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 M:S N:S V1:S V2: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 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U61 1) 0.002/0.002 (U71 1) 0.002/0.002 (U72 1) 0.002/0.002 (isNat) 0.002/0.002 (plus 1 2) 0.002/0.002 (x 1 2) 0.002/0.002 (0) 0.002/0.002 (fSNonEmpty) 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,V2:S) -> U12(isNat(V2:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt) -> tt 0.002/0.002 U31(tt,V2:S) -> U32(isNat(V2:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,N:S) -> N:S 0.002/0.002 U51(tt,M:S,N:S) -> U52(isNat(N:S),M:S,N:S) 0.002/0.002 U52(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.002/0.002 U61(tt) -> 0 0.002/0.002 U71(tt,M:S,N:S) -> U72(isNat(N:S),M:S,N:S) 0.002/0.002 U72(tt,M:S,N:S) -> plus(x(N:S,M:S),N:S) 0.002/0.002 isNat(plus(V1:S,V2:S)) -> U11(isNat(V1:S),V2:S) 0.002/0.002 isNat(x(V1:S,V2:S)) -> U31(isNat(V1:S),V2:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNat(V1:S)) 0.002/0.002 plus(N:S,0) -> U41(isNat(N:S),N:S) 0.002/0.002 plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 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 M:S N:S V1:S V2: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 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U61 1) 0.002/0.002 (U71 1) 0.002/0.002 (U72 1) 0.002/0.002 (isNat) 0.002/0.002 (plus 1 2) 0.002/0.002 (x 1 2) 0.002/0.002 (0) 0.002/0.002 (fSNonEmpty) 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,V2:S) -> U12(isNat(V2:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt) -> tt 0.002/0.002 U31(tt,V2:S) -> U32(isNat(V2:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,N:S) -> N:S 0.002/0.002 U51(tt,M:S,N:S) -> U52(isNat(N:S),M:S,N:S) 0.002/0.002 U52(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.002/0.002 U61(tt) -> 0 0.002/0.002 U71(tt,M:S,N:S) -> U72(isNat(N:S),M:S,N:S) 0.002/0.002 U72(tt,M:S,N:S) -> plus(x(N:S,M:S),N:S) 0.002/0.002 isNat(plus(V1:S,V2:S)) -> U11(isNat(V1:S),V2:S) 0.002/0.002 isNat(x(V1:S,V2:S)) -> U31(isNat(V1:S),V2:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNat(V1:S)) 0.002/0.002 plus(N:S,0) -> U41(isNat(N:S),N:S) 0.002/0.002 plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 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 M:S N:S V1:S V2: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 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U61 1) 0.002/0.002 (U71 1) 0.002/0.002 (U72 1) 0.002/0.002 (isNat) 0.002/0.002 (plus 1 2) 0.002/0.002 (x 1 2) 0.002/0.002 (0) 0.002/0.002 (fSNonEmpty) 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,V2:S) -> U12(isNat(V2:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt) -> tt 0.002/0.002 U31(tt,V2:S) -> U32(isNat(V2:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,N:S) -> N:S 0.002/0.002 U51(tt,M:S,N:S) -> U52(isNat(N:S),M:S,N:S) 0.002/0.002 U52(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.002/0.002 U61(tt) -> 0 0.002/0.002 