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 (2ndsneg 1 2) 0.002/0.002 (2ndspos 1 2) 0.002/0.002 (from 1) 0.002/0.002 (pi 1) 0.002/0.002 (plus 1 2) 0.002/0.002 (square 1) 0.002/0.002 (times 1 2) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (cons2 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (negrecip 1) 0.002/0.002 (posrecip 1) 0.002/0.002 (rcons 1 2) 0.002/0.002 (rnil) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 2ndsneg(0,Z:S) -> rnil 0.002/0.002 2ndsneg(s(N:S),cons(X:S,Z:S)) -> 2ndsneg(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(negrecip(Y:S),2ndspos(N:S,Z:S)) 0.002/0.002 2ndspos(0,Z:S) -> rnil 0.002/0.002 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 plus(0,Y:S) -> Y:S 0.002/0.002 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 square(X:S) -> times(X:S,X:S) 0.002/0.002 times(0,Y:S) -> 0 0.002/0.002 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y: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 (2ndsneg 1 2) 0.002/0.002 (2ndspos 1 2) 0.002/0.002 (from 1) 0.002/0.002 (pi 1) 0.002/0.002 (plus 1 2) 0.002/0.002 (square 1) 0.002/0.002 (times 1 2) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (cons2 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (negrecip 1) 0.002/0.002 (posrecip 1) 0.002/0.002 (rcons 1 2) 0.002/0.002 (rnil) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 2ndsneg(0,Z:S) -> rnil 0.002/0.002 2ndsneg(s(N:S),cons(X:S,Z:S)) -> 2ndsneg(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(negrecip(Y:S),2ndspos(N:S,Z:S)) 0.002/0.002 2ndspos(0,Z:S) -> rnil 0.002/0.002 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 plus(0,Y:S) -> Y:S 0.002/0.002 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 square(X:S) -> times(X:S,X:S) 0.002/0.002 times(0,Y:S) -> 0 0.002/0.002 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y: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 (2ndsneg 1 2) 0.002/0.002 (2ndspos 1 2) 0.002/0.002 (from 1) 0.002/0.002 (pi 1) 0.002/0.002 (plus 1 2) 0.002/0.002 (square 1) 0.002/0.002 (times 1 2) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (cons2 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (negrecip 1) 0.002/0.002 (posrecip 1) 0.002/0.002 (rcons 1 2) 0.002/0.002 (rnil) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 2ndsneg(0,Z:S) -> rnil 0.002/0.002 2ndsneg(s(N:S),cons(X:S,Z:S)) -> 2ndsneg(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(negrecip(Y:S),2ndspos(N:S,Z:S)) 0.002/0.002 2ndspos(0,Z:S) -> rnil 0.002/0.002 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 plus(0,Y:S) -> Y:S 0.002/0.002 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 square(X:S) -> times(X:S,X:S) 0.002/0.002 times(0,Y:S) -> 0 0.002/0.002 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 2 (l' :-> r') => 2ndsneg(s(N:S),cons(X:S,Z:S)) -> 2ndsneg(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> 2ndsneg(s(N:S),cons(X:S,Z:S))} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(2ndsneg(s(N:S),cons(X:S,Z:S)),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(2ndsneg(s(N:S),cons2(X:S,Z:S)),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 3 (l' :-> r') => 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(negrecip(Y:S),2ndspos(N:S,Z:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S)))} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(rcons(negrecip(Y:S),2ndspos(N:S,Z:S)),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 4 (l' :-> r') => 2ndspos(0,Z:S) -> rnil 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> 2ndspos(0,Z:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(2ndspos(0,Z:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(rnil,x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 5 (l' :-> r') => 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> 2ndspos(s(N:S),cons(X:S,Z:S))} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(2ndspos(s(N:S),cons(X:S,Z:S)),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(2ndspos(s(N:S),cons2(X:S,Z:S)),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 6 (l' :-> r') => 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S)))} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 7 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> from(X:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(from(X:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(cons(X:S,from(s(X:S))),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 8 (l' :-> r') => pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> pi(X:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(pi(X:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(2ndspos(X:S,from(0)),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 9 (l' :-> r') => plus(0,Y:S) -> Y:S 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> plus(0,Y:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(plus(0,Y:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(Y:S,x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 10 (l' :-> r') => plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> plus(s(X:S),Y:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(plus(s(X:S),Y:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(s(plus(X:S,Y:S)),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 11 (l' :-> r') => square(X:S) -> times(X:S,X:S) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> square(X:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(square(X:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(times(X:S,X:S),x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 12 (l' :-> r') => times(0,Y:S) -> 0 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> times(0,Y:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(times(0,Y:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(0,x8: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) => 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)) 0.002/0.002 Rule 13 (l' :-> r') => times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S)) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [2,1] 0.002/0.002 Sigma => {x7:S -> times(s(X:S),Y:S)} 0.002/0.002 s => 2ndsneg(s(x6:S),cons2(times(s(X:S),Y:S),x8:S)) 0.002/0.002 t => 2ndsneg(s(x6:S),cons(plus(Y:S,times(X:S,Y:S)),x8: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 5 (l' :-> r') => 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> 2ndspos(s(N:S),cons(X:S,Z:S))} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(2ndspos(s(N:S),cons(X:S,Z:S)),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(2ndspos(s(N:S),cons2(X:S,Z:S)),x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 6 (l' :-> r') => 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S)))} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)),x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 7 