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 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 (half 1) 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 1) 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 half(dbl(X:S:S)) -> X:S:S 0.001/0.001 half(num0) -> num0 0.001/0.001 half(s(num0)) -> num0 0.001/0.001 half(s(s(X:S:S))) -> s(half(X: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 (half 1) 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 1) 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 half(dbl(X:S:S)) -> X:S:S 0.001/0.001 half(num0) -> num0 0.001/0.001 half(s(num0)) -> num0 0.001/0.001 half(s(s(X:S:S))) -> s(half(X: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 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 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 (half 1) 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 1) 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 half(dbl(X:S:S)) -> X:S:S 0.001/0.001 half(num0) -> num0 0.001/0.001 half(s(num0)) -> num0 0.001/0.001 half(s(s(X:S:S))) -> s(half(X: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 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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 6 (l' :-> r') => 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 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> first(s(X:S:S),cons(Y:S:S,Z:S:S))} 0.001/0.001 s => cons(x12:S,first(first(s(X:S:S),cons(Y:S:S,Z:S:S)),x13:S)) 0.001/0.001 t => first(s(cons(Y:S:S,first(X:S:S,Z:S:S))),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 7 (l' :-> r') => half(dbl(X:S:S)) -> X:S:S 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> half(dbl(X:S:S))} 0.001/0.001 s => cons(x12:S,first(half(dbl(X:S:S)),x13:S)) 0.001/0.001 t => first(s(X:S:S),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 8 (l' :-> r') => half(num0) -> num0 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> half(num0)} 0.001/0.001 s => cons(x12:S,first(half(num0),x13:S)) 0.001/0.001 t => first(s(num0),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 9 (l' :-> r') => half(s(num0)) -> num0 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> half(s(num0))} 0.001/0.001 s => cons(x12:S,first(half(s(num0)),x13:S)) 0.001/0.001 t => first(s(num0),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 10 (l' :-> r') => half(s(s(X:S:S))) -> s(half(X:S:S)) 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> half(s(s(X:S:S)))} 0.001/0.001 s => cons(x12:S,first(half(s(s(X:S:S))),x13:S)) 0.001/0.001 t => first(s(s(half(X:S:S))),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 11 (l' :-> r') => sqr(num0) -> num0 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> sqr(num0)} 0.001/0.001 s => cons(x12:S,first(sqr(num0),x13:S)) 0.001/0.001 t => first(s(num0),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 12 (l' :-> r') => sqr(s(X:S:S)) -> s(add(sqr(X:S:S),dbl(X:S:S))) 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> sqr(s(X:S:S))} 0.001/0.001 s => cons(x12:S,first(sqr(s(X:S:S)),x13:S)) 0.001/0.001 t => first(s(s(add(sqr(X:S:S),dbl(X:S:S)))),cons(x12:S,x13: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(x11:S),cons(x12:S,x13:S)) -> cons(x12:S,first(x11:S,x13:S)) 0.001/0.001 Rule 13 (l' :-> r') => terms(N:S:S) -> cons(recip(sqr(N:S:S)),terms(s(N:S:S))) 0.001/0.001 Var => x11:S 0.001/0.001 Pos x11:S in l => [1,1] 0.001/0.001 Sigma => {x11:S -> terms(N:S:S)} 0.001/0.001 s => cons(x12:S,first(terms(N:S:S),x13:S)) 0.001/0.001 t => first(s(cons(recip(sqr(N:S:S)),terms(s(N:S:S)))),cons(x12:S,x13: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) => sqr(s(x16:S)) -> s(add(sqr(x16:S),dbl(x16:S))) 0.001/0.001 Rule 12 (l' :-> r') => sqr(s(X:S:S)) -> s(add(sqr(X:S:S),dbl(X:S:S))) 0.001/0.001 Var => x16:S 0.001/0.001 Pos x16:S in l => [1,1] 0.001/0.001 Sigma => {x16:S -> sqr(s(X:S:S))} 0.001/0.001 s => s(add(sqr(sqr(s(X:S:S))),dbl(sqr(s(X:S:S))))) 0.001/0.001 t => sqr(s(s(add(sqr(X:S:S),dbl(X:S:S))))) 0.001/0.001 NW => 0 0.001/0.001 0.001/0.001 0.001/0.001 ->Extended u-Critical Pair: 0.001/0.001 Rule 1 (l :-> r) => sqr(s(x16:S)) -> s(add(sqr(x16:S),dbl(x16:S))) 0.001/0.001 Rule 13 (l' :-> r') => terms(N:S:S) -> cons(recip(sqr(N:S:S)),terms(s(N:S:S))) 0.001/0.001 Var => x16:S 0.001/0.001 Pos x16:S in l => [1,1] 0.001/0.001 Sigma => {x16:S -> terms(N:S:S)} 0.001/0.001 s => s(add(sqr(terms(N:S:S)),dbl(terms(N:S:S)))) 