U71(tt,M:S,N:S) -> U72(isNat(N:S),M:S,N:S) 0.002/0.002 U72(tt,M:S,N:S) -> plus(x(N:S,M:S),N:S) 0.002/0.002 isNat(plus(V1:S,V2:S)) -> U11(isNat(V1:S),V2:S) 0.002/0.002 isNat(x(V1:S,V2:S)) -> U31(isNat(V1:S),V2:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNat(V1:S)) 0.002/0.002 plus(N:S,0) -> U41(isNat(N:S),N:S) 0.002/0.002 plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 ) 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 Extended u-Critical Pairs NonLHRV Processor [JLAMP21]: 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x21:S,0) -> U41(isNat(x21:S),x21:S) 0.002/0.002 Rule 16 (l' :-> r') => plus(N:S,0) -> U41(isNat(N:S),N:S) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> plus(N:S,0)} 0.002/0.002 s => U41(isNat(plus(N:S,0)),plus(N:S,0)) 0.002/0.002 t => plus(U41(isNat(N:S),N:S),0) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x21:S,0) -> U41(isNat(x21:S),x21:S) 0.002/0.002 Rule 17 (l' :-> r') => plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> plus(N:S,s(M:S))} 0.002/0.002 s => U41(isNat(plus(N:S,s(M:S))),plus(N:S,s(M:S))) 0.002/0.002 t => plus(U51(isNat(M:S),M:S,N:S),0) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x21:S,0) -> U41(isNat(x21:S),x21:S) 0.002/0.002 Rule 18 (l' :-> r') => x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> x(N:S,0)} 0.002/0.002 s => U41(isNat(x(N:S,0)),x(N:S,0)) 0.002/0.002 t => plus(U61(isNat(N:S)),0) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x21:S,0) -> U41(isNat(x21:S),x21:S) 0.002/0.002 Rule 19 (l' :-> r') => x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> x(N:S,s(M:S))} 0.002/0.002 s => U41(isNat(x(N:S,s(M:S))),x(N:S,s(M:S))) 0.002/0.002 t => plus(U71(isNat(M:S),M:S,N:S),0) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x23:S,s(x22:S)) -> U51(isNat(x22:S),x22:S,x23:S) 0.002/0.002 Rule 17 (l' :-> r') => plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 Var => x22:S 0.002/0.002 Pos x22:S in l => [2,1] 0.002/0.002 Sigma => {x22:S -> plus(N:S,s(M:S))} 0.002/0.002 s => U51(isNat(plus(N:S,s(M:S))),plus(N:S,s(M:S)),x23:S) 0.002/0.002 t => plus(x23:S,s(U51(isNat(M:S),M:S,N:S))) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x23:S,s(x22:S)) -> U51(isNat(x22:S),x22:S,x23:S) 0.002/0.002 Rule 18 (l' :-> r') => x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 Var => x22:S 0.002/0.002 Pos x22:S in l => [2,1] 0.002/0.002 Sigma => {x22:S -> x(N:S,0)} 0.002/0.002 s => U51(isNat(x(N:S,0)),x(N:S,0),x23:S) 0.002/0.002 t => plus(x23:S,s(U61(isNat(N:S)))) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => plus(x23:S,s(x22:S)) -> U51(isNat(x22:S),x22:S,x23:S) 0.002/0.002 Rule 19 (l' :-> r') => x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 Var => x22:S 0.002/0.002 Pos x22:S in l => [2,1] 0.002/0.002 Sigma => {x22:S -> x(N:S,s(M:S))} 0.002/0.002 s => U51(isNat(x(N:S,s(M:S))),x(N:S,s(M:S)),x23:S) 0.002/0.002 t => plus(x23:S,s(U71(isNat(M:S),M:S,N:S))) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => x(x24:S,0) -> U61(isNat(x24:S)) 