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 N:S X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1 2) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms 1) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (recip 1) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(0,X:S) -> X:S 0.001/0.001 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.001/0.001 dbl(0) -> 0 0.001/0.001 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.001/0.001 first(0,X:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 sqr(0) -> 0 0.001/0.001 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.001/0.001 terms(N:S) -> cons(recip(sqr(N:S)),terms(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 N:S X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1 2) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms 1) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (recip 1) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(0,X:S) -> X:S 0.001/0.001 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.001/0.001 dbl(0) -> 0 0.001/0.001 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.001/0.001 first(0,X:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 sqr(0) -> 0 0.001/0.001 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.001/0.001 terms(N:S) -> cons(recip(sqr(N:S)),terms(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 N:S X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1 2) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms 1) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (recip 1) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(0,X:S) -> X:S 0.001/0.001 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.001/0.001 dbl(0) -> 0 0.001/0.001 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.001/0.001 first(0,X:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 sqr(0) -> 0 0.001/0.001 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.001/0.001 terms(N:S) -> cons(recip(sqr(N:S)),terms(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) => first(s(x10:S),cons(x11:S,x12:S)) -> cons(x11:S,first(x10:S,x12:S)) 0.001/0.001 Rule 6 (l' :-> r') => first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 Var => x10:S 0.001/0.001 Pos x10:S in l => [1,1] 0.001/0.001 Sigma => {x10:S -> first(s(X:S),cons(Y:S,Z:S))} 0.001/0.001 s => cons(x11:S,first(first(s(X:S),cons(Y:S,Z:S)),x12:S)) 0.001/0.001 t => first(s(cons(Y:S,first(X:S,Z:S))),cons(x11:S,x12:S)) 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) => first(s(x10:S),cons(x11:S,x12:S)) -> cons(x11:S,first(x10:S,x12:S)) 0.001/0.001 Rule 7 (l' :-> r') => sqr(0) -> 0 0.001/0.001 Var => x10:S 0.001/0.001 Pos x10:S in l => [1,1] 0.001/0.001 Sigma => {x10:S -> sqr(0)} 0.001/0.001 s => cons(x11:S,first(sqr(0),x12:S)) 0.001/0.001 t => first(s(0),cons(x11:S,x12:S)) 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) => first(s(x10:S),cons(x11:S,x12:S)) -> cons(x11:S,first(x10:S,x12:S)) 0.001/0.001 Rule 8 (l' :-> r') => sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.001/0.001 Var => x10:S 0.001/0.001 Pos x10:S in l => [1,1] 0.001/0.001 Sigma => {x10:S -> sqr(s(X:S))} 0.001/0.001 s => cons(x11:S,first(sqr(s(X:S)),x12:S)) 0.001/0.001 t => first(s(s(add(sqr(X:S),dbl(X:S)))),cons(x11:S,x12:S)) 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) => first(s(x10:S),cons(x11:S,x12:S)) -> cons(x11:S,first(x10:S,x12:S)) 0.001/0.001 Rule 9 (l' :-> r') => terms(N:S) -> cons(recip(sqr(N:S)),terms(s(N:S))) 0.001/0.001 Var => x10:S 0.001/0.001 Pos x10:S in l => [1,1] 0.001/0.001 Sigma => {x10:S -> terms(N:S)} 0.001/0.001 s => cons(x11:S,first(terms(N:S),x12:S)) 0.001/0.001 t => first(s(cons(recip(sqr(N:S)),terms(s(N:S)))),cons(x11:S,x12:S)) 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) => terms(x14:S) -> cons(recip(sqr(x14:S)),terms(s(x14:S))) 0.001/0.001 Rule 9 (l' :-> r') => terms(N:S) -> cons(recip(sqr(N:S)),terms(s(N:S))) 0.001/0.001 Var => x14:S 0.001/0.001 Pos x14:S in l => [1] 0.001/0.001 Sigma => {x14:S -> terms(N:S)} 0.001/0.001 s => cons(recip(sqr(terms(N:S))),terms(s(terms(N:S)))) 0.001/0.001 t => terms(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.001/0.001 NW => 0 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 N:S