0.002/0.002 NO 0.002/0.002 0.002/0.002 Problem 1: 0.002/0.002 0.002/0.002 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 Confluence Problem: 0.002/0.002 (VAR vNonEmpty:S v_NonEmpty:S:S L:S:S N:S:S V:S:S V1:S:S V2:S:S X:S:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (U11 1) 0.002/0.002 (U12 1) 0.002/0.002 (U21 1) 0.002/0.002 (U22 1) 0.002/0.002 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U42 1) 0.002/0.002 (U43 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U53 1) 0.002/0.002 (U61 1) 0.002/0.002 (and 1) 0.002/0.002 (isNat 1) 0.002/0.002 (isNatIList 1) 0.002/0.002 (isNatIListKind 1) 0.002/0.002 (isNatKind 1) 0.002/0.002 (isNatList 1) 0.002/0.002 (length 1) 0.002/0.002 (zeros) 0.002/0.002 (cons) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (num0) 0.002/0.002 (s) 0.002/0.002 (tt) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 U11(tt,V1:S:S) -> U12(isNatList(V1:S:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S:S) -> U22(isNat(V1:S:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S:S) -> U32(isNatList(V:S:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S:S,V2:S:S) -> U42(isNat(V1:S:S),V2:S:S) 0.002/0.002 U42(tt,V2:S:S) -> U43(isNatIList(V2:S:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S:S,V2:S:S) -> U52(isNat(V1:S:S),V2:S:S) 0.002/0.002 U52(tt,V2:S:S) -> U53(isNatList(V2:S:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S:S) -> s(length(L:S:S)) 0.002/0.002 and(tt,X:S:S) -> X:S:S 0.002/0.002 isNat(length(V1:S:S)) -> U11(isNatIListKind(V1:S:S),V1:S:S) 0.002/0.002 isNat(num0) -> tt 0.002/0.002 isNat(s(V1:S:S)) -> U21(isNatKind(V1:S:S),V1:S:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S:S,V2:S:S)) -> U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatIList(V:S:S) -> U31(isNatIListKind(V:S:S),V:S:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S:S,V2:S:S)) -> and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S:S)) -> isNatIListKind(V1:S:S) 0.002/0.002 isNatKind(num0) -> tt 0.002/0.002 isNatKind(s(V1:S:S)) -> isNatKind(V1:S:S) 0.002/0.002 isNatList(cons(V1:S:S,V2:S:S)) -> U51(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S:S,L:S:S)) -> U61(and(and(isNatList(L:S:S),isNatIListKind(L:S:S)),and(isNat(N:S:S),isNatKind(N:S:S))),L:S:S) 0.002/0.002 length(nil) -> num0 0.002/0.002 zeros -> cons(num0,zeros) 0.002/0.002 ) 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 0.002/0.002 Problem 1: 0.002/0.002 0.002/0.002 CleanTRS Processor: 0.002/0.002 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 Confluence Problem: 0.002/0.002 (VAR vNonEmpty:S v_NonEmpty:S:S L:S:S N:S:S V:S:S V1:S:S V2:S:S X:S:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (U11 1) 0.002/0.002 (U12 1) 0.002/0.002 (U21 1) 0.002/0.002 (U22 1) 0.002/0.002 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U42 1) 0.002/0.002 (U43 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U53 1) 0.002/0.002 (U61 1) 0.002/0.002 (and 1) 0.002/0.002 (isNat 1) 0.002/0.002 (isNatIList 1) 0.002/0.002 (isNatIListKind 1) 0.002/0.002 (isNatKind 1) 0.002/0.002 (isNatList 1) 0.002/0.002 (length 1) 0.002/0.002 (zeros) 0.002/0.002 (cons) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (num0) 0.002/0.002 (s) 0.002/0.002 (tt) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 U11(tt,V1:S:S) -> U12(isNatList(V1:S:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S:S) -> U22(isNat(V1:S:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S:S) -> U32(isNatList(V:S:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S:S,V2:S:S) -> U42(isNat(V1:S:S),V2:S:S) 0.002/0.002 U42(tt,V2:S:S) -> U43(isNatIList(V2:S:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S:S,V2:S:S) -> U52(isNat(V1:S:S),V2:S:S) 0.002/0.002 U52(tt,V2:S:S) -> U53(isNatList(V2:S:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S:S) -> s(length(L:S:S)) 0.002/0.002 and(tt,X:S:S) -> X:S:S 0.002/0.002 isNat(length(V1:S:S)) -> U11(isNatIListKind(V1:S:S),V1:S:S) 0.002/0.002 isNat(num0) -> tt 0.002/0.002 isNat(s(V1:S:S)) -> U21(isNatKind(V1:S:S),V1:S:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S:S,V2:S:S)) -> U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatIList(V:S:S) -> U31(isNatIListKind(V:S:S),V:S:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S:S,V2:S:S)) -> and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S:S)) -> isNatIListKind(V1:S:S) 0.002/0.002 isNatKind(num0) -> tt 0.002/0.002 isNatKind(s(V1:S:S)) -> isNatKind(V1:S:S) 0.002/0.002 isNatList(cons(V1:S:S,V2:S:S)) -> U51(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S:S,L:S:S)) -> U61(and(and(isNatList(L:S:S),isNatIListKind(L:S:S)),and(isNat(N:S:S),isNatKind(N:S:S))),L:S:S) 0.002/0.002 length(nil) -> num0 0.002/0.002 zeros -> cons(num0,zeros) 0.002/0.002 ) 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 0.002/0.002 Problem 1: 0.002/0.002 0.002/0.002 Modular Confluence Combinations Decomposition Processor: 0.002/0.002 It is a CTRS -> No modular confluence 0.002/0.002 0.002/0.002 Problem 1: 0.002/0.002 CS-TRS Processor: 0.002/0.002 R is a CS-TRS 0.002/0.002 0.002/0.002 Problem 1: 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 Confluence Problem: 0.002/0.002 (VAR vNonEmpty:S v_NonEmpty:S:S L:S:S N:S:S V:S:S V1:S:S V2:S:S X:S:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (U11 1) 0.002/0.002 (U12 1) 0.002/0.002 (U21 1) 0.002/0.002 (U22 1) 0.002/0.002 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U42 1) 0.002/0.002 (U43 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U53 1) 0.002/0.002 (U61 1) 0.002/0.002 (and 1) 0.002/0.002 (isNat 1) 0.002/0.002 (isNatIList 1) 0.002/0.002 (isNatIListKind 1) 0.002/0.002 (isNatKind 1) 0.002/0.002 (isNatList 1) 0.002/0.002 (length 1) 0.002/0.002 (zeros) 0.002/0.002 (cons) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (num0) 0.002/0.002 (s) 0.002/0.002 (tt) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 U11(tt,V1:S:S) -> U12(isNatList(V1:S:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S:S) -> U22(isNat(V1:S:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S:S) -> U32(isNatList(V:S:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S:S,V2:S:S) -> U42(isNat(V1:S:S),V2:S:S) 0.002/0.002 U42(tt,V2:S:S) -> U43(isNatIList(V2:S:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S:S,V2:S:S) -> U52(isNat(V1:S:S),V2:S:S) 