0.001/0.001 YES 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 v_NonEmpty:S:S N:S:S X:S:S Y:S:S Z:S:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms) 0.001/0.001 (cons) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (num0) 0.001/0.001 (recip 1) 0.001/0.001 (s) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(num0,X:S:S) -> X:S:S 0.001/0.001 add(s(X:S:S),Y:S:S) -> s(add(X:S:S,Y:S:S)) 0.001/0.001 dbl(num0) -> num0 0.001/0.001 dbl(s(X:S:S)) -> s(s(dbl(X:S:S))) 0.001/0.001 first(num0,X:S:S) -> nil 0.001/0.001 first(s(X:S:S),cons(Y:S:S,Z:S:S)) -> cons(Y:S:S,first(X:S:S,Z:S:S)) 0.001/0.001 sqr(num0) -> num0 0.001/0.001 sqr(s(X:S:S)) -> s(add(sqr(X:S:S),dbl(X:S:S))) 0.001/0.001 terms(N:S:S) -> cons(recip(sqr(N:S:S)),terms(s(N:S: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 v_NonEmpty:S:S N:S:S X:S:S Y:S:S Z:S:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms) 0.001/0.001 (cons) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (num0) 0.001/0.001 (recip 1) 0.001/0.001 (s) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(num0,X:S:S) -> X:S:S 0.001/0.001 add(s(X:S:S),Y:S:S) -> s(add(X:S:S,Y:S:S)) 0.001/0.001 dbl(num0) -> num0 0.001/0.001 dbl(s(X:S:S)) -> s(s(dbl(X:S:S))) 0.001/0.001 first(num0,X:S:S) -> nil 0.001/0.001 first(s(X:S:S),cons(Y:S:S,Z:S:S)) -> cons(Y:S:S,first(X:S:S,Z:S:S)) 0.001/0.001 sqr(num0) -> num0 0.001/0.001 sqr(s(X:S:S)) -> s(add(sqr(X:S:S),dbl(X:S:S))) 0.001/0.001 terms(N:S:S) -> cons(recip(sqr(N:S:S)),terms(s(N:S: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 Left-Homogeneous u-Replacing Variables Processor: 0.001/0.001 R satisfies LHRV condition 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 v_NonEmpty:S:S N:S:S X:S:S Y:S:S Z:S:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (add 1) 0.001/0.001 (dbl 1) 0.001/0.001 (first 1 2) 0.001/0.001 (sqr 1) 0.001/0.001 (terms) 0.001/0.001 (cons) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (num0) 0.001/0.001 (recip 1) 0.001/0.001 (s) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 add(num0,X:S:S) -> X:S:S 0.001/0.001 add(s(X:S:S),Y:S:S) -> s(add(X:S:S,Y:S:S)) 0.001/0.001 dbl(num0) -> num0 0.001/0.001 dbl(s(X:S:S)) -> s(s(dbl(X:S:S))) 0.001/0.001 first(num0,X:S:S) -> nil 0.001/0.001 first(s(X:S:S),cons(Y:S:S,Z:S:S)) -> cons(Y:S:S,first(X:S:S,Z:S:S)) 0.001/0.001 sqr(num0) -> num0 0.001/0.001 sqr(s(X:S:S)) -> s(add(sqr(X:S:S),dbl(X:S:S))) 0.001/0.001 terms(N:S:S) -> cons(recip(sqr(N:S:S)),terms(s(N:S:S))) 0.001/0.001 ) 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(num0,X:S:S) -> X:S:S 0.001/0.001 add(s(X:S:S),Y:S:S) -> s(add(X:S:S,Y:S:S)) 0.001/0.001 dbl(num0) -> num0 0.001/0.001 dbl(s(X:S:S)) -> s(s(dbl(X:S:S))) 0.001/0.001 first(num0,X:S:S) -> nil 0.001/0.001 