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> from(X:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(from(X:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(cons(X:S,from(s(X:S))),x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 8 (l' :-> r') => pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> pi(X:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(pi(X:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(2ndspos(X:S,from(0)),x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 9 (l' :-> r') => plus(0,Y:S) -> Y:S 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> plus(0,Y:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(plus(0,Y:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(Y:S,x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 10 (l' :-> r') => plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> plus(s(X:S),Y:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(plus(s(X:S),Y:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(s(plus(X:S,Y:S)),x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 11 (l' :-> r') => square(X:S) -> times(X:S,X:S) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> square(X:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(square(X:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(times(X:S,X:S),x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 12 (l' :-> r') => times(0,Y:S) -> 0 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> times(0,Y:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(times(0,Y:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(0,x16: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) => 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)) 0.002/0.002 Rule 13 (l' :-> r') => times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S)) 0.002/0.002 Var => x15:S 0.002/0.002 Pos x15:S in l => [2,1] 0.002/0.002 Sigma => {x15:S -> times(s(X:S),Y:S)} 0.002/0.002 s => 2ndspos(s(x14:S),cons2(times(s(X:S),Y:S),x16:S)) 0.002/0.002 t => 2ndspos(s(x14:S),cons(plus(Y:S,times(X:S,Y:S)),x16: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) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 7 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> from(X:S)} 0.002/0.002 s => cons(from(X:S),from(s(from(X:S)))) 0.002/0.002 t => from(cons(X:S,from(s(X:S)))) 0.002/0.002 NW => 0 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 8 (l' :-> r') => pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> pi(X:S)} 0.002/0.002 s => cons(pi(X:S),from(s(pi(X:S)))) 0.002/0.002 t => from(2ndspos(X:S,from(0))) 0.002/0.002 NW => 0 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 9 (l' :-> r') => plus(0,Y:S) -> Y:S 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> plus(0,Y:S)} 0.002/0.002 s => cons(plus(0,Y:S),from(s(plus(0,Y:S)))) 0.002/0.002 t => from(Y:S) 0.002/0.002 NW => 0 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 10 (l' :-> r') => plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> plus(s(X:S),Y:S)} 0.002/0.002 s => cons(plus(s(X:S),Y:S),from(s(plus(s(X:S),Y:S)))) 0.002/0.002 t => from(s(plus(X:S,Y:S))) 0.002/0.002 NW => 0 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 11 (l' :-> r') => square(X:S) -> times(X:S,X:S) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> square(X:S)} 0.002/0.002 s => cons(square(X:S),from(s(square(X:S)))) 0.002/0.002 t => from(times(X:S,X:S)) 0.002/0.002 NW => 0 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 12 (l' :-> r') => times(0,Y:S) -> 0 