60 isNilNil(Cons(Nil, Cons(x, xs))) -> False 14.29/4.60 isNilNil(Cons(Cons(x, xs), Nil)) -> False 14.29/4.60 isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False 14.29/4.60 nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs))) 14.29/4.60 number17 -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) 14.29/4.60 goal(x) -> nestdec(x) 14.29/4.60 14.29/4.60 Types: 14.29/4.60 dec :: Nil:Cons -> Nil:Cons 14.29/4.60 Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons 14.29/4.60 Nil :: Nil:Cons 14.29/4.60 isNilNil :: Nil:Cons -> True:False 14.29/4.60 True :: True:False 14.29/4.60 False :: True:False 14.29/4.60 nestdec :: Nil:Cons -> Nil:Cons 14.29/4.60 number17 :: Nil:Cons 14.29/4.60 goal :: Nil:Cons -> Nil:Cons 14.29/4.60 hole_Nil:Cons1_0 :: Nil:Cons 14.29/4.60 hole_True:False2_0 :: True:False 14.29/4.60 gen_Nil:Cons3_0 :: Nat -> Nil:Cons 14.29/4.60 14.29/4.60 14.29/4.60 Lemmas: 14.29/4.60 dec(gen_Nil:Cons3_0(+(1, n5_0))) -> gen_Nil:Cons3_0(0), rt in Omega(1 + n5_0) 14.29/4.60 14.29/4.60 14.29/4.60 Generator Equations: 14.29/4.60 gen_Nil:Cons3_0(0) <=> Nil 14.29/4.60 gen_Nil:Cons3_0(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons3_0(x)) 14.29/4.60 14.29/4.60 14.29/4.60 The following defined symbols remain to be analysed: 14.29/4.60 nestdec 14.56/4.64 EOF