0.002/0.002 Rule 18 (l' :-> r') => x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 Var => x24:S 0.002/0.002 Pos x24:S in l => [1] 0.002/0.002 Sigma => {x24:S -> x(N:S,0)} 0.002/0.002 s => U61(isNat(x(N:S,0))) 0.002/0.002 t => x(U61(isNat(N:S)),0) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => x(x24:S,0) -> U61(isNat(x24:S)) 0.002/0.002 Rule 19 (l' :-> r') => x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 Var => x24:S 0.002/0.002 Pos x24:S in l => [1] 0.002/0.002 Sigma => {x24:S -> x(N:S,s(M:S))} 0.002/0.002 s => U61(isNat(x(N:S,s(M:S)))) 0.002/0.002 t => x(U71(isNat(M:S),M:S,N:S),0) 0.002/0.002 NW => 1 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => x(x26:S,s(x25:S)) -> U71(isNat(x25:S),x25:S,x26:S) 0.002/0.002 Rule 19 (l' :-> r') => x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 Var => x25:S 0.002/0.002 Pos x25:S in l => [2,1] 0.002/0.002 Sigma => {x25:S -> x(N:S,s(M:S))} 0.002/0.002 s => U71(isNat(x(N:S,s(M:S))),x(N:S,s(M:S)),x26:S) 0.002/0.002 t => x(x26:S,s(U71(isNat(M:S),M:S,N:S))) 0.002/0.002 NW => 1 0.002/0.002 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 M:S N:S V1:S V2: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 (U31 1) 0.002/0.002 (U32 1) 0.002/0.002 (U41 1) 0.002/0.002 (U51 1) 0.002/0.002 (U52 1) 0.002/0.002 (U61 1) 0.002/0.002 (U71 1) 0.002/0.002 (U72 1) 0.002/0.002 (isNat) 0.002/0.002 (plus 1 2) 0.002/0.002 (x 1 2) 0.002/0.002 (0) 0.002/0.002 (fSNonEmpty) 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,V2:S) -> U12(isNat(V2:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt) -> tt 0.002/0.002 U31(tt,V2:S) -> U32(isNat(V2:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,N:S) -> N:S 0.002/0.002 U51(tt,M:S,N:S) -> U52(isNat(N:S),M:S,N:S) 0.002/0.002 U52(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.002/0.002 U61(tt) -> 0 0.002/0.002 U71(tt,M:S,N:S) -> U72(isNat(N:S),M:S,N:S) 0.002/0.002 U72(tt,M:S,N:S) -> plus(x(N:S,M:S),N:S) 0.002/0.002 isNat(plus(V1:S,V2:S)) -> U11(isNat(V1:S),V2:S) 0.002/0.002 isNat(x(V1:S,V2:S)) -> U31(isNat(V1:S),V2:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNat(V1:S)) 0.002/0.002 plus(N:S,0) -> U41(isNat(N:S),N:S) 0.002/0.002 plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 ) 0.002/0.002 Critical Pairs: 0.002/0.002 => Not trivial, Not overlay, NW1, N1 0.002/0.002 => Not trivial, Not overlay, NW1, N2 0.002/0.002 => Not trivial, Not overlay, NW1, N3 0.002/0.002 => Not trivial, Not overlay, NW1, N4 0.002/0.002 => Not trivial, Not overlay, NW1, N5 0.002/0.002 => Not trivial, Not overlay, NW1, N6 0.002/0.002 => Not trivial, Not overlay, NW1, N7 0.002/0.002 => Not trivial, Not overlay, NW1, N8 0.002/0.002 => Not trivial, Not overlay, NW1, N9 0.002/0.002 => Not trivial, Not overlay, NW1, N10 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,V2:S) -> U12(isNat(V2:S)) 0.002/0.002 U12(tt) -> tt 0.002/0.002 U21(tt) -> tt 0.002/0.002 U31(tt,V2:S) -> U32(isNat(V2:S)) 0.002/0.002 U32(tt) -> tt 0.002/0.002 U41(tt,N:S) -> N:S 