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> times(0,Y:S)} 0.002/0.002 s => cons(times(0,Y:S),from(s(times(0,Y:S)))) 0.002/0.002 t => from(0) 0.002/0.002 NW => 0 0.002/0.002 0.002/0.002 0.002/0.002 ->Extended u-Critical Pair: 0.002/0.002 Rule 1 (l :-> r) => from(x21:S) -> cons(x21:S,from(s(x21:S))) 0.002/0.002 Rule 13 (l' :-> r') => times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S)) 0.002/0.002 Var => x21:S 0.002/0.002 Pos x21:S in l => [1] 0.002/0.002 Sigma => {x21:S -> times(s(X:S),Y:S)} 0.002/0.002 s => cons(times(s(X:S),Y:S),from(s(times(s(X:S),Y:S)))) 0.002/0.002 t => from(plus(Y:S,times(X:S,Y: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 (2ndsneg 1 2) 0.002/0.002 (2ndspos 1 2) 0.002/0.002 (from 1) 0.002/0.002 (pi 1) 0.002/0.002 (plus 1 2) 0.002/0.002 (square 1) 0.002/0.002 (times 1 2) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (cons2 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (negrecip 1) 0.002/0.002 (posrecip 1) 0.002/0.002 (rcons 1 2) 0.002/0.002 (rnil) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 2ndsneg(0,Z:S) -> rnil 0.002/0.002 2ndsneg(s(N:S),cons(X:S,Z:S)) -> 2ndsneg(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(negrecip(Y:S),2ndspos(N:S,Z:S)) 0.002/0.002 2ndspos(0,Z:S) -> rnil 0.002/0.002 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 plus(0,Y:S) -> Y:S 0.002/0.002 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 square(X:S) -> times(X:S,X:S) 0.002/0.002 times(0,Y:S) -> 0 0.002/0.002 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S)) 0.002/0.002 ) 0.002/0.002 Critical Pairs: 0.002/0.002 <2ndsneg(s(x6:S),cons2(2ndsneg(s(N:S),cons(X:S,Z:S)),x8:S)),2ndsneg(s(x6:S),cons(2ndsneg(s(N:S),cons2(X:S,Z:S)),x8:S))> => Not trivial, Not overlay, NW1, N1 0.002/0.002 <2ndsneg(s(x6:S),cons2(2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))),x8:S)),2ndsneg(s(x6:S),cons(rcons(negrecip(Y:S),2ndspos(N:S,Z:S)),x8:S))> => Not trivial, Not overlay, NW1, N2 0.002/0.002 <2ndsneg(s(x6:S),cons2(2ndspos(0,Z:S),x8:S)),2ndsneg(s(x6:S),cons(rnil,x8:S))> => Not trivial, Not overlay, NW1, N3 0.002/0.002 <2ndsneg(s(x6:S),cons2(2ndspos(s(N:S),cons(X:S,Z:S)),x8:S)),2ndsneg(s(x6:S),cons(2ndspos(s(N:S),cons2(X:S,Z:S)),x8:S))> => Not trivial, Not overlay, NW1, N4 0.002/0.002 <2ndsneg(s(x6:S),cons2(2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))),x8:S)),2ndsneg(s(x6:S),cons(rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)),x8:S))> => Not trivial, Not overlay, NW1, N5 0.002/0.002 <2ndsneg(s(x6:S),cons2(from(X:S),x8:S)),2ndsneg(s(x6:S),cons(cons(X:S,from(s(X:S))),x8:S))> => Not trivial, Not overlay, NW1, N6 0.002/0.002 <2ndsneg(s(x6:S),cons2(pi(X:S),x8:S)),2ndsneg(s(x6:S),cons(2ndspos(X:S,from(0)),x8:S))> => Not trivial, Not overlay, NW1, N7 0.002/0.002 <2ndsneg(s(x6:S),cons2(plus(0,Y:S),x8:S)),2ndsneg(s(x6:S),cons(Y:S,x8:S))> => Not trivial, Not overlay, NW1, N8 0.002/0.002 <2ndsneg(s(x6:S),cons2(plus(s(X:S),Y:S),x8:S)),2ndsneg(s(x6:S),cons(s(plus(X:S,Y:S)),x8:S))> => Not trivial, Not overlay, NW1, N9 0.002/0.002 <2ndsneg(s(x6:S),cons2(square(X:S),x8:S)),2ndsneg(s(x6:S),cons(times(X:S,X:S),x8:S))> => Not trivial, Not overlay, NW1, N10 0.002/0.002 <2ndsneg(s(x6:S),cons2(times(0,Y:S),x8:S)),2ndsneg(s(x6:S),cons(0,x8:S))> => Not trivial, Not overlay, NW1, N11 0.002/0.002 <2ndsneg(s(x6:S),cons2(times(s(X:S),Y:S),x8:S)),2ndsneg(s(x6:S),cons(plus(Y:S,times(X:S,Y:S)),x8:S))> => Not trivial, Not overlay, NW1, N12 0.002/0.002 <2ndspos(s(x14:S),cons2(2ndspos(s(N:S),cons(X:S,Z:S)),x16:S)),2ndspos(s(x14:S),cons(2ndspos(s(N:S),cons2(X:S,Z:S)),x16:S))> => Not trivial, Not overlay, NW1, N13 0.002/0.002 <2ndspos(s(x14:S),cons2(2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))),x16:S)),2ndspos(s(x14:S),cons(rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)),x16:S))> => Not trivial, Not overlay, NW1, N14 0.002/0.002 <2ndspos(s(x14:S),cons2(from(X:S),x16:S)),2ndspos(s(x14:S),cons(cons(X:S,from(s(X:S))),x16:S))> => Not trivial, Not overlay, NW1, N15 0.002/0.002 <2ndspos(s(x14:S),cons2(pi(X:S),x16:S)),2ndspos(s(x14:S),cons(2ndspos(X:S,from(0)),x16:S))> => Not trivial, Not overlay, NW1, N16 0.002/0.002 <2ndspos(s(x14:S),cons2(plus(0,Y:S),x16:S)),2ndspos(s(x14:S),cons(Y:S,x16:S))> => Not trivial, Not overlay, NW1, N17 0.002/0.002 <2ndspos(s(x14:S),cons2(plus(s(X:S),Y:S),x16:S)),2ndspos(s(x14:S),cons(s(plus(X:S,Y:S)),x16:S))> => Not trivial, Not overlay, NW1, N18 0.002/0.002 <2ndspos(s(x14:S),cons2(square(X:S),x16:S)),2ndspos(s(x14:S),cons(times(X:S,X:S),x16:S))> => Not trivial, Not overlay, NW1, N19 0.002/0.002 <2ndspos(s(x14:S),cons2(times(0,Y:S),x16:S)),2ndspos(s(x14:S),cons(0,x16:S))> => Not trivial, Not overlay, NW1, N20 0.002/0.002 <2ndspos(s(x14:S),cons2(times(s(X:S),Y:S),x16:S)),2ndspos(s(x14:S),cons(plus(Y:S,times(X:S,Y:S)),x16:S))> => Not trivial, Not overlay, NW1, N21 0.002/0.002 => Not trivial, Not overlay, NW0, N22 0.002/0.002 => Not trivial, Not overlay, NW0, N23 0.002/0.002 => Not trivial, Not overlay, NW0, N24 0.002/0.002 => Not trivial, Not overlay, NW0, N25 0.002/0.002 => Not trivial, Not overlay, NW0, N26 0.002/0.002 => Not trivial, Not overlay, NW0, N27 0.002/0.002 => Not trivial, Not overlay, NW0, N28 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 Huet Levy Processor: 0.002/0.002 -> Rules: 0.002/0.002 2ndsneg(0,Z:S) -> rnil 0.002/0.002 2ndsneg(s(N:S),cons(X:S,Z:S)) -> 2ndsneg(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(negrecip(Y:S),2ndspos(N:S,Z:S)) 0.002/0.002 2ndspos(0,Z:S) -> rnil 0.002/0.002 2ndspos(s(N:S),cons(X:S,Z:S)) -> 2ndspos(s(N:S),cons2(X:S,Z:S)) 0.002/0.002 2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))) -> rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 pi(X:S) -> 2ndspos(X:S,from(0)) 0.002/0.002 plus(0,Y:S) -> Y:S 0.002/0.002 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S)) 0.002/0.002 square(X:S) -> times(X:S,X:S) 0.002/0.002 times(0,Y:S) -> 0 0.002/0.002 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S)) 0.002/0.002 -> Vars: 0.002/0.002 Z, N, X, Z, N, X, Y, Z, Z, N, X, Z, N, X, Y, Z, X, X, Y, X, Y, X, Y, X, Y 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [Z], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 2, UV-LActive: [N, X], UV-RActive: [N, Z], UV-LFrozen: [Z], UV-RFrozen: [X]) 0.002/0.002 (UV-RuleId: 3, UV-LActive: [N, Y], UV-RActive: [N, Y, Z], UV-LFrozen: [X, Z], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 4, UV-LActive: [Z], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 5, UV-LActive: [N, X], UV-RActive: [N, Z], UV-LFrozen: [Z], UV-RFrozen: [X]) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [N, Y], UV-RActive: [N, Y, Z], UV-LFrozen: [X, Z], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 7, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: [X]) 0.002/0.002 (UV-RuleId: 8, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 9, UV-LActive: [Y], UV-RActive: [Y], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 10, UV-LActive: [X, Y], UV-RActive: [X, Y], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 11, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 12, UV-LActive: [Y], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 13, UV-LActive: [X, Y], UV-RActive: [X, Y], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 -> FVars: 0.002/0.002 x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29 0.002/0.002 -> PVars: 0.002/0.002 Z: [x5, x8, x12, x13, x16, x20], N: [x6, x9, x14, x17], X: [x7, x10, x15, x18, x21, x22, x24, x26, x28], Y: [x11, x19, x23, x25, x27, x29] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: 2ndsneg(0,x5:S) -> rnil, id: 1, possubterms: 2ndsneg(0,x5:S)->[], 0->[1]) 0.002/0.002 (rule: 2ndsneg(s(x6:S),cons(x7:S,x8:S)) -> 2ndsneg(s(x6:S),cons2(x7:S,x8:S)), id: 2, possubterms: 2ndsneg(s(x6:S),cons(x7:S,x8:S))->[], s(x6:S)->[1], cons(x7:S,x8:S)->[2]) 0.002/0.002 (rule: 2ndsneg(s(x9:S),cons2(x10:S,cons(x11:S,x12:S))) -> rcons(negrecip(x11:S),2ndspos(x9:S,x12:S)), id: 3, possubterms: 2ndsneg(s(x9:S),cons2(x10:S,cons(x11:S,x12:S)))->[], s(x9:S)->[1], cons2(x10:S,cons(x11:S,x12:S))->[2], cons(x11:S,x12:S)->[2, 2]) 0.002/0.002 (rule: 2ndspos(0,x13:S) -> rnil, id: 4, possubterms: 2ndspos(0,x13:S)->[], 0->[1]) 0.002/0.002 (rule: 2ndspos(s(x14:S),cons(x15:S,x16:S)) -> 2ndspos(s(x14:S),cons2(x15:S,x16:S)), id: 5, possubterms: 2ndspos(s(x14:S),cons(x15:S,x16:S))->[], s(x14:S)->[1], cons(x15:S,x16:S)->[2]) 0.002/0.002 (rule: 2ndspos(s(x17:S),cons2(x18:S,cons(x19:S,x20:S))) -> rcons(posrecip(x19:S),2ndsneg(x17:S,x20:S)), id: 6, possubterms: 2ndspos(s(x17:S),cons2(x18:S,cons(x19:S,x20:S)))->[], s(x17:S)->[1], cons2(x18:S,cons(x19:S,x20:S))->[2], cons(x19:S,x20:S)->[2, 2]) 0.002/0.002 (rule: from(x21:S) -> cons(x21:S,from(s(x21:S))), id: 7, possubterms: from(x21:S)->[]) 0.002/0.002 (rule: pi(x22:S) -> 2ndspos(x22:S,from(0)), id: 8, possubterms: pi(x22:S)->[]) 0.002/0.002 (rule: plus(0,x23:S) -> x23:S, id: 9, possubterms: plus(0,x23:S)->[], 0->[1]) 0.002/0.002 (rule: plus(s(x24:S),x25:S) -> s(plus(x24:S,x25:S)), id: 10, possubterms: plus(s(x24:S),x25:S)->[], s(x24:S)->[1]) 0.002/0.002 (rule: square(x26:S) -> times(x26:S,x26:S), id: 11, possubterms: square(x26:S)->[]) 0.002/0.002 (rule: times(0,x27:S) -> 0, id: 12, possubterms: times(0,x27:S)->[], 0->[1]) 0.002/0.002 (rule: times(s(x28:S),x29:S) -> plus(x29:S,times(x28:S,x29:S)), id: 13, possubterms: times(s(x28:S),x29:S)->[], s(x28:S)->[1]) 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, NW0, N1 0.002/0.002 => Not trivial, Not overlay, NW0, N2 0.002/0.002 <2ndspos(s(N:S),cons2(square(X:S),Z:S)),2ndspos(s(N:S),cons(times(X:S,X:S),Z:S))> => Not trivial, Not overlay, NW1, N3 0.002/0.002 <2ndspos(s(N:S),cons2(plus(0,Y:S),Z:S)),2ndspos(s(N:S),cons(Y:S,Z:S))> => Not trivial, Not overlay, NW1, N4 0.002/0.002 <2ndspos(s(N:S),cons2(pi(X:S),Z:S)),2ndspos(s(N:S),cons(2ndspos(X:S,from(0)),Z:S))> => Not trivial, Not overlay, NW1, N5 0.002/0.002 => Not trivial, Not overlay, NW0, N6 0.002/0.002 <2ndspos(s(N:S),cons2(from(X:S),Z:S)),2ndspos(s(N:S),cons(cons(X:S,from(s(X:S))),Z:S))> => Not trivial, Not overlay, NW1, N7 0.002/0.002 => Not trivial, Not overlay, NW0, N8 0.002/0.002 => Not trivial, Not overlay, NW0, N9 0.002/0.002 <2ndspos(s(N:S),cons2(times(s(X:S),Y:S),Z:S)),2ndspos(s(N:S),cons(plus(Y:S,times(X:S,Y:S)),Z:S))> => Not trivial, Not overlay, NW1, N10 0.002/0.002 <2ndsneg(s(N:S),cons2(2ndsneg(s(N:S),cons2(X:S,cons(Y:S,Z:S))),Z:S)),2ndsneg(s(N:S),cons(rcons(negrecip(Y:S),2ndspos(N:S,Z:S)),Z:S))> => Not trivial, Not overlay, NW1, N11 0.002/0.002 <2ndspos(s(N:S),cons2(2ndspos(s(N:S),cons(X:S,Z:S)),Z:S)),2ndspos(s(N:S),cons(2ndspos(s(N:S),cons2(X:S,Z:S)),Z:S))> => Not trivial, Not overlay, NW1, N12 0.002/0.002 => Not trivial, Not overlay, NW0, N13 0.002/0.002 <2ndsneg(s(N:S),cons2(2ndspos(0,Z:S),Z:S)),2ndsneg(s(N:S),cons(rnil,Z:S))> => Not trivial, Not overlay, NW1, N14 0.002/0.002 <2ndsneg(s(N:S),cons2(plus(s(X:S),Y:S),Z:S)),2ndsneg(s(N:S),cons(s(plus(X:S,Y:S)),Z:S))> => Not trivial, Not overlay, NW1, N15 0.002/0.002 => Not trivial, Not overlay, NW0, N16 0.002/0.002 <2ndspos(s(N:S),cons2(times(0,Y:S),Z:S)),2ndspos(s(N:S),cons(0,Z:S))> => Not trivial, Not overlay, NW1, N17 0.002/0.002 <2ndsneg(s(N:S),cons2(times(0,Y:S),Z:S)),2ndsneg(s(N:S),cons(0,Z:S))> => Not trivial, Not overlay, NW1, N18 0.002/0.002 <2ndspos(s(N:S),cons2(2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))),Z:S)),2ndspos(s(N:S),cons(rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)),Z:S))> => Not trivial, Not overlay, NW1, N19 0.002/0.002 <2ndsneg(s(N:S),cons2(2ndsneg(s(N:S),cons(X:S,Z:S)),Z:S)),2ndsneg(s(N:S),cons(2ndsneg(s(N:S),cons2(X:S,Z:S)),Z:S))> => Not trivial, Not overlay, NW1, N20 0.002/0.002 <2ndsneg(s(N:S),cons2(pi(X:S),Z:S)),2ndsneg(s(N:S),cons(2ndspos(X:S,from(0)),Z:S))> => Not trivial, Not overlay, NW1, N21 0.002/0.002 <2ndspos(s(N:S),cons2(plus(s(X:S),Y:S),Z:S)),2ndspos(s(N:S),cons(s(plus(X:S,Y:S)),Z:S))> => Not trivial, Not overlay, NW1, N22 0.002/0.002 <2ndsneg(s(N:S),cons2(2ndspos(s(N:S),cons2(X:S,cons(Y:S,Z:S))),Z:S)),2ndsneg(s(N:S),cons(rcons(posrecip(Y:S),2ndsneg(N:S,Z:S)),Z:S))> => Not trivial, Not overlay, NW1, N23 0.002/0.002 <2ndsneg(s(N:S),cons2(plus(0,Y:S),Z:S)),2ndsneg(s(N:S),cons(Y:S,Z:S))> => Not trivial, Not overlay, NW1, N24 0.002/0.002 <2ndsneg(s(N:S),cons2(from(X:S),Z:S)),2ndsneg(s(N:S),cons(cons(X:S,from(s(X:S))),Z:S))> => Not trivial, Not overlay, NW1, N25 0.002/0.002 <2ndsneg(s(N:S),cons2(2ndspos(s(N:S),cons(X:S,Z:S)),Z:S)),2ndsneg(s(N:S),cons(2ndspos(s(N:S),cons2(X:S,Z:S)),Z:S))> => Not trivial, Not overlay, NW1, N26 0.002/0.002 <2ndsneg(s(N:S),cons2(square(X:S),Z:S)),2ndsneg(s(N:S),cons(times(X:S,X:S),Z:S))> => Not trivial, Not overlay, NW1, N27 0.002/0.002 <2ndsneg(s(N:S),cons2(times(s(X:S),Y:S),Z:S)),2ndsneg(s(N:S),cons(plus(Y:S,times(X:S,Y:S)),Z:S))> => Not trivial, Not overlay, NW1, N28 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: cons(from(X:S),from(s(from(X:S)))) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('cons(from(X:S),from(s(from(X:S))))', D0) 0.002/0.002 ID: 1 => ('cons(cons(X:S,from(s(X:S))),from(s(from(X:S))))', D1, R7, P[1], S{x21:S -> X:S}), NR: 'cons(X:S,from(s(X:S)))' 0.002/0.002 t: from(cons(X:S,from(s(X:S)))) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('from(cons(X:S,from(s(X:S))))', D0) 0.002/0.002 ID: 1 => ('cons(cons(X:S,from(s(X:S))),from(s(cons(X:S,from(s(X:S))))))', D1, R7, P[], S{x21:S -> cons(X:S,from(s(X:S)))}), NR: 'cons(cons(X:S,from(s(X:S))),from(s(cons(X:S,from(s(X:S))))))' 0.002/0.002 cons(from(X:S),from(s(from(X:S)))) ->* no union *<- from(cons(X:S,from(s(X:S)))) 0.002/0.002 "Not joinable" 0.002/0.002 0.002/0.002 The problem is not joinable. 0.002/0.002 0.02user 0.00system 0:00.02elapsed 95%CPU (0avgtext+0avgdata 12504maxresident)k 0.002/0.002 0inputs+0outputs (0major+1432minor)pagefaults 0swaps