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 L:S N:S V:S V1:S V2:S X: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) 0.002/0.002 (isNatIList) 0.002/0.002 (isNatIListKind) 0.002/0.002 (isNatKind) 0.002/0.002 (isNatList) 0.002/0.002 (length 1) 0.002/0.002 (zeros) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s 1) 0.002/0.002 (tt) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 U11(tt,V1:S) -> U12(isNatList(V1:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S) -> U22(isNat(V1:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S) -> U32(isNatList(V:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S,V2:S) -> U42(isNat(V1:S),V2:S) 0.002/0.002 U42(tt,V2:S) -> U43(isNatIList(V2:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S,V2:S) -> U52(isNat(V1:S),V2:S) 0.002/0.002 U52(tt,V2:S) -> U53(isNatList(V2:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S) -> s(length(L:S)) 0.002/0.002 and(tt,X:S) -> X:S 0.002/0.002 isNat(length(V1:S)) -> U11(isNatIListKind(V1:S),V1:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNatKind(V1:S),V1:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S,V2:S)) -> U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatIList(V:S) -> U31(isNatIListKind(V:S),V:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S,V2:S)) -> and(isNatKind(V1:S),isNatIListKind(V2:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S)) -> isNatIListKind(V1:S) 0.002/0.002 isNatKind(0) -> tt 0.002/0.002 isNatKind(s(V1:S)) -> isNatKind(V1:S) 0.002/0.002 isNatList(cons(V1:S,V2:S)) -> U51(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S,L:S)) -> U61(and(and(isNatList(L:S),isNatIListKind(L:S)),and(isNat(N:S),isNatKind(N:S))),L:S) 0.002/0.002 length(nil) -> 0 0.002/0.002 zeros -> cons(0,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 L:S N:S V:S V1:S V2:S X: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) 0.002/0.002 (isNatIList) 0.002/0.002 (isNatIListKind) 0.002/0.002 (isNatKind) 0.002/0.002 (isNatList) 0.002/0.002 (length 1) 0.002/0.002 (zeros) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s 1) 0.002/0.002 (tt) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 U11(tt,V1:S) -> U12(isNatList(V1:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S) -> U22(isNat(V1:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S) -> U32(isNatList(V:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S,V2:S) -> U42(isNat(V1:S),V2:S) 0.002/0.002 U42(tt,V2:S) -> U43(isNatIList(V2:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S,V2:S) -> U52(isNat(V1:S),V2:S) 0.002/0.002 U52(tt,V2:S) -> U53(isNatList(V2:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S) -> s(length(L:S)) 0.002/0.002 and(tt,X:S) -> X:S 0.002/0.002 isNat(length(V1:S)) -> U11(isNatIListKind(V1:S),V1:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNatKind(V1:S),V1:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S,V2:S)) -> U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatIList(V:S) -> U31(isNatIListKind(V:S),V:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S,V2:S)) -> and(isNatKind(V1:S),isNatIListKind(V2:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S)) -> isNatIListKind(V1:S) 0.002/0.002 isNatKind(0) -> tt 0.002/0.002 isNatKind(s(V1:S)) -> isNatKind(V1:S) 0.002/0.002 isNatList(cons(V1:S,V2:S)) -> U51(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S,L:S)) -> U61(and(and(isNatList(L:S),isNatIListKind(L:S)),and(isNat(N:S),isNatKind(N:S))),L:S) 0.002/0.002 length(nil) -> 0 0.002/0.002 zeros -> cons(0,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 L:S N:S V:S V1:S V2:S X: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) 0.002/0.002 (isNatIList) 0.002/0.002 (isNatIListKind) 0.002/0.002 (isNatKind) 0.002/0.002 (isNatList) 0.002/0.002 (length 1) 0.002/0.002 (zeros) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s 1) 0.002/0.002 (tt) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 U11(tt,V1:S) -> U12(isNatList(V1:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S) -> U22(isNat(V1:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S) -> U32(isNatList(V:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S,V2:S) -> U42(isNat(V1:S),V2:S) 0.002/0.002 U42(tt,V2:S) -> U43(isNatIList(V2:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S,V2:S) -> U52(isNat(V1:S),V2:S) 0.002/0.002 U52(tt,V2:S) -> U53(isNatList(V2:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S) -> s(length(L:S)) 0.002/0.002 and(tt,X:S) -> X:S 0.002/0.002 