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 N:S X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (add 1 2) 0.002/0.002 (dbl 1) 0.002/0.002 (first 1 2) 0.002/0.002 (sqr 1) 0.002/0.002 (terms 1) 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 (recip 1) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 dbl(0) -> 0 0.002/0.002 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.002/0.002 first(0,X:S) -> nil 0.002/0.002 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.002/0.002 sqr(0) -> 0 0.002/0.002 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.002/0.002 terms(N:S) -> cons(recip(sqr(N:S)),terms(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 N:S X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (add 1 2) 0.002/0.002 (dbl 1) 0.002/0.002 (first 1 2) 0.002/0.002 (sqr 1) 0.002/0.002 (terms 1) 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 (recip 1) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 dbl(0) -> 0 0.002/0.002 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.002/0.002 first(0,X:S) -> nil 0.002/0.002 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.002/0.002 sqr(0) -> 0 0.002/0.002 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.002/0.002 terms(N:S) -> cons(recip(sqr(N:S)),terms(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 N:S X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (add 1 2) 0.002/0.002 (dbl 1) 0.002/0.002 (first 1 2) 0.002/0.002 (sqr 1) 0.002/0.002 (terms 1) 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 (recip 1) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 dbl(0) -> 0 0.002/0.002 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.002/0.002 first(0,X:S) -> nil 0.002/0.002 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.002/0.002 sqr(0) -> 0 0.002/0.002 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.002/0.002 terms(N:S) -> cons(recip(sqr(N:S)),terms(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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 2 (l' :-> r') => add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> add(s(X:S),Y:S)} 0.002/0.002 s => s(add(x6:S,add(s(X:S),Y:S))) 0.002/0.002 t => add(s(x6:S),s(add(X:S,Y: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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 3 (l' :-> r') => dbl(0) -> 0 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> dbl(0)} 0.002/0.002 s => s(add(x6:S,dbl(0))) 0.002/0.002 t => add(s(x6: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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 4 (l' :-> r') => dbl(s(X:S)) -> s(s(dbl(X:S))) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> dbl(s(X:S))} 0.002/0.002 s => s(add(x6:S,dbl(s(X:S)))) 0.002/0.002 t => add(s(x6:S),s(s(dbl(X: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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 5 (l' :-> r') => first(0,X:S) -> nil 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> first(0,X:S)} 0.002/0.002 s => s(add(x6:S,first(0,X:S))) 0.002/0.002 t => add(s(x6:S),nil) 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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 6 (l' :-> r') => first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> first(s(X:S),cons(Y:S,Z:S))} 0.002/0.002 s => s(add(x6:S,first(s(X:S),cons(Y:S,Z:S)))) 0.002/0.002 t => add(s(x6:S),cons(Y:S,first(X:S,Z: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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 7 (l' :-> r') => sqr(0) -> 0 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> sqr(0)} 0.002/0.002 s => s(add(x6:S,sqr(0))) 0.002/0.002 t => add(s(x6: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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 8 (l' :-> r') => sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> sqr(s(X:S))} 0.002/0.002 s => s(add(x6:S,sqr(s(X:S)))) 0.002/0.002 t => add(s(x6:S),s(add(sqr(X:S),dbl(X: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) => add(s(x6:S),x7:S) -> s(add(x6:S,x7:S)) 0.002/0.002 Rule 9 (l' :-> r') => terms(N:S) -> cons(recip(sqr(N:S)),terms(s(N:S))) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2] 0.002/0.002 Sigma => {x7:S -> terms(N:S)} 0.002/0.002 s => s(add(x6:S,terms(N:S))) 0.002/0.002 