0.001/0.001 NO 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 0.001/0.001 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 Confluence Problem: 0.001/0.001 (VAR vNonEmpty:S M:S N:S V1:S V2:S X:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (U11 1) 0.001/0.001 (U21 1) 0.001/0.001 (and 1) 0.001/0.001 (isNat) 0.001/0.001 (plus 1 2) 0.001/0.001 (0) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (s 1) 0.001/0.001 (tt) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 U11(tt,N:S) -> N:S 0.001/0.001 U21(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.001/0.001 and(tt,X:S) -> X:S 0.001/0.001 isNat(plus(V1:S,V2:S)) -> and(isNat(V1:S),isNat(V2:S)) 0.001/0.001 isNat(0) -> tt 0.001/0.001 isNat(s(V1:S)) -> isNat(V1:S) 0.001/0.001 plus(N:S,0) -> U11(isNat(N:S),N:S) 0.001/0.001 plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 ) 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 0.001/0.001 CleanTRS Processor: 0.001/0.001 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 Confluence Problem: 0.001/0.001 (VAR vNonEmpty:S M:S N:S V1:S V2:S X:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (U11 1) 0.001/0.001 (U21 1) 0.001/0.001 (and 1) 0.001/0.001 (isNat) 0.001/0.001 (plus 1 2) 0.001/0.001 (0) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (s 1) 0.001/0.001 (tt) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 U11(tt,N:S) -> N:S 0.001/0.001 U21(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.001/0.001 and(tt,X:S) -> X:S 0.001/0.001 isNat(plus(V1:S,V2:S)) -> and(isNat(V1:S),isNat(V2:S)) 0.001/0.001 isNat(0) -> tt 0.001/0.001 isNat(s(V1:S)) -> isNat(V1:S) 0.001/0.001 plus(N:S,0) -> U11(isNat(N:S),N:S) 0.001/0.001 plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 ) 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 0.001/0.001 Modular Confluence Combinations Decomposition Processor: 0.001/0.001 It is a CTRS -> No modular confluence 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 CS-TRS Processor: 0.001/0.001 R is a CS-TRS 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 Confluence Problem: 0.001/0.001 (VAR vNonEmpty:S M:S N:S V1:S V2:S X:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (U11 1) 0.001/0.001 (U21 1) 0.001/0.001 (and 1) 0.001/0.001 (isNat) 0.001/0.001 (plus 1 2) 0.001/0.001 (0) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (s 1) 0.001/0.001 (tt) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 U11(tt,N:S) -> N:S 0.001/0.001 U21(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.001/0.001 and(tt,X:S) -> X:S 0.001/0.001 isNat(plus(V1:S,V2:S)) -> and(isNat(V1:S),isNat(V2:S)) 0.001/0.001 isNat(0) -> tt 0.001/0.001 isNat(s(V1:S)) -> isNat(V1:S) 0.001/0.001 plus(N:S,0) -> U11(isNat(N:S),N:S) 0.001/0.001 plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 ) 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 0.001/0.001 Extended u-Critical Pairs NonLHRV Processor [JLAMP21]: 0.001/0.001 ->Extended u-Critical Pair: 0.001/0.001 Rule 1 (l :-> r) => plus(x13:S,0) -> U11(isNat(x13:S),x13:S) 0.001/0.001 Rule 7 (l' :-> r') => plus(N:S,0) -> U11(isNat(N:S),N:S) 0.001/0.001 Var => x13:S 0.001/0.001 Pos x13:S in l => [1] 0.001/0.001 Sigma => {x13:S -> plus(N:S,0)} 0.001/0.001 s => U11(isNat(plus(N:S,0)),plus(N:S,0)) 0.001/0.001 