0.002/0.002 U51(tt,M:S,N:S) -> U52(isNat(N:S),M:S,N:S) 0.002/0.002 U52(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.002/0.002 U61(tt) -> 0 0.002/0.002 U71(tt,M:S,N:S) -> U72(isNat(N:S),M:S,N:S) 0.002/0.002 U72(tt,M:S,N:S) -> plus(x(N:S,M:S),N:S) 0.002/0.002 isNat(plus(V1:S,V2:S)) -> U11(isNat(V1:S),V2:S) 0.002/0.002 isNat(x(V1:S,V2:S)) -> U31(isNat(V1:S),V2:S) 0.002/0.002 isNat(0) -> tt 0.002/0.002 isNat(s(V1:S)) -> U21(isNat(V1:S)) 0.002/0.002 plus(N:S,0) -> U41(isNat(N:S),N:S) 0.002/0.002 plus(N:S,s(M:S)) -> U51(isNat(M:S),M:S,N:S) 0.002/0.002 x(N:S,0) -> U61(isNat(N:S)) 0.002/0.002 x(N:S,s(M:S)) -> U71(isNat(M:S),M:S,N:S) 0.002/0.002 -> Vars: 0.002/0.002 V2, V2, N, M, N, M, N, M, N, M, N, V1, V2, V1, V2, V1, N, M, N, N, M, N 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V2], UV-RFrozen: [V2]) 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: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 4, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V2], UV-RFrozen: [V2]) 0.002/0.002 (UV-RuleId: 5, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [N], UV-LFrozen: [N], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 7, UV-LActive: [], UV-RActive: [], UV-LFrozen: [M, N], UV-RFrozen: [M, N]) 0.002/0.002 (UV-RuleId: 8, UV-LActive: [], UV-RActive: [M, N], UV-LFrozen: [M, N], 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: [], UV-LFrozen: [M, N], UV-RFrozen: [M, N]) 0.002/0.002 (UV-RuleId: 11, UV-LActive: [], UV-RActive: [M, N], UV-LFrozen: [M, N], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 12, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 13, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.002/0.002 (UV-RuleId: 14, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], 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: [N], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [N]) 0.002/0.002 (UV-RuleId: 17, UV-LActive: [M, N], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [M, N]) 0.002/0.002 (UV-RuleId: 18, UV-LActive: [N], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [N]) 0.002/0.002 (UV-RuleId: 19, UV-LActive: [M, N], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [M, N]) 0.002/0.002 -> FVars: 0.002/0.002 x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26 0.002/0.002 -> PVars: 0.002/0.002 V2: [x5, x6, x17, x19], N: [x7, x9, x11, x13, x15, x21, x23, x24, x26], M: [x8, x10, x12, x14, x22, x25], V1: [x16, x18, x20] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: U11(tt,x5:S) -> U12(isNat(x5:S)), id: 1, possubterms: U11(tt,x5: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) -> tt, id: 3, possubterms: U21(tt)->[], tt->[1]) 0.002/0.002 (rule: U31(tt,x6:S) -> U32(isNat(x6:S)), id: 4, possubterms: U31(tt,x6:S)->[], tt->[1]) 0.002/0.002 (rule: U32(tt) -> tt, id: 5, possubterms: U32(tt)->[], tt->[1]) 0.002/0.002 (rule: U41(tt,x7:S) -> x7:S, id: 6, possubterms: U41(tt,x7:S)->[], tt->[1]) 0.002/0.002 (rule: U51(tt,x8:S,x9:S) -> U52(isNat(x9:S),x8:S,x9:S), id: 