t => add(s(x6:S),cons(recip(sqr(N:S)),terms(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) => terms(x14:S) -> cons(recip(sqr(x14:S)),terms(s(x14:S))) 0.002/0.002 Rule 9 (l' :-> r') => terms(N:S) -> cons(recip(sqr(N:S)),terms(s(N:S))) 0.002/0.002 Var => x14:S 0.002/0.002 Pos x14:S in l => [1] 0.002/0.002 Sigma => {x14:S -> terms(N:S)} 0.002/0.002 s => cons(recip(sqr(terms(N:S))),terms(s(terms(N:S)))) 0.002/0.002 t => terms(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.002/0.002 NW => 0 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 N:S X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (add 1 2) 0.002/0.002 (dbl 1) 0.002/0.002 (first 1 2) 0.002/0.002 (sqr 1) 0.002/0.002 (terms 1) 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 (recip 1) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 dbl(0) -> 0 0.002/0.002 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.002/0.002 first(0,X:S) -> nil 0.002/0.002 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.002/0.002 sqr(0) -> 0 0.002/0.002 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.002/0.002 terms(N:S) -> cons(recip(sqr(N:S)),terms(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, NW0, N9 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 Huet Levy Processor: 0.002/0.002 -> Rules: 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 dbl(0) -> 0 0.002/0.002 dbl(s(X:S)) -> s(s(dbl(X:S))) 0.002/0.002 first(0,X:S) -> nil 0.002/0.002 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.002/0.002 sqr(0) -> 0 0.002/0.002 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.002/0.002 terms(N:S) -> cons(recip(sqr(N:S)),terms(s(N:S))) 0.002/0.002 -> Vars: 0.002/0.002 X, X, Y, X, X, X, Y, Z, X, N 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 2, UV-LActive: [Y], UV-RActive: [], UV-LFrozen: [X], UV-RFrozen: [X, Y]) 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: [X], UV-RFrozen: [X]) 0.002/0.002 (UV-RuleId: 5, UV-LActive: [X], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [Y], UV-RActive: [Y], UV-LFrozen: [X, Z], UV-RFrozen: [X, Z]) 0.002/0.002 (UV-RuleId: 7, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 8, UV-LActive: [], UV-RActive: [], UV-LFrozen: [X], UV-RFrozen: [X]) 0.002/0.002 (UV-RuleId: 9, UV-LActive: [N], UV-RActive: [N], UV-LFrozen: [], UV-RFrozen: [N]) 0.002/0.002 -> FVars: 0.002/0.002 x5, x6, x7, x8, x9, x10, x11, x12, x13, x14 0.002/0.002 -> PVars: 0.002/0.002 X: [x5, x6, x8, x9, x10, x13], Y: [x7, x11], Z: [x12], N: [x14] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: add(0,x5:S) -> x5:S, id: 1, possubterms: add(0,x5:S)->[], 0->[1]) 0.002/0.002 (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]) 0.002/0.002 (rule: dbl(0) -> 0, id: 3, possubterms: dbl(0)->[], 0->[1]) 0.002/0.002 (rule: dbl(s(x8:S)) -> s(s(dbl(x8:S))), id: 4, possubterms: dbl(s(x8:S))->[], s(x8:S)->[1]) 0.002/0.002 (rule: first(0,x9:S) -> nil, id: 5, possubterms: first(0,x9:S)->[], 0->[1]) 0.002/0.002 (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]) 0.002/0.002 (rule: sqr(0) -> 0, id: 7, possubterms: sqr(0)->[], 0->[1]) 0.002/0.002 (rule: sqr(s(x13:S)) -> s(add(sqr(x13:S),dbl(x13:S))), id: 8, possubterms: sqr(s(x13:S))->[], s(x13:S)->[1]) 0.002/0.002 (rule: terms(x14:S) -> cons(recip(sqr(x14:S)),terms(s(x14:S))), id: 9, possubterms: terms(x14:S)->[]) 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, NW0, 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 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: s(add(X:S,terms(N:S))) 0.002/0.002 Nodes: [0] 0.002/0.002 Edges: [] 0.002/0.002 ID: 0 => ('s(add(X:S,terms(N:S)))', D0) 0.002/0.002 t: add(s(X:S),cons(recip(sqr(N:S)),terms(s(N:S)))) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('add(s(X:S),cons(recip(sqr(N:S)),terms(s(N:S))))', D0) 0.002/0.002 ID: 1 => ('s(add(X:S,cons(recip(sqr(N:S)),terms(s(N:S)))))', D1, R2, P[], S{x6:S -> X:S, x7:S -> cons(recip(sqr(N:S)),terms(s(N:S)))}), NR: 's(add(X:S,cons(recip(sqr(N:S)),terms(s(N:S)))))' 0.002/0.002 s(add(X:S,terms(N:S))) ->* no union *<- add(s(X:S),cons(recip(sqr(N:S)),terms(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 60%CPU (0avgtext+0avgdata 10500maxresident)k 0.002/0.002 0inputs+0outputs (0major+1017minor)pagefaults 0swaps