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 M:S N:S X:S Y:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2 3) 0.002/0.002 (nats 1) 0.002/0.002 (sieve 1) 0.002/0.002 (zprimes) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(cons(X:S,Y:S),0,M:S) -> cons(0,filter(Y:S,M:S,M:S)) 0.002/0.002 filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 zprimes -> sieve(nats(s(s(0)))) 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 M:S N:S X:S Y:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2 3) 0.002/0.002 (nats 1) 0.002/0.002 (sieve 1) 0.002/0.002 (zprimes) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(cons(X:S,Y:S),0,M:S) -> cons(0,filter(Y:S,M:S,M:S)) 0.002/0.002 filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 zprimes -> sieve(nats(s(s(0)))) 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 M:S N:S X:S Y:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2 3) 0.002/0.002 (nats 1) 0.002/0.002 (sieve 1) 0.002/0.002 (zprimes) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(cons(X:S,Y:S),0,M:S) -> cons(0,filter(Y:S,M:S,M:S)) 0.002/0.002 filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 zprimes -> sieve(nats(s(s(0)))) 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(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)) 0.002/0.002 Rule 1 (l' :-> r') => filter(cons(X:S,Y:S),0,M:S) -> cons(0,filter(Y:S,M:S,M:S)) 0.002/0.002 Var => x5:S 0.002/0.002 Pos x5:S in l => [3] 0.002/0.002 Sigma => {x5:S -> filter(cons(X:S,Y:S),0,M:S)} 0.002/0.002 s => cons(0,filter(x7:S,filter(cons(X:S,Y:S),0,M:S),filter(cons(X:S,Y:S),0,M:S))) 0.002/0.002 t => filter(cons(x6:S,x7:S),0,cons(0,filter(Y:S,M:S,M: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) => filter(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)) 0.002/0.002 Rule 2 (l' :-> r') => filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 Var => x5:S 0.002/0.002 Pos x5:S in l => [3] 0.002/0.002 Sigma => {x5:S -> filter(cons(X:S,Y:S),s(N:S),M:S)} 0.002/0.002 s => cons(0,filter(x7:S,filter(cons(X:S,Y:S),s(N:S),M:S),filter(cons(X:S,Y:S),s(N:S),M:S))) 0.002/0.002 t => filter(cons(x6:S,x7:S),0,cons(X:S,filter(Y:S,N:S,M: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) => filter(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)) 0.002/0.002 Rule 3 (l' :-> r') => nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 Var => x5:S 0.002/0.002 Pos x5:S in l => [3] 0.002/0.002 Sigma => {x5:S -> nats(N:S)} 0.002/0.002 s => cons(0,filter(x7:S,nats(N:S),nats(N:S))) 0.002/0.002 t => filter(cons(x6:S,x7:S),0,cons(N:S,nats(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) => filter(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)) 0.002/0.002 Rule 4 (l' :-> r') => sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 Var => x5:S 0.002/0.002 Pos x5:S in l => [3] 0.002/0.002 Sigma => {x5:S -> sieve(cons(0,Y:S))} 0.002/0.002 s => cons(0,filter(x7:S,sieve(cons(0,Y:S)),sieve(cons(0,Y:S)))) 0.002/0.002 t => filter(cons(x6:S,x7:S),0,cons(0,sieve(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) => filter(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)) 0.002/0.002 Rule 5 (l' :-> r') => sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 Var => x5:S 0.002/0.002 Pos x5:S in l => [3] 0.002/0.002 Sigma => {x5:S -> sieve(cons(s(N:S),Y:S))} 0.002/0.002 s => cons(0,filter(x7:S,sieve(cons(s(N:S),Y:S)),sieve(cons(s(N:S),Y:S)))) 0.002/0.002 t => filter(cons(x6:S,x7:S),0,cons(s(N:S),sieve(filter(Y:S,N: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) => filter(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)) 0.002/0.002 Rule 6 (l' :-> r') => zprimes -> sieve(nats(s(s(0)))) 0.002/0.002 Var => x5:S 0.002/0.002 Pos x5:S in l => [3] 0.002/0.002 Sigma => {x5:S -> zprimes} 0.002/0.002 s => cons(0,filter(x7:S,zprimes,zprimes)) 0.002/0.002 t => filter(cons(x6:S,x7:S),0,sieve(nats(s(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) => filter(cons(x10:S,x11:S),s(x9:S),x8:S) -> cons(x10:S,filter(x11:S,x9:S,x8:S)) 0.002/0.002 Rule 2 (l' :-> r') => filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 Var => x8:S 0.002/0.002 Pos x8:S in l => [3] 0.002/0.002 Sigma => {x8:S -> filter(cons(X:S,Y:S),s(N:S),M:S)} 0.002/0.002 s => cons(x10:S,filter(x11:S,x9:S,filter(cons(X:S,Y:S),s(N:S),M:S))) 0.002/0.002 t => filter(cons(x10:S,x11:S),s(x9:S),cons(X:S,filter(Y:S,N:S,M: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) => filter(cons(x10:S,x11:S),s(x9:S),x8:S) -> cons(x10:S,filter(x11:S,x9:S,x8:S)) 0.002/0.002 Rule 3 (l' :-> r') => nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 Var => x8:S 0.002/0.002 Pos x8:S in l => [3] 0.002/0.002 Sigma => {x8:S -> nats(N:S)} 0.002/0.002 s => cons(x10:S,filter(x11:S,x9:S,nats(N:S))) 0.002/0.002 t => filter(cons(x10:S,x11:S),s(x9:S),cons(N:S,nats(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) => filter(cons(x10:S,x11:S),s(x9:S),x8:S) -> cons(x10:S,filter(x11:S,x9:S,x8:S)) 0.002/0.002 Rule 4 (l' :-> r') => sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 Var => x8:S 0.002/0.002 Pos x8:S in l => [3] 0.002/0.002 Sigma => {x8:S -> sieve(cons(0,Y:S))} 0.002/0.002 s => cons(x10:S,filter(x11:S,x9:S,sieve(cons(0,Y:S)))) 0.002/0.002 t => filter(cons(x10:S,x11:S),s(x9:S),cons(0,sieve(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) => filter(cons(x10:S,x11:S),s(x9:S),x8:S) -> cons(x10:S,filter(x11:S,x9:S,x8:S)) 0.002/0.002 Rule 5 (l' :-> r') => sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 Var => x8:S 0.002/0.002 Pos x8:S in l => [3] 0.002/0.002 Sigma => {x8:S -> sieve(cons(s(N:S),Y:S))} 0.002/0.002 s => cons(x10:S,filter(x11:S,x9:S,sieve(cons(s(N:S),Y:S)))) 0.002/0.002 t => filter(cons(x10:S,x11:S),s(x9:S),cons(s(N:S),sieve(filter(Y:S,N: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) => filter(cons(x10:S,x11:S),s(x9:S),x8:S) -> cons(x10:S,filter(x11:S,x9:S,x8:S)) 0.002/0.002 Rule 6 (l' :-> r') => zprimes -> sieve(nats(s(s(0)))) 0.002/0.002 Var => x8:S 0.002/0.002 Pos x8:S in l => [3] 0.002/0.002 Sigma => {x8:S -> zprimes} 0.002/0.002 s => cons(x10:S,filter(x11:S,x9:S,zprimes)) 0.002/0.002 t => filter(cons(x10:S,x11:S),s(x9:S),sieve(nats(s(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) => nats(x12:S) -> cons(x12:S,nats(s(x12:S))) 0.002/0.002 Rule 3 (l' :-> r') => nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 Var => x12:S 0.002/0.002 Pos x12:S in l => [1] 0.002/0.002 Sigma => {x12:S -> nats(N:S)} 0.002/0.002 s => cons(nats(N:S),nats(s(nats(N:S)))) 0.002/0.002 t => nats(cons(N:S,nats(s(N: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) => nats(x12:S) -> cons(x12:S,nats(s(x12:S))) 0.002/0.002 Rule 4 (l' :-> r') => sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 Var => x12:S 0.002/0.002 Pos x12:S in l => [1] 0.002/0.002 Sigma => {x12:S -> sieve(cons(0,Y:S))} 0.002/0.002 s => cons(sieve(cons(0,Y:S)),nats(s(sieve(cons(0,Y:S))))) 0.002/0.002 t => nats(cons(0,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) => nats(x12:S) -> cons(x12:S,nats(s(x12:S))) 0.002/0.002 Rule 5 (l' :-> r') => sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 Var => x12:S 0.002/0.002 Pos x12:S in l => [1] 0.002/0.002 Sigma => {x12:S -> sieve(cons(s(N:S),Y:S))} 0.002/0.002 s => cons(sieve(cons(s(N:S),Y:S)),nats(s(sieve(cons(s(N:S),Y:S))))) 0.002/0.002 t => nats(cons(s(N:S),sieve(filter(Y:S,N:S,N: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) => nats(x12:S) -> cons(x12:S,nats(s(x12:S))) 0.002/0.002 Rule 6 (l' :-> r') => zprimes -> sieve(nats(s(s(0)))) 0.002/0.002 Var => x12:S 0.002/0.002 Pos x12:S in l => [1] 0.002/0.002 Sigma => {x12:S -> zprimes} 0.002/0.002 s => cons(zprimes,nats(s(zprimes))) 0.002/0.002 t => nats(sieve(nats(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) => sieve(cons(s(x14:S),x15:S)) -> cons(s(x14:S),sieve(filter(x15:S,x14:S,x14:S))) 0.002/0.002 Rule 5 (l' :-> r') => sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 Var => x14:S 0.002/0.002 Pos x14:S in l => [1,1,1] 0.002/0.002 Sigma => {x14:S -> sieve(cons(s(N:S),Y:S))} 0.002/0.002 s => cons(s(sieve(cons(s(N:S),Y:S))),sieve(filter(x15:S,sieve(cons(s(N:S),Y:S)),sieve(cons(s(N:S),Y:S))))) 0.002/0.002 t => sieve(cons(s(cons(s(N:S),sieve(filter(Y:S,N:S,N: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(s(x14:S),x15:S)) -> cons(s(x14:S),sieve(filter(x15:S,x14:S,x14:S))) 0.002/0.002 Rule 6 (l' :-> r') => zprimes -> sieve(nats(s(s(0)))) 0.002/0.002 Var => x14:S 0.002/0.002 Pos x14:S in l => [1,1,1] 0.002/0.002 Sigma => {x14:S -> zprimes} 0.002/0.002 s => cons(s(zprimes),sieve(filter(x15:S,zprimes,zprimes))) 0.002/0.002 t => sieve(cons(s(sieve(nats(s(s(0))))),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 M:S N:S X:S Y:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (filter 1 2 3) 0.002/0.002 (nats 1) 0.002/0.002 (sieve 1) 0.002/0.002 (zprimes) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (s 1) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 filter(cons(X:S,Y:S),0,M:S) -> cons(0,filter(Y:S,M:S,M:S)) 0.002/0.002 filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 zprimes -> sieve(nats(s(s(0)))) 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, NW1, N9 0.002/0.002 => Not trivial, Not overlay, NW1, N10 0.002/0.002 => Not trivial, Not overlay, NW1, 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(cons(X:S,Y