first(s(X:S:S),cons(Y:S:S,Z:S:S)) -> cons(Y:S:S,first(X:S:S,Z:S:S)) 0.001/0.001 sqr(num0) -> num0 0.001/0.001 sqr(s(X:S:S)) -> s(add(sqr(X:S:S),dbl(X:S:S))) 0.001/0.001 terms(N:S:S) -> cons(recip(sqr(N:S:S)),terms(s(N:S:S))) 0.001/0.001 -> Vars: 0.001/0.001 X:S, X:S, Y:S, X:S, X:S, X:S, Y:S, Z:S, X:S, N:S 0.001/0.001 -> UVars: 0.001/0.001 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [X:S], UV-LFrozen: [X:S], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 2, UV-LActive: [], UV-RActive: [], UV-LFrozen: [X:S, Y:S], UV-RFrozen: [X:S, Y:S]) 0.001/0.001 (UV-RuleId: 3, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 4, UV-LActive: [], UV-RActive: [], UV-LFrozen: [X:S], UV-RFrozen: [X:S]) 0.001/0.001 (UV-RuleId: 5, UV-LActive: [X:S], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [], UV-LFrozen: [X:S, Y:S, Z:S], UV-RFrozen: [X:S, Y:S, Z:S]) 0.001/0.001 (UV-RuleId: 7, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 8, UV-LActive: [], UV-RActive: [], UV-LFrozen: [X:S], UV-RFrozen: [X:S]) 0.001/0.001 (UV-RuleId: 9, UV-LActive: [], UV-RActive: [], UV-LFrozen: [N:S], UV-RFrozen: [N:S]) 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 X:S: [x6, x7, x9, x10, x11, x14], Y:S: [x8, x12], Z:S: [x13], N:S: [x15] 0.001/0.001 0.001/0.001 -> Rlps: 0.001/0.001 (rule: add(num0,x6:S) -> x6:S, id: 1, possubterms: add(num0,x6:S)->[], num0->[1]) 0.001/0.001 (rule: add(s(x7:S),x8:S) -> s(add(x7:S,x8:S)), id: 2, possubterms: add(s(x7:S),x8:S)->[], s(x7:S)->[1]) 0.001/0.001 (rule: dbl(num0) -> num0, id: 3, possubterms: dbl(num0)->[], num0->[1]) 0.001/0.001 (rule: dbl(s(x9:S)) -> s(s(dbl(x9:S))), id: 4, possubterms: dbl(s(x9:S))->[], s(x9:S)->[1]) 0.001/0.001 (rule: first(num0,x10:S) -> nil, id: 5, possubterms: first(num0,x10:S)->[], num0->[1]) 0.001/0.001 (rule: first(s(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)), id: 6, possubterms: first(s(x11:S),cons(x12:S,x13:S))->[], s(x11:S)->[1], cons(x12:S,x13:S)->[2]) 0.001/0.001 (rule: sqr(num0) -> num0, id: 7, possubterms: sqr(num0)->[], num0->[1]) 0.001/0.001 (rule: sqr(s(x14:S)) -> s(add(sqr(x14:S),dbl(x14:S))), id: 8, possubterms: sqr(s(x14:S))->[], s(x14:S)->[1]) 0.001/0.001 (rule: terms(x15:S) -> cons(recip(sqr(x15:S)),terms(s(x15:S))), id: 9, possubterms: terms(x15: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 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 Weakly orthogonal, Almost orthogonal, Orthogonal 0.001/0.001 Huet-Levy confluent, Not Newman confluent 0.001/0.001 R is a CS-TRS, Left-homogeneous u-replacing variables 0.001/0.001 0.001/0.001 0.001/0.001 The problem is joinable. 0.001/0.001 0.00user 0.00system 0:00.01elapsed 69%CPU (0avgtext+0avgdata 8560maxresident)k 0.001/0.001 8inputs+0outputs (0major+687minor)pagefaults 0swaps