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 X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2) 0.002/0.002 (from 1) 0.002/0.002 (head 1) 0.002/0.002 (if 1) 0.002/0.002 (primes) 0.002/0.002 (sieve 1) 0.002/0.002 (tail 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (divides 1 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (false) 0.002/0.002 (s 1) 0.002/0.002 (true) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(s(s(X:S)),cons(Y:S,Z:S)) -> if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 head(cons(X:S,Y:S)) -> X:S 0.002/0.002 if(false,X:S,Y:S) -> Y:S 0.002/0.002 if(true,X:S,Y:S) -> X:S 0.002/0.002 primes -> sieve(from(s(s(0)))) 0.002/0.002 sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 tail(cons(X:S,Y: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 X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2) 0.002/0.002 (from 1) 0.002/0.002 (head 1) 0.002/0.002 (if 1) 0.002/0.002 (primes) 0.002/0.002 (sieve 1) 0.002/0.002 (tail 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (divides 1 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (false) 0.002/0.002 (s 1) 0.002/0.002 (true) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(s(s(X:S)),cons(Y:S,Z:S)) -> if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 head(cons(X:S,Y:S)) -> X:S 0.002/0.002 if(false,X:S,Y:S) -> Y:S 0.002/0.002 if(true,X:S,Y:S) -> X:S 0.002/0.002 primes -> sieve(from(s(s(0)))) 0.002/0.002 sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 tail(cons(X:S,Y: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 X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2) 0.002/0.002 (from 1) 0.002/0.002 (head 1) 0.002/0.002 (if 1) 0.002/0.002 (primes) 0.002/0.002 (sieve 1) 0.002/0.002 (tail 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (divides 1 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (false) 0.002/0.002 (s 1) 0.002/0.002 (true) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(s(s(X:S)),cons(Y:S,Z:S)) -> if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 head(cons(X:S,Y:S)) -> X:S 0.002/0.002 if(false,X:S,Y:S) -> Y:S 0.002/0.002 if(true,X:S,Y:S) -> X:S 0.002/0.002 primes -> sieve(from(s(s(0)))) 0.002/0.002 sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 tail(cons(X:S,Y: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 1 (l' :-> r') => filter(s(s(X:S)),cons(Y:S,Z:S)) -> if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> filter(s(s(X:S)),cons(Y:S,Z:S))} 0.002/0.002 s => if(divides(s(s(filter(s(s(X:S)),cons(Y:S,Z:S)))),x5:S),filter(s(s(filter(s(s(X:S)),cons(Y:S,Z:S)))),x6:S),cons(x5:S,filter(filter(s(s(X:S)),cons(Y:S,Z:S)),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))))),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 2 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> from(X:S)} 0.002/0.002 s => if(divides(s(s(from(X:S))),x5:S),filter(s(s(from(X:S))),x6:S),cons(x5:S,filter(from(X:S),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(cons(X:S,from(s(X:S))))),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 3 (l' :-> r') => head(cons(X:S,Y:S)) -> X:S 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> head(cons(X:S,Y:S))} 0.002/0.002 s => if(divides(s(s(head(cons(X:S,Y:S)))),x5:S),filter(s(s(head(cons(X:S,Y:S)))),x6:S),cons(x5:S,filter(head(cons(X:S,Y:S)),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(X:S)),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 4 (l' :-> r') => if(false,X:S,Y:S) -> Y:S 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> if(false,X:S,Y:S)} 0.002/0.002 s => if(divides(s(s(if(false,X:S,Y:S))),x5:S),filter(s(s(if(false,X:S,Y:S))),x6:S),cons(x5:S,filter(if(false,X:S,Y:S),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(Y:S)),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 5 (l' :-> r') => if(true,X:S,Y:S) -> X:S 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> if(true,X:S,Y:S)} 0.002/0.002 s => if(divides(s(s(if(true,X:S,Y:S))),x5:S),filter(s(s(if(true,X:S,Y:S))),x6:S),cons(x5:S,filter(if(true,X:S,Y:S),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(X:S)),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 6 (l' :-> r') => primes -> sieve(from(s(s(0)))) 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> primes} 0.002/0.002 s => if(divides(s(s(primes)),x5:S),filter(s(s(primes)),x6:S),cons(x5:S,filter(primes,sieve(x5:S)))) 0.002/0.002 t => filter(s(s(sieve(from(s(s(0)))))),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 7 (l' :-> r') => sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> sieve(cons(X:S,Y:S))} 0.002/0.002 s => if(divides(s(s(sieve(cons(X:S,Y:S)))),x5:S),filter(s(s(sieve(cons(X:S,Y:S)))),x6:S),cons(x5:S,filter(sieve(cons(X:S,Y:S)),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(cons(X:S,filter(X:S,sieve(Y:S))))),cons(x5:S,x6: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) => filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))) 0.002/0.002 Rule 8 (l' :-> r') => tail(cons(X:S,Y:S)) -> Y:S 0.002/0.002 Var => x4:S 0.002/0.002 Pos x4:S in l => [1,1,1] 0.002/0.002 Sigma => {x4:S -> tail(cons(X:S,Y:S))} 0.002/0.002 s => if(divides(s(s(tail(cons(X:S,Y:S)))),x5:S),filter(s(s(tail(cons(X:S,Y:S)))),x6:S),cons(x5:S,filter(tail(cons(X:S,Y:S)),sieve(x5:S)))) 0.002/0.002 t => filter(s(s(Y:S)),cons(x5:S,x6: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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 2 (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 => [1] 0.002/0.002 Sigma => {x7: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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 3 (l' :-> r') => head(cons(X:S,Y:S)) -> X:S 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> head(cons(X:S,Y:S))} 0.002/0.002 s => cons(head(cons(X:S,Y:S)),from(s(head(cons(X:S,Y:S))))) 0.002/0.002 t => from(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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 4 (l' :-> r') => if(false,X:S,Y:S) -> Y:S 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> if(false,X:S,Y:S)} 0.002/0.002 s => cons(if(false,X:S,Y:S),from(s(if(false,X:S,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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 5 (l' :-> r') => if(true,X:S,Y:S) -> X:S 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> if(true,X:S,Y:S)} 0.002/0.002 s => cons(if(true,X:S,Y:S),from(s(if(true,X:S,Y:S)))) 0.002/0.002 t => from(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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 6 (l' :-> r') => primes -> sieve(from(s(s(0)))) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> primes} 0.002/0.002 s => cons(primes,from(s(primes))) 0.002/0.002 t => from(sieve(from(s(s(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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 7 (l' :-> r') => sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> sieve(cons(X:S,Y:S))} 0.002/0.002 s => cons(sieve(cons(X:S,Y:S)),from(s(sieve(cons(X:S,Y:S))))) 0.002/0.002 t => from(cons(X:S,filter(X:S,sieve(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(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 8 (l' :-> r') => tail(cons(X:S,Y:S)) -> Y:S 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> tail(cons(X:S,Y:S))} 0.002/0.002 s => cons(tail(cons(X:S,Y:S)),from(s(tail(cons(X:S,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) => sieve(cons(x14:S,x15:S)) -> cons(x14:S,filter(x14:S,sieve(x15:S))) 0.002/0.002 Rule 7 (l' :-> r') => sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 Var => x14:S 0.002/0.002 Pos x14:S in l => [1,1] 0.002/0.002 Sigma => {x14:S -> sieve(cons(X:S,Y:S))} 0.002/0.002 s => cons(sieve(cons(X:S,Y:S)),filter(sieve(cons(X:S,Y:S)),sieve(x15:S))) 0.002/0.002 t => sieve(cons(cons(X:S,filter(X:S,sieve(Y:S))),x15: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) => sieve(cons(x14:S,x15:S)) -> cons(x14:S,filter(x14:S,sieve(x15:S))) 0.002/0.002 Rule 8 (l' :-> r') => tail(cons(X:S,Y:S)) -> Y:S 0.002/0.002 Var => x14:S 0.002/0.002 Pos x14:S in l => [1,1] 0.002/0.002 Sigma => {x14:S -> tail(cons(X:S,Y:S))} 0.002/0.002 s => cons(tail(cons(X:S,Y:S)),filter(tail(cons(X:S,Y:S)),sieve(x15:S))) 0.002/0.002 t => sieve(cons(Y:S,x15: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 X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2) 0.002/0.002 (from 1) 0.002/0.002 (head 1) 0.002/0.002 (if 1) 0.002/0.002 (primes) 0.002/0.002 (sieve 1) 0.002/0.002 (tail 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (divides 1 2) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (false) 0.002/0.002 (s 1) 0.002/0.002 (true) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(s(s(X:S)),cons(Y:S,Z:S)) -> if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 head(cons(X:S,Y:S)) -> X:S 0.002/0.002 if(false,X:S,Y:S) -> Y:S 0.002/0.002 if(true,X:S,Y:S) -> X:S 0.002/0.002 primes -> sieve(from(s(s(0)))) 0.002/0.002 sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 tail(cons(X:S,Y:S)) -> Y:S 0.002/0.002 ) 0.002/0.002 Critical Pairs: 0.002/0.002 => Not trivial, Not overlay, NW0, N1 0.002/0.002 => Not trivial, Not overlay, NW0, N2 0.002/0.002 => Not trivial, Not overlay, NW0, N3 0.002/0.002 => Not trivial, Not overlay, NW0, N4 0.002/0.002 => Not trivial, Not overlay, NW0, N5 0.002/0.002 => Not trivial, Not overlay, NW0, N6 0.002/0.002 => Not trivial, Not overlay, NW0, 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 => Not trivial, Not overlay, NW0, N10 0.002/0.002 => Not trivial, Not overlay, NW0, N11 0.002/0.002 => Not trivial, Not overlay, NW0, N12 0.002/0.002 => Not trivial, Not overlay, NW0, N13 0.002/0.002 => Not trivial, Not overlay, NW0, N14 0.002/0.002 => Not trivial, Not overlay, NW0, N15 0.002/0.002 => Not trivial, Not overlay, NW0, N16 0.002/0.002 => Not trivial, Not overlay, NW0, N17 0.002/0.002 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 Huet Levy Processor: 0.002/0.002 -> Rules: 0.002/0.002 filter(s(s(X:S)),cons(Y:S,Z:S)) -> if(divides(s(s(X:S)),Y:S),filter(s(s(X:S)),Z:S),cons(Y:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 head(cons(X:S,Y:S)) -> X:S 0.002/0.002 if(false,X:S,Y:S) -> Y:S 0.002/0.002 if(true,X:S,Y:S) -> X:S 0.002/0.002 primes -> sieve(from(s(s(0)))) 0.002/0.002 sieve(cons(X:S,Y:S)) -> cons(X:S,filter(X:S,sieve(Y:S))) 0.002/0.002 tail(cons(X:S,Y:S)) -> Y:S 0.002/0.002 -> Vars: 0.002/0.002 X, Y, Z, X, X, Y, X, Y, X, Y, X, Y, X, Y 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [X, Y], UV-RActive: [X, Y], UV-LFrozen: [Z], UV-RFrozen: [X, Y, Z]) 0.002/0.002 (UV-RuleId: 2, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: [X]) 0.002/0.002 (UV-RuleId: 3, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [Y], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 