7, possubterms: U51(tt,x8:S,x9:S)->[], tt->[1]) 0.002/0.002 (rule: U52(tt,x10:S,x11:S) -> s(plus(x11:S,x10:S)), id: 8, possubterms: U52(tt,x10:S,x11:S)->[], tt->[1]) 0.002/0.002 (rule: U61(tt) -> 0, id: 9, possubterms: U61(tt)->[], tt->[1]) 0.002/0.002 (rule: U71(tt,x12:S,x13:S) -> U72(isNat(x13:S),x12:S,x13:S), id: 10, possubterms: U71(tt,x12:S,x13:S)->[], tt->[1]) 0.002/0.002 (rule: U72(tt,x14:S,x15:S) -> plus(x(x15:S,x14:S),x15:S), id: 11, possubterms: U72(tt,x14:S,x15:S)->[], tt->[1]) 0.002/0.002 (rule: isNat(plus(x16:S,x17:S)) -> U11(isNat(x16:S),x17:S), id: 12, possubterms: isNat(plus(x16:S,x17:S))->[]) 0.002/0.002 (rule: isNat(x(x18:S,x19:S)) -> U31(isNat(x18:S),x19:S), id: 13, possubterms: isNat(x(x18:S,x19:S))->[]) 0.002/0.002 (rule: isNat(0) -> tt, id: 14, possubterms: isNat(0)->[]) 0.002/0.002 (rule: isNat(s(x20:S)) -> U21(isNat(x20:S)), id: 15, possubterms: isNat(s(x20:S))->[]) 0.002/0.002 (rule: plus(x21:S,0) -> U41(isNat(x21:S),x21:S), id: 16, possubterms: plus(x21:S,0)->[], 0->[2]) 0.002/0.002 (rule: plus(x23:S,s(x22:S)) -> U51(isNat(x22:S),x22:S,x23:S), id: 17, possubterms: plus(x23:S,s(x22:S))->[], s(x22:S)->[2]) 0.002/0.002 (rule: x(x24:S,0) -> U61(isNat(x24:S)), id: 18, possubterms: x(x24:S,0)->[], 0->[2]) 0.002/0.002 (rule: x(x26:S,s(x25:S)) -> U71(isNat(x25:S),x25:S,x26:S), id: 19, possubterms: x(x26:S,s(x25:S))->[], s(x25:S)->[2]) 0.002/0.002 0.002/0.002 -> Unifications: 0.002/0.002 0.002/0.002 0.002/0.002 -> Critical pairs info: 0.002/0.002 => Not trivial, Not overlay, NW1, N1 0.002/0.002 => Not trivial, Not overlay, NW1, N2 0.002/0.002 => Not trivial, Not overlay, NW1, N3 0.002/0.002 => Not trivial, Not overlay, NW1, N4 0.002/0.002 => Not trivial, Not overlay, NW1, N5 0.002/0.002 => Not trivial, Not overlay, NW1, N6 0.002/0.002 => Not trivial, Not overlay, NW1, N7 0.002/0.002 => Not trivial, Not overlay, NW1, N8 0.002/0.002 => Not trivial, Not overlay, NW1, N9 0.002/0.002 => Not trivial, Not overlay, NW1, N10 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: U51(isNat(plus(N:S,s(M:S))),plus(N:S,s(M:S)),N:S) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('U51(isNat(plus(N:S,s(M:S))),plus(N:S,s(M:S)),N:S)', D0) 0.002/0.002 ID: 1 => ('U51(U11(isNat(N:S),s(M:S)),plus(N:S,s(M:S)),N:S)', D1, R12, P[1], S{x16:S -> N:S, x17:S -> s(M:S)}), NR: 'U11(isNat(N:S),s(M:S))' 0.002/0.002 t: plus(N:S,s(U51(isNat(M:S),M:S,N:S))) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('plus(N:S,s(U51(isNat(M:S),M:S,N:S)))', D0) 0.002/0.002 ID: 1 => ('U51(isNat(U51(isNat(M:S),M:S,N:S)),U51(isNat(M:S),M:S,N:S),N:S)', D1, R17, P[], S{x22:S -> U51(isNat(M:S),M:S,N:S), x23:S -> N:S}), NR: 'U51(isNat(U51(isNat(M:S),M:S,N:S)),U51(isNat(M:S),M:S,N:S),N:S)' 0.002/0.002 U51(isNat(plus(N:S,s(M:S))),plus(N:S,s(M:S)),N:S) ->* no union *<- plus(N:S,s(U51(isNat(M:S),M:S,N: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 72%CPU (0avgtext+0avgdata 11368maxresident)k 0.002/0.002 0inputs+0outputs (0major+1129minor)pagefaults 0swaps