0.002/0.002 U52(tt,V2:S:S) -> U53(isNatList(V2:S:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S:S) -> s(length(L:S:S)) 0.002/0.002 and(tt,X:S:S) -> X:S:S 0.002/0.002 isNat(length(V1:S:S)) -> U11(isNatIListKind(V1:S:S),V1:S:S) 0.002/0.002 isNat(num0) -> tt 0.002/0.002 isNat(s(V1:S:S)) -> U21(isNatKind(V1:S:S),V1:S:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S:S,V2:S:S)) -> U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatIList(V:S:S) -> U31(isNatIListKind(V:S:S),V:S:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S:S,V2:S:S)) -> and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S:S)) -> isNatIListKind(V1:S:S) 0.002/0.002 isNatKind(num0) -> tt 0.002/0.002 isNatKind(s(V1:S:S)) -> isNatKind(V1:S:S) 0.002/0.002 isNatList(cons(V1:S:S,V2:S:S)) -> U51(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S:S,L:S:S)) -> U61(and(and(isNatList(L:S:S),isNatIListKind(L:S:S)),and(isNat(N:S:S),isNatKind(N:S:S))),L:S:S) 0.002/0.002 length(nil) -> num0 0.002/0.002 zeros -> cons(num0,zeros) 0.002/0.002 ) 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 Huet Levy Processor: 0.002/0.002 -> Rules: 0.002/0.002 U11(tt,V1:S:S) -> U12(isNatList(V1:S:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S:S) -> U22(isNat(V1:S:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S:S) -> U32(isNatList(V:S:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S:S,V2:S:S) -> U42(isNat(V1:S:S),V2:S:S) 0.002/0.002 U42(tt,V2:S:S) -> U43(isNatIList(V2:S:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S:S,V2:S:S) -> U52(isNat(V1:S:S),V2:S:S) 0.002/0.002 U52(tt,V2:S:S) -> U53(isNatList(V2:S:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S:S) -> s(length(L:S:S)) 0.002/0.002 and(tt,X:S:S) -> X:S:S 0.002/0.002 isNat(length(V1:S:S)) -> U11(isNatIListKind(V1:S:S),V1:S:S) 0.002/0.002 isNat(num0) -> tt 0.002/0.002 isNat(s(V1:S:S)) -> U21(isNatKind(V1:S:S),V1:S:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S:S,V2:S:S)) -> U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatIList(V:S:S) -> U31(isNatIListKind(V:S:S),V:S:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S:S,V2:S:S)) -> and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S:S)) -> isNatIListKind(V1:S:S) 0.002/0.002 isNatKind(num0) -> tt 0.002/0.002 isNatKind(s(V1:S:S)) -> isNatKind(V1:S:S) 0.002/0.002 isNatList(cons(V1:S:S,V2:S:S)) -> U51(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S:S,L:S:S)) -> U61(and(and(isNatList(L:S:S),isNatIListKind(L:S:S)),and(isNat(N:S:S),isNatKind(N:S:S))),L:S:S) 0.002/0.002 length(nil) -> num0 0.002/0.002 zeros -> cons(num0,zeros) 0.002/0.002 -> Vars: 0.002/0.002 V1:S, V1:S, V:S, V1:S, V2:S, V2:S, V1:S, V2:S, V2:S, L:S, X:S, V1:S, V1:S, V1:S, V2:S, V:S, V1:S, V2:S, V1:S, V1:S, V1:S, V2:S, L:S, N:S 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 2, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 3, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 4, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 5, UV-LActive: [], UV-RActive: [V:S], UV-LFrozen: [V:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 7, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S, V2:S], UV-RFrozen: [V2:S]) 0.002/0.002 (UV-RuleId: 8, UV-LActive: [], UV-RActive: [V2:S], UV-LFrozen: [V2:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 9, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 10, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S, V2:S], UV-RFrozen: [V2:S]) 0.002/0.002 (UV-RuleId: 11, UV-LActive: [], UV-RActive: [V2:S], UV-LFrozen: [V2:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 12, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 13, UV-LActive: [], UV-RActive: [], UV-LFrozen: [L:S], UV-RFrozen: [L:S]) 0.002/0.002 (UV-RuleId: 14, UV-LActive: [], UV-RActive: [X:S], UV-LFrozen: [X:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 15, UV-LActive: [V1:S], UV-RActive: [V1:S], UV-LFrozen: [], UV-RFrozen: [V1:S]) 0.002/0.002 (UV-RuleId: 16, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 17, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S], UV-RFrozen: [V1:S]) 0.002/0.002 (UV-RuleId: 18, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 19, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S, V2:S], UV-RFrozen: [V1:S, V2:S]) 0.002/0.002 (UV-RuleId: 20, UV-LActive: [V:S], UV-RActive: [V:S], UV-LFrozen: [], UV-RFrozen: [V:S]) 0.002/0.002 (UV-RuleId: 21, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 22, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S, V2:S], UV-RFrozen: [V2:S]) 0.002/0.002 (UV-RuleId: 23, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 24, UV-LActive: [V1:S], UV-RActive: [V1:S], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 25, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 26, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 27, UV-LActive: [], UV-RActive: [V1:S], UV-LFrozen: [V1:S, V2:S], UV-RFrozen: [V1:S, V2:S]) 0.002/0.002 (UV-RuleId: 28, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 29, UV-LActive: [], UV-RActive: [L:S], UV-LFrozen: [L:S, N:S], UV-RFrozen: [L:S, N:S]) 0.002/0.002 (UV-RuleId: 30, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 31, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 -> FVars: 0.002/0.002 x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31 0.002/0.002 -> PVars: 0.002/0.002 V1:S: [x8, x9, x11, x14, x19, x20, x21, x24, x26, x27, x28], V:S: [x10, x23], V2:S: [x12, x13, x15, x16, x22, x25, x29], L:S: [x17, x30], X:S: [x18], N:S: [x31] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: U11(tt,x8:S) -> U12(isNatList(x8:S)), id: 1, possubterms: U11(tt,x8:S)->[], tt->[1]) 0.002/0.002 (rule: U12(tt) -> tt, id: 2, possubterms: U12(tt)->[], tt->[1]) 0.002/0.002 (rule: U21(tt,x9:S) -> U22(isNat(x9:S)), id: 3, possubterms: U21(tt,x9:S)->[], tt->[1]) 0.002/0.002 (rule: U22(tt) -> tt, id: 4, possubterms: U22(tt)->[], tt->[1]) 0.002/0.002 (rule: U31(tt,x10:S) -> U32(isNatList(x10:S)), id: 5, possubterms: U31(tt,x10:S)->[], tt->[1]) 0.002/0.002 (rule: U32(tt) -> tt, id: 6, possubterms: U32(tt)->[], tt->[1]) 0.002/0.002 (rule: U41(tt,x11:S,x12:S) -> U42(isNat(x11:S),x12:S), id: 7, possubterms: U41(tt,x11:S,x12:S)->[], tt->[1]) 0.002/0.002 (rule: U42(tt,x13:S) -> U43(isNatIList(x13:S)), id: 8, possubterms: U42(tt,x13:S)->[], tt->[1]) 0.002/0.002 (rule: U43(tt) -> tt, id: 9, possubterms: U43(tt)->[], tt->[1]) 0.002/0.002 (rule: U51(tt,x14:S,x15:S) -> U52(isNat(x14:S),x15:S), id: 10, possubterms: U51(tt,x14:S,x15:S)->[], tt->[1]) 0.002/0.002 (rule: U52(tt,x16:S) -> U53(isNatList(x16:S)), id: 11, possubterms: U52(tt,x16:S)->[], tt->[1]) 0.002/0.002 (rule: U53(tt) -> tt, id: 12, possubterms: U53(tt)->[], tt->[1]) 0.002/0.002 (rule: U61(tt,x17:S) -> s(length(x17:S)), id: 13, possubterms: U61(tt,x17:S)->[], tt->[1]) 0.002/0.002 (rule: and(tt,x18:S) -> x18:S, id: 14, possubterms: and(tt,x18:S)->[], tt->[1]) 0.002/0.002 (rule: isNat(length(x19:S)) -> U11(isNatIListKind(x19:S),x19:S), id: 15, possubterms: isNat(length(x19:S))->[], length(x19:S)->[1]) 0.002/0.002 (rule: isNat(num0) -> tt, id: 16, possubterms: isNat(num0)->[], num0->[1]) 0.002/0.002 (rule: isNat(s(x20:S)) -> U21(isNatKind(x20:S),x20:S), id: 17, possubterms: isNat(s(x20:S))->[], s(x20:S)->[1]) 0.002/0.002 (rule: isNatIList(zeros) -> tt, id: 18, possubterms: isNatIList(zeros)->[], zeros->[1]) 0.002/0.002 (rule: isNatIList(cons(x21:S,x22:S)) -> U41(and(isNatKind(x21:S),isNatIListKind(x22:S)),x21:S,x22:S), id: 19, possubterms: isNatIList(cons(x21:S,x22:S))->[], cons(x21:S,x22:S)->[1]) 0.002/0.002 (rule: isNatIList(x23:S) -> U31(isNatIListKind(x23:S),x23:S), id: 20, possubterms: isNatIList(x23:S)->[]) 0.002/0.002 (rule: isNatIListKind(zeros) -> tt, id: 21, possubterms: isNatIListKind(zeros)->[], zeros->[1]) 0.002/0.002 (rule: isNatIListKind(cons(x24:S,x25:S)) -> and(isNatKind(x24:S),isNatIListKind(x25:S)), id: 22, possubterms: isNatIListKind(cons(x24:S,x25:S))->[], cons(x24:S,x25:S)->[1]) 0.002/0.002 (rule: isNatIListKind(nil) -> tt, id: 23, possubterms: isNatIListKind(nil)->[], nil->[1]) 0.002/0.002 (rule: isNatKind(length(x26:S)) -> isNatIListKind(x26:S), id: 24, possubterms: isNatKind(length(x26:S))->[], length(x26:S)->[1]) 0.002/0.002 (rule: isNatKind(num0) -> tt, id: 25, possubterms: isNatKind(num0)->[], num0->[1]) 0.002/0.002 (rule: isNatKind(s(x27:S)) -> isNatKind(x27:S), id: 26, possubterms: isNatKind(s(x27:S))->[], s(x27:S)->[1]) 0.002/0.002 (rule: isNatList(cons(x28:S,x29:S)) -> U51(and(isNatKind(x28:S),isNatIListKind(x29:S)),x28:S,x29:S), id: 27, possubterms: isNatList(cons(x28:S,x29:S))->[], cons(x28:S,x29:S)->[1]) 0.002/0.002 (rule: isNatList(nil) -> tt, id: 28, possubterms: isNatList(nil)->[], nil->[1]) 0.002/0.002 (rule: length(cons(x31:S,x30:S)) -> U61(and(and(isNatList(x30:S),isNatIListKind(x30:S)),and(isNat(x31:S),isNatKind(x31:S))),x30:S), id: 29, possubterms: length(cons(x31:S,x30:S))->[], cons(x31:S,x30:S)->[1]) 0.002/0.002 (rule: length(nil) -> num0, id: 30, possubterms: length(nil)->[], nil->[1]) 0.002/0.002 (rule: zeros -> cons(num0,zeros), id: 31, possubterms: zeros->[]) 0.002/0.002 0.002/0.002 -> Unifications: 0.002/0.002 (R15 unifies with R29 at p: [1], l: isNat(length(x19:S)), lp: length(x19:S), sig: {x19:S -> cons(N:S:S,L:S:S)}, l': length(cons(N:S:S,L:S:S)), r: U11(isNatIListKind(x19:S),x19:S), r': U61(and(and(isNatList(L:S:S),isNatIListKind(L:S:S)),and(isNat(N:S:S),isNatKind(N:S:S))),L:S:S)) 0.002/0.002 (R15 unifies with