t => plus(U11(isNat(N:S),N:S),0) 0.001/0.001 NW => 1 0.001/0.001 0.001/0.001 0.001/0.001 ->Extended u-Critical Pair: 0.001/0.001 Rule 1 (l :-> r) => plus(x13:S,0) -> U11(isNat(x13:S),x13:S) 0.001/0.001 Rule 8 (l' :-> r') => plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 Var => x13:S 0.001/0.001 Pos x13:S in l => [1] 0.001/0.001 Sigma => {x13:S -> plus(N:S,s(M:S))} 0.001/0.001 s => U11(isNat(plus(N:S,s(M:S))),plus(N:S,s(M:S))) 0.001/0.001 t => plus(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S),0) 0.001/0.001 NW => 1 0.001/0.001 0.001/0.001 0.001/0.001 ->Extended u-Critical Pair: 0.001/0.001 Rule 1 (l :-> r) => plus(x15:S,s(x14:S)) -> U21(and(isNat(x14:S),isNat(x15:S)),x14:S,x15:S) 0.001/0.001 Rule 8 (l' :-> r') => plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 Var => x14:S 0.001/0.001 Pos x14:S in l => [2,1] 0.001/0.001 Sigma => {x14:S -> plus(N:S,s(M:S))} 0.001/0.001 s => U21(and(isNat(plus(N:S,s(M:S))),isNat(x15:S)),plus(N:S,s(M:S)),x15:S) 0.001/0.001 t => plus(x15:S,s(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S))) 0.001/0.001 NW => 1 0.001/0.001 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 Confluence Problem: 0.001/0.001 (VAR vNonEmpty:S M:S N:S V1:S V2:S X:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (U11 1) 0.001/0.001 (U21 1) 0.001/0.001 (and 1) 0.001/0.001 (isNat) 0.001/0.001 (plus 1 2) 0.001/0.001 (0) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (s 1) 0.001/0.001 (tt) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 U11(tt,N:S) -> N:S 0.001/0.001 U21(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.001/0.001 and(tt,X:S) -> X:S 0.001/0.001 isNat(plus(V1:S,V2:S)) -> and(isNat(V1:S),isNat(V2:S)) 0.001/0.001 isNat(0) -> tt 0.001/0.001 isNat(s(V1:S)) -> isNat(V1:S) 0.001/0.001 plus(N:S,0) -> U11(isNat(N:S),N:S) 0.001/0.001 plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 ) 0.001/0.001 Critical Pairs: 0.001/0.001 => Not trivial, Not overlay, NW1, N1 0.001/0.001 => Not trivial, Not overlay, NW1, N2 0.001/0.001 => Not trivial, Not overlay, NW1, N3 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 0.001/0.001 Huet Levy Processor: 0.001/0.001 -> Rules: 0.001/0.001 U11(tt,N:S) -> N:S 0.001/0.001 U21(tt,M:S,N:S) -> s(plus(N:S,M:S)) 0.001/0.001 and(tt,X:S) -> X:S 0.001/0.001 isNat(plus(V1:S,V2:S)) -> and(isNat(V1:S),isNat(V2:S)) 0.001/0.001 isNat(0) -> tt 0.001/0.001 isNat(s(V1:S)) -> isNat(V1:S) 0.001/0.001 plus(N:S,0) -> U11(isNat(N:S),N:S) 0.001/0.001 plus(N:S,s(M:S)) -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S) 0.001/0.001 -> Vars: 0.001/0.001 N, M, N, X, V1, V2, V1, N, M, N 0.001/0.001 -> UVars: 0.001/0.001 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [N], UV-LFrozen: [N], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 2, UV-LActive: [], UV-RActive: [M, N], UV-LFrozen: [M, N], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 3, UV-LActive: [], UV-RActive: [X], UV-LFrozen: [X], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 4, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1, V2], UV-RFrozen: [V1, V2]) 0.001/0.001 (UV-RuleId: 5, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [], UV-LFrozen: [V1], UV-RFrozen: [V1]) 0.001/0.001 (UV-RuleId: 7, UV-LActive: [N], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [N]) 0.001/0.001 (UV-RuleId: 8, UV-LActive: [M, N], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [M, N]) 0.001/0.001 -> FVars: 0.001/0.001 x6, x7, x8, x9, x10, x11, x12, x13, x14, x15 0.001/0.001 -> PVars: 0.001/0.001 N: [x6, x8, x13, x15], M: [x7, x14], X: [x9], V1: [x10, x12], V2: [x11] 0.001/0.001 0.001/0.001 -> Rlps: 0.001/0.001 (rule: U11(tt,x6:S) -> x6:S, id: 1, possubterms: U11(tt,x6:S)->[], tt->[1]) 0.001/0.001 (rule: U21(tt,x7:S,x8:S) -> s(plus(x8:S,x7:S)), id: 2, possubterms: U21(tt,x7:S,x8:S)->[], tt->[1]) 0.001/0.001 (rule: and(tt,x9:S) -> x9:S, id: 3, possubterms: and(tt,x9:S)->[], tt->[1]) 0.001/0.001 (rule: isNat(plus(x10:S,x11:S)) -> and(isNat(x10:S),isNat(x11:S)), id: 4, possubterms: isNat(plus(x10:S,x11:S))->[]) 0.001/0.001 (rule: isNat(0) -> tt, id: 5, possubterms: isNat(0)->[]) 0.001/0.001 (rule: isNat(s(x12:S)) -> isNat(x12:S), id: 6, possubterms: isNat(s(x12:S))->[]) 0.001/0.001 (rule: plus(x13:S,0) -> U11(isNat(x13:S),x13:S), id: 7, possubterms: plus(x13:S,0)->[], 0->[2]) 0.001/0.001 (rule: plus(x15:S,s(x14:S)) -> U21(and(isNat(x14:S),isNat(x15:S)),x14:S,x15:S), id: 8, possubterms: plus(x15:S,s(x14:S))->[], s(x14:S)->[2]) 0.001/0.001 0.001/0.001 -> Unifications: 0.001/0.001 0.001/0.001 0.001/0.001 -> Critical pairs info: 0.001/0.001 => Not trivial, Not overlay, NW1, N1 0.001/0.001 => Not trivial, Not overlay, NW1, N2 0.001/0.001 => Not trivial, Not overlay, NW1, N3 0.001/0.001 0.001/0.001 -> Problem conclusions: 0.001/0.001 Left linear, Not right linear, Not linear 0.001/0.001 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.001/0.001 Not Huet-Levy confluent, Not Newman confluent 0.001/0.001 R is a CS-TRS, Not left-homogeneous u-replacing variables 0.001/0.001 0.001/0.001 0.001/0.001 Problem 1: 0.001/0.001 No Convergence Brute Force Processor: 0.001/0.001 -> Rewritings: 0.001/0.001 s: U21(and(isNat(plus(N:S,s(M:S))),isNat(N:S)),plus(N:S,s(M:S)),N:S) 0.001/0.001 Nodes: [0,1] 0.001/0.001 Edges: [(0,1)] 0.001/0.001 ID: 0 => ('U21(and(isNat(plus(N:S,s(M:S))),isNat(N:S)),plus(N:S,s(M:S)),N:S)', D0) 0.001/0.001 ID: 1 => ('U21(and(and(isNat(N:S),isNat(s(M:S))),isNat(N:S)),plus(N:S,s(M:S)),N:S)', D1, R4, P[1, 1], S{x10:S -> N:S, x11:S -> s(M:S)}), NR: 'and(isNat(N:S),isNat(s(M:S)))' 0.001/0.001 t: plus(N:S,s(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S))) 0.001/0.001 Nodes: [0,1] 0.001/0.001 Edges: [(0,1)] 0.001/0.001 ID: 0 => ('plus(N:S,s(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)))', D0) 0.001/0.001 ID: 1 => ('U21(and(isNat(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)),isNat(N:S)),U21(and(isNat(M:S),isNat(N:S)),M:S,N:S),N:S)', D1, R8, P[], S{x14:S -> U21(and(isNat(M:S),isNat(N:S)),M:S,N:S), x15:S -> N:S}), NR: 'U21(and(isNat(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S)),isNat(N:S)),U21(and(isNat(M:S),isNat(N:S)),M:S,N:S),N:S)' 0.001/0.001 U21(and(isNat(plus(N:S,s(M:S))),isNat(N:S)),plus(N:S,s(M:S)),N:S) ->* no union *<- plus(N:S,s(U21(and(isNat(M:S),isNat(N:S)),M:S,N:S))) 0.001/0.001 "Not joinable" 0.001/0.001 0.001/0.001 The problem is not joinable. 0.001/0.001 0.00user 0.00system 0:00.01elapsed 50%CPU (0avgtext+0avgdata 10128maxresident)k 0.001/0.001 0inputs+0outputs (0major+847minor)pagefaults 0swaps