isNat(length(V1:S)) -> U11(isNatIListKind(V1:S),V1:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNatKind(V1:S),V1:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S,V2:S)) -> U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatIList(V:S) -> U31(isNatIListKind(V:S),V:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S,V2:S)) -> and(isNatKind(V1:S),isNatIListKind(V2:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S)) -> isNatIListKind(V1:S) 0.002/0.002 isNatKind(0) -> tt 0.002/0.002 isNatKind(s(V1:S)) -> isNatKind(V1:S) 0.002/0.002 isNatList(cons(V1:S,V2:S)) -> U51(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S,L:S)) -> U61(and(and(isNatList(L:S),isNatIListKind(L:S)),and(isNat(N:S),isNatKind(N:S))),L:S) 0.002/0.002 length(nil) -> 0 0.002/0.002 zeros -> cons(0,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) -> U12(isNatList(V1:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt,V1:S) -> U22(isNat(V1:S)) 0.002/0.002 U22(tt) -> tt 0.002/0.002 U31(tt,V:S) -> U32(isNatList(V:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,V1:S,V2:S) -> U42(isNat(V1:S),V2:S) 0.002/0.002 U42(tt,V2:S) -> U43(isNatIList(V2:S)) 0.002/0.002 U43(tt) -> tt 0.002/0.002 U51(tt,V1:S,V2:S) -> U52(isNat(V1:S),V2:S) 0.002/0.002 U52(tt,V2:S) -> U53(isNatList(V2:S)) 0.002/0.002 U53(tt) -> tt 0.002/0.002 U61(tt,L:S) -> s(length(L:S)) 0.002/0.002 and(tt,X:S) -> X:S 0.002/0.002 isNat(length(V1:S)) -> U11(isNatIListKind(V1:S),V1:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNatKind(V1:S),V1:S) 0.002/0.002 isNatIList(zeros) -> tt 0.002/0.002 isNatIList(cons(V1:S,V2:S)) -> U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatIList(V:S) -> U31(isNatIListKind(V:S),V:S) 0.002/0.002 isNatIListKind(zeros) -> tt 0.002/0.002 isNatIListKind(cons(V1:S,V2:S)) -> and(isNatKind(V1:S),isNatIListKind(V2:S)) 0.002/0.002 isNatIListKind(nil) -> tt 0.002/0.002 isNatKind(length(V1:S)) -> isNatIListKind(V1:S) 0.002/0.002 isNatKind(0) -> tt 0.002/0.002 isNatKind(s(V1:S)) -> isNatKind(V1:S) 0.002/0.002 isNatList(cons(V1:S,V2:S)) -> U51(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 isNatList(nil) -> tt 0.002/0.002 length(cons(N:S,L:S)) -> U61(and(and(isNatList(L:S),isNatIListKind(L:S)),and(isNat(N:S),isNatKind(N:S))),L:S) 0.002/0.002 length(nil) -> 0 0.002/0.002 zeros -> cons(0,zeros) 0.002/0.002 -> Vars: 0.002/0.002 V1, V1, V, V1, V2, V2, V1, V2, V2, L, X, V1, V1, V1, V2, V, V1, V2, V1, V1, V1, V2, L, N 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 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: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 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: [], UV-LFrozen: [V], UV-RFrozen: [V]) 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: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 8, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V2], UV-RFrozen: [V2]) 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: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 11, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V2], UV-RFrozen: [V2]) 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: [L], UV-LFrozen: [L], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 14, UV-LActive: [], UV-RActive: [X], UV-LFrozen: [X], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 15, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 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: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 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: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 20, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V], UV-RFrozen: [V]) 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: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 23, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 24, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 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: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 0.002/0.002 (UV-RuleId: 27, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 28, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 29, UV-LActive: [N], UV-RActive: [], UV-LFrozen: [L], UV-RFrozen: [L, N]) 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 x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30 0.002/0.002 -> PVars: 0.002/0.002 V1: [x7, x8, x10, x13, x18, x19, x20, x23, x25, x26, x27], V: [x9, x22], V2: [x11, x12, x14, x15, x21, x24, x28], L: [x16, x29], X: [x17], N: [x30] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: U11(tt,x7:S) -> U12(isNatList(x7:S)), id: 1, possubterms: U11(tt,x7: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,x8:S) -> U22(isNat(x8:S)), id: 3, possubterms: U21(tt,x8: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,x9:S) -> U32(isNatList(x9:S)), id: 5, possubterms: U31(tt,x9: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,x10:S,x11:S) -> U42(isNat(x10:S),x11:S), id: 7, possubterms: U41(tt,x10:S,x11:S)->[], tt->[1]) 0.002/0.002 (rule: U42(tt,x12:S) -> U43(isNatIList(x12:S)), id: 8, possubterms: U42(tt,x12: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,x13:S,x14:S) -> U52(isNat(x13:S),x14:S), id: 10, possubterms: U51(tt,x13:S,x14:S)->[], tt->[1]) 0.002/0.002 (rule: U52(tt,x15:S) -> U53(isNatList(x15:S)), id: 11, possubterms: U52(tt,x15: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,x16:S) -> s(length(x16:S)), id: 13, possubterms: U61(tt,x16:S)->[], tt->[1]) 0.002/0.002 (rule: and(tt,x17:S) -> x17:S, id: 14, possubterms: and(tt,x17:S)->[], tt->[1]) 0.002/0.002 (rule: isNat(length(x18:S)) -> U11(isNatIListKind(x18:S),x18:S), id: 15, possubterms: isNat(length(x18:S))->[]) 0.002/0.002 (rule: isNat(0) -> tt, id: 16, possubterms: isNat(0)->[]) 0.002/0.002 (rule: isNat(s(x19:S)) -> U21(isNatKind(x19:S),x19:S), id: 17, possubterms: isNat(s(x19:S))->[]) 0.002/0.002 (rule: isNatIList(zeros) -> tt, id: 18, possubterms: isNatIList(zeros)->[]) 0.002/0.002 (rule: isNatIList(cons(x20:S,x21:S)) -> U41(and(isNatKind(x20:S),isNatIListKind(x21:S)),x20:S,x21:S), id: 19, possubterms: isNatIList(cons(x20:S,x21:S))->[]) 0.002/0.002 (rule: isNatIList(x22:S) -> U31(isNatIListKind(x22:S),x22:S), id: 20, possubterms: isNatIList(x22:S)->[]) 0.002/0.002 (rule: isNatIListKind(zeros) -> tt, id: 21, possubterms: isNatIListKind(zeros)->[]) 0.002/0.002 (rule: isNatIListKind(cons(x23:S,x24:S)) -> and(isNatKind(x23:S),isNatIListKind(x24:S)), id: 22, possubterms: isNatIListKind(cons(x23:S,x24:S))->[]) 0.002/0.002 (rule: isNatIListKind(nil) -> tt, id: 23, possubterms: isNatIListKind(nil)->[]) 0.002/0.002 (rule: isNatKind(length(x25:S)) -> isNatIListKind(x25:S), id: 24, possubterms: isNatKind(length(x25:S))->[]) 0.002/0.002 (rule: isNatKind(0) -> tt, id: 25, possubterms: isNatKind(0)->[]) 0.002/0.002 (rule: isNatKind(s(x26:S)) -> isNatKind(x26:S), id: 26, possubterms: isNatKind(s(x26:S))->[]) 0.002/0.002 (rule: isNatList(cons(x27:S,x28:S)) -> U51(and(isNatKind(x27:S),isNatIListKind(x28:S)),x27:S,x28:S), id: 27, possubterms: isNatList(cons(x27:S,x28:S))->[]) 0.002/0.002 (rule: isNatList(nil) -> tt, id: 28, possubterms: isNatList(nil)->[]) 0.002/0.002 (rule: length(cons(x30:S,x29:S)) -> U61(and(and(isNatList(x29:S),isNatIListKind(x29:S)),and(isNat(x30:S),isNatKind(x30:S))),x29:S), id: 29, possubterms: length(cons(x30:S,x29:S))->[], cons(x30:S,x29:S)->[1]) 0.002/0.002 (rule: length(nil) -> 0, id: 30, possubterms: length(nil)->[], nil->[1]) 0.002/0.002 (rule: zeros -> cons(0,zeros), id: 31, possubterms: zeros->[]) 0.002/0.002 0.002/0.002 -> Unifications: 0.002/0.002 (R20 unifies with R18 at p: [], l: isNatIList(x22:S), lp: isNatIList(x22:S), sig: {x22:S -> zeros}, l': isNatIList(zeros), r: U31(isNatIListKind(x22:S),x22:S), r': tt) 0.002/0.002 (R20 unifies with R19 at p: [], l: isNatIList(x22:S), lp: isNatIList(x22:S), sig: {x22:S -> cons(V1:S,V2:S)}, l': isNatIList(cons(V1:S,V2:S)), r: U31(isNatIListKind(x22:S),x22:S), r': U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S)) 0.002/0.002 0.002/0.002 -> Critical pairs info: 0.002/0.002 => Not trivial, Overlay, NW0, N1 0.002/0.002 => Not trivial, Overlay, NW0, N2 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),isNatIListKind(V2:S)),V1:S,V2:S) 0.002/0.002 Nodes: [0] 0.002/0.002 Edges: [] 0.002/0.002 ID: 0 => ('U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S)', D0) 0.002/0.002 t: U31(isNatIListKind(cons(V1:S,V2:S)),cons(V1:S,V2: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,V2:S)),cons(V1:S,V2:S))', D0) 0.002/0.002 ID: 1 => ('U31(and(isNatKind(V1:S),isNatIListKind(V2:S)),cons(V1:S,V2:S))', D1, R22, P[1], S{x23:S -> V1:S, x24:S -> V2:S}), NR: 'and(isNatKind(V1:S),isNatIListKind(V2:S))' 0.002/0.002 U41(and(isNatKind(V1:S),isNatIListKind(V2:S)),V1:S,V2:S) ->* no union *<- U31(isNatIListKind(cons(V1:S,V2:S)),cons(V1:S,V2:S)) 0.002/0.002 "Not joinable" 0.002/0.002 0.002/0.002 The problem is not joinable. 0.002/0.002 0.00user 0.00system 0:00.02elapsed 62%CPU (0avgtext+0avgdata 11408maxresident)k 0.002/0.002 0inputs+0outputs (0major+1068minor)pagefaults 0swaps