X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1 2) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms 1) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (recip 1) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(0,X:S) -> X:S 0.001/0.001 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.001/0.001 dbl(0) -> 0 0.001/0.001 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.001/0.001 first(0,X:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 sqr(0) -> 0 0.001/0.001 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.001/0.001 terms(N:S) -> cons(recip(sqr(N:S)),terms(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 => Not trivial, Not overlay, NW1, N4 0.001/0.001 => Not trivial, Not overlay, NW0, N5 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 0.001/0.001 Huet Levy Processor: 0.001/0.001 -> Rules: 0.001/0.001 add(0,X:S) -> X:S 0.001/0.001 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.001/0.001 dbl(0) -> 0 0.001/0.001 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.001/0.001 first(0,X:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 sqr(0) -> 0 0.001/0.001 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.001/0.001 terms(N:S) -> cons(recip(sqr(N:S)),terms(s(N:S))) 0.001/0.001 -> Vars: 0.001/0.001 X, X, Y, X, X, X, Y, Z, X, N 0.001/0.001 -> UVars: 0.001/0.001 (UV-RuleId: 1, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 2, UV-LActive: [X, Y], UV-RActive: [X, Y], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 3, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 4, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 5, UV-LActive: [X], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 6, UV-LActive: [X, Y], UV-RActive: [Y], UV-LFrozen: [Z], UV-RFrozen: [X, Z]) 0.001/0.001 (UV-RuleId: 7, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 8, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 9, UV-LActive: [N], UV-RActive: [N], UV-LFrozen: [], UV-RFrozen: [N]) 0.001/0.001 -> FVars: 0.001/0.001 x5, x6, x7, x8, x9, x10, x11, x12, x13, x14 0.001/0.001 -> PVars: 0.001/0.001 X: [x5, x6, x8, x9, x10, x13], Y: [x7, x11], Z: [x12], N: [x14] 0.001/0.001 0.001/0.001 -> Rlps: 0.001/0.001 (rule: add(0,x5:S) -> x5:S, id: 1, possubterms: add(0,x5:S)->[], 0->[1]) 0.001/0.001 (rule: add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)), id: 2, possubterms: add(s(x6:S),x7:S)->[], s(x6:S)->[1]) 0.001/0.001 (rule: dbl(0) -> 0, id: 3, possubterms: dbl(0)->[], 0->[1]) 0.001/0.001 (rule: dbl(s(x8:S)) -> s(s(dbl(x8:S))), id: 4, possubterms: dbl(s(x8:S))->[], s(x8:S)->[1]) 0.001/0.001 (rule: first(0,x9:S) -> nil, id: 5, possubterms: first(0,x9:S)->[], 0->[1]) 0.001/0.001 (rule: first(s(x10:S),cons(x11:S,x12:S)) -> cons(x11:S,first(x10:S,x12:S)), id: 6, possubterms: first(s(x10:S),cons(x11:S,x12:S))->[], s(x10:S)->[1], cons(x11:S,x12:S)->[2]) 0.001/0.001 (rule: sqr(0) -> 0, id: 7, possubterms: sqr(0)->[], 0->[1]) 0.001/0.001 (rule: sqr(s(x13:S)) -> s(add(sqr(x13:S),dbl(x13:S))), id: 8, possubterms: sqr(s(x13:S))->[], s(x13:S)->[1]) 0.001/0.001 (rule: terms(x14:S) -> cons(recip(sqr(x14:S)),terms(s(x14:S))), id: 9, possubterms: terms(x14:S)->[]) 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 => Not trivial, Not overlay, NW0, N4 0.001/0.001 => Not trivial, Not overlay, NW1, N5 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: cons(Y:S,first(sqr(s(X:S)),Z:S)) 0.001/0.001 Nodes: [0] 0.001/0.001 Edges: [] 0.001/0.001 ID: 0 => ('cons(Y:S,first(sqr(s(X:S)),Z:S))', D0) 0.001/0.001 t: first(s(s(add(sqr(X:S),dbl(X:S)))),cons(Y:S,Z:S)) 0.001/0.001 Nodes: [0,1] 0.001/0.001 Edges: [(0,1)] 0.001/0.001 ID: 0 => ('first(s(s(add(sqr(X:S),dbl(X:S)))),cons(Y:S,Z:S))', D0) 0.001/0.001 ID: 1 => ('cons(Y:S,first(s(add(sqr(X:S),dbl(X:S))),Z:S))', D1, R6, P[], S{x10:S -> s(add(sqr(X:S),dbl(X:S))), x11:S -> Y:S, x12:S -> Z:S}), NR: 'cons(Y:S,first(s(add(sqr(X:S),dbl(X:S))),Z:S))' 0.001/0.001 cons(Y:S,first(sqr(s(X:S)),Z:S)) ->* no union *<- first(s(s(add(sqr(X:S),dbl(X:S)))),cons(Y:S,Z:S)) 0.001/0.001 "Not joinable" 0.001/0.001 0.001/0.001 The problem is not joinable. 0.001/0.001 0.01user 0.00system 0:00.01elapsed 84%CPU (0avgtext+0avgdata 10648maxresident)k 0.001/0.001 0inputs+0outputs (0major+963minor)pagefaults 0swaps