0.001/0.001 t => sqr(s(cons(recip(sqr(N:S:S)),terms(s(N:S: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 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 (half 1) 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 1) 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 half(dbl(X:S:S)) -> X:S:S 0.001/0.001 half(num0) -> num0 0.001/0.001 half(s(num0)) -> num0 0.001/0.001 half(s(s(X:S:S))) -> s(half(X: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 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, NW1, N5 0.001/0.001 => Not trivial, Not overlay, NW1, N6 0.001/0.001 => Not trivial, Not overlay, NW1, N7 0.001/0.001 => Not trivial, Not overlay, NW1, N8 0.001/0.001 => Not trivial, Not overlay, NW0, N9 0.001/0.001 => Not trivial, Not overlay, NW0, N10 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 half(dbl(X:S:S)) -> X:S:S 0.001/0.001 half(num0) -> num0 0.001/0.001 half(s(num0)) -> num0 0.001/0.001 half(s(s(X:S:S))) -> s(half(X: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, X: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: [X:S], UV-RActive: [X:S], UV-LFrozen: [Y:S], UV-RFrozen: [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: [X:S], UV-RActive: [X:S], UV-LFrozen: [], UV-RFrozen: []) 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: [X:S], UV-RActive: [], UV-LFrozen: [Y:S, Z:S], UV-RFrozen: [X:S, Y:S, Z:S]) 0.001/0.001 (UV-RuleId: 7, UV-LActive: [X:S], UV-RActive: [X:S], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 8, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 9, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 10, UV-LActive: [X:S], UV-RActive: [X:S], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 11, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 12, UV-LActive: [X:S], UV-RActive: [X:S], UV-LFrozen: [], UV-RFrozen: [X:S]) 0.001/0.001 (UV-RuleId: 13, 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, x16, x17 0.001/0.001 -> PVars: 0.001/0.001 X:S: [x6, x7, x9, x10, x11, x14, x15, x16], Y:S: [x8, x12], Z:S: [x13], N:S: [x17] 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: half(dbl(x14:S)) -> x14:S, id: 7, possubterms: half(dbl(x14:S))->[], dbl(x14:S)->[1]) 0.001/0.001 (rule: half(num0) -> num0, id: 8, possubterms: half(num0)->[], num0->[1]) 0.001/0.001 (rule: half(s(num0)) -> num0, id: 9, possubterms: half(s(num0))->[], s(num0)->[1], num0->[1, 1]) 0.001/0.001 (rule: half(s(s(x15:S))) -> s(half(x15:S)), id: 10, possubterms: half(s(s(x15:S)))->[], s(s(x15:S))->[1], s(x15:S)->[1, 1]) 0.001/0.001 (rule: sqr(num0) -> num0, id: 11, possubterms: sqr(num0)->[], num0->[1]) 0.001/0.001 (rule: sqr(s(x16:S)) -> s(add(sqr(x16:S),dbl(x16:S))), id: 12, possubterms: sqr(s(x16:S))->[], s(x16:S)->[1]) 0.001/0.001 (rule: terms(x17:S) -> cons(recip(sqr(x17:S)),terms(s(x17:S))), id: 13, possubterms: terms(x17:S)->[]) 0.001/0.001 0.001/0.001 -> Unifications: 0.001/0.001 (R7 unifies with R3 at p: [1], l: half(dbl(x14:S)), lp: dbl(x14:S), sig: {x14:S -> num0}, l': dbl(num0), r: x14:S, r': num0) 0.001/0.001 (R7 unifies with R4 at p: [1], l: half(dbl(x14:S)), lp: dbl(x14:S), sig: {x14:S -> s(X:S:S)}, l': dbl(s(X:S:S)), r: x14:S, r': s(s(dbl(X:S:S)))) 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, NW0, 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 => Not trivial, Not overlay, NW1, N6 0.001/0.001 => Not trivial, Not overlay, NW1, N7 0.001/0.001 => Not trivial, Not overlay, NW1, N8 0.001/0.001 => Not trivial, Not overlay, NW0, N9 0.001/0.001 => Not trivial, Not overlay, NW0, N10 0.001/0.001 => Not trivial, Not overlay, NW1, N11 0.001/0.001 => Not trivial, Not overlay, NW1, N12 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:S,first(half(s(num0)),Z:S:S)) 0.001/0.001 Nodes: [0] 0.001/0.001 Edges: [] 0.001/0.001 ID: 0 => ('cons(Y:S:S,first(half(s(num0)),Z:S:S))', D0) 0.001/0.001 t: first(s(num0),cons(Y:S:S,Z:S:S)) 0.001/0.001 Nodes: [0,1] 0.001/0.001 Edges: [(0,1)] 0.001/0.001 ID: 0 => ('first(s(num0),cons(Y:S:S,Z:S:S))', D0) 0.001/0.001 ID: 1 => ('cons(Y:S:S,first(num0,Z:S:S))', D1, R6, P[], S{x11:S -> num0, x12:S -> Y:S:S, x13:S -> Z:S:S}), NR: 'cons(Y:S:S,first(num0,Z:S:S))' 0.001/0.001 cons(Y:S:S,first(half(s(num0)),Z:S:S)) ->* no union *<- first(s(num0),cons(Y:S:S,Z:S: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 91%CPU (0avgtext+0avgdata 10960maxresident)k 0.001/0.001 8inputs+0outputs (0major+1090minor)pagefaults 0swaps