4, UV-LActive: [], UV-RActive: [Y], UV-LFrozen: [X, Y], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 5, UV-LActive: [], UV-RActive: [X], UV-LFrozen: [X, Y], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 7, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [Y], UV-RFrozen: [X, Y]) 0.002/0.002 (UV-RuleId: 8, UV-LActive: [X], UV-RActive: [Y], UV-LFrozen: [Y], UV-RFrozen: []) 0.002/0.002 -> FVars: 0.002/0.002 x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17 0.002/0.002 -> PVars: 0.002/0.002 X: [x4, x7, x8, x10, x12, x14, x16], Y: [x5, x9, x11, x13, x15, x17], Z: [x6] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: filter(s(s(x4:S)),cons(x5:S,x6:S)) -> if(divides(s(s(x4:S)),x5:S),filter(s(s(x4:S)),x6:S),cons(x5:S,filter(x4:S,sieve(x5:S)))), id: 1, possubterms: filter(s(s(x4:S)),cons(x5:S,x6:S))->[], s(s(x4:S))->[1], s(x4:S)->[1, 1], cons(x5:S,x6:S)->[2]) 0.002/0.002 (rule: from(x7:S) -> cons(x7:S,from(s(x7:S))), id: 2, possubterms: from(x7:S)->[]) 0.002/0.002 (rule: head(cons(x8:S,x9:S)) -> x8:S, id: 3, possubterms: head(cons(x8:S,x9:S))->[], cons(x8:S,x9:S)->[1]) 0.002/0.002 (rule: if(false,x10:S,x11:S) -> x11:S, id: 4, possubterms: if(false,x10:S,x11:S)->[], false->[1]) 0.002/0.002 (rule: if(true,x12:S,x13:S) -> x12:S, id: 5, possubterms: if(true,x12:S,x13:S)->[], true->[1]) 0.002/0.002 (rule: primes -> sieve(from(s(s(0)))), id: 6, possubterms: primes->[]) 0.002/0.002 (rule: sieve(cons(x14:S,x15:S)) -> cons(x14:S,filter(x14:S,sieve(x15:S))), id: 7, possubterms: sieve(cons(x14:S,x15:S))->[], cons(x14:S,x15:S)->[1]) 0.002/0.002 (rule: tail(cons(x16:S,x17:S)) -> x17:S, id: 8, possubterms: tail(cons(x16:S,x17:S))->[], cons(x16:S,x17: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 => Not trivial, Not overlay, NW0, N3 0.002/0.002 => Not trivial, Not overlay, NW0, N4 0.002/0.002 => Not trivial, Not overlay, NW0, N5 0.002/0.002 => Not trivial, Not overlay, NW0, N6 0.002/0.002 => Not trivial, Not overlay, NW0, 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 => Not trivial, Not overlay, NW0, N10 0.002/0.002 => Not trivial, Not overlay, NW0, N11 0.002/0.002 => Not trivial, Not overlay, NW0, N12 0.002/0.002 => Not trivial, Not overlay, NW0, N13 0.002/0.002 => Not trivial, Not overlay, NW0, N14 0.002/0.002 => Not trivial, Not overlay, NW0, N15 0.002/0.002 => Not trivial, Not overlay, NW0, N16 0.002/0.002 => Not trivial, Not overlay, NW0, N17 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(sieve(cons(X:S,Y:S)),from(s(sieve(cons(X:S,Y:S))))) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('cons(sieve(cons(X:S,Y:S)),from(s(sieve(cons(X:S,Y:S)))))', D0) 0.002/0.002 ID: 1 => ('cons(cons(X:S,filter(X:S,sieve(Y:S))),from(s(sieve(cons(X:S,Y:S)))))', D1, R7, P[1], S{x14:S -> X:S, x15:S -> Y:S}), NR: 'cons(X:S,filter(X:S,sieve(Y:S)))' 0.002/0.002 t: from(cons(X:S,filter(X:S,sieve(Y: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,filter(X:S,sieve(Y:S))))', D0) 0.002/0.002 ID: 1 => ('cons(cons(X:S,filter(X:S,sieve(Y:S))),from(s(cons(X:S,filter(X:S,sieve(Y:S))))))', D1, R2, P[], S{x7:S -> cons(X:S,filter(X:S,sieve(Y:S)))}), NR: 'cons(cons(X:S,filter(X:S,sieve(Y:S))),from(s(cons(X:S,filter(X:S,sieve(Y:S))))))' 0.002/0.002 cons(sieve(cons(X:S,Y:S)),from(s(sieve(cons(X:S,Y:S))))) ->* no union *<- from(cons(X:S,filter(X:S,sieve(Y:S)))) 0.002/0.002 "Not joinable" 0.002/0.002 0.002/0.002 The problem is not joinable. 0.002/0.002 0.01user 0.00system 0:00.02elapsed 86%CPU (0avgtext+0avgdata 12056maxresident)k 0.002/0.002 0inputs+0outputs (0major+1315minor)pagefaults 0swaps