R30 at p: [1], l: isNat(length(x19:S)), lp: length(x19:S), sig: {x19:S -> nil}, l': length(nil), r: U11(isNatIListKind(x19:S),x19:S), r': num0) 0.002/0.002 (R18 unifies with R31 at p: [1], l: isNatIList(zeros), lp: zeros, sig: {}, l': zeros, r: tt, r': cons(num0,zeros)) 0.002/0.002 (R20 unifies with R18 at p: [], l: isNatIList(x23:S), lp: isNatIList(x23:S), sig: {x23:S -> zeros}, l': isNatIList(zeros), r: U31(isNatIListKind(x23:S),x23:S), r': tt) 0.002/0.002 (R20 unifies with R19 at p: [], l: isNatIList(x23:S), lp: isNatIList(x23:S), sig: {x23:S -> cons(V1:S:S,V2:S:S)}, l': isNatIList(cons(V1:S:S,V2:S:S)), r: U31(isNatIListKind(x23:S),x23:S), r': U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S)) 0.002/0.002 (R21 unifies with R31 at p: [1], l: isNatIListKind(zeros), lp: zeros, sig: {}, l': zeros, r: tt, r': cons(num0,zeros)) 0.002/0.002 (R24 unifies with R29 at p: [1], l: isNatKind(length(x26:S)), lp: length(x26:S), sig: {x26:S -> cons(N:S:S,L:S:S)}, l': length(cons(N:S:S,L:S:S)), r: isNatIListKind(x26:S), r': U61(and(and(isNatList(L:S:S),isNatIListKind(L:S:S)),and(isNat(N:S:S),isNatKind(N:S:S))),L:S:S)) 0.002/0.002 (R24 unifies with R30 at p: [1], l: isNatKind(length(x26:S)), lp: length(x26:S), sig: {x26:S -> nil}, l': length(nil), r: isNatIListKind(x26:S), r': num0) 0.002/0.002 0.002/0.002 -> Critical pairs info: 0.002/0.002 => Not trivial, Not overlay, NW0, N1 0.002/0.002 => Not trivial, Overlay, NW0, N2 0.002/0.002 => Not trivial, Not overlay, NW0, N3 0.002/0.002 => Not trivial, Overlay, NW0, N4 0.002/0.002 => Not trivial, Not overlay, NW0, N5 0.002/0.002 => Not trivial, Not overlay, NW0, N6 0.002/0.002 => Not trivial, Not overlay, NW0, N7 0.002/0.002 => Not trivial, Not overlay, NW0, N8 0.002/0.002 0.002/0.002 -> Problem conclusions: 0.002/0.002 Left linear, Not right linear, Not linear 0.002/0.002 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.002/0.002 Not Huet-Levy confluent, Not Newman confluent 0.002/0.002 R is a CS-TRS, Not left-homogeneous u-replacing variables 0.002/0.002 0.002/0.002 0.002/0.002 Problem 1: 0.002/0.002 No Convergence Brute Force Processor: 0.002/0.002 -> Rewritings: 0.002/0.002 s: U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) 0.002/0.002 Nodes: [0] 0.002/0.002 Edges: [] 0.002/0.002 ID: 0 => ('U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S)', D0) 0.002/0.002 t: U31(isNatIListKind(cons(V1:S:S,V2:S:S)),cons(V1:S:S,V2:S:S)) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('U31(isNatIListKind(cons(V1:S:S,V2:S:S)),cons(V1:S:S,V2:S:S))', D0) 0.002/0.002 ID: 1 => ('U31(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),cons(V1:S:S,V2:S:S))', D1, R22, P[1], S{x24:S -> V1:S:S, x25:S -> V2:S:S}), NR: 'and(isNatKind(V1:S:S),isNatIListKind(V2:S:S))' 0.002/0.002 U41(and(isNatKind(V1:S:S),isNatIListKind(V2:S:S)),V1:S:S,V2:S:S) ->* no union *<- U31(isNatIListKind(cons(V1:S:S,V2:S:S)),cons(V1:S:S,V2:S:S)) 0.002/0.002 "Not joinable" 0.002/0.002 0.002/0.002 The problem is not joinable. 0.002/0.002 0.01user 0.00system 0:00.02elapsed 68%CPU (0avgtext+0avgdata 11772maxresident)k 0.002/0.002 8inputs+0outputs (0major+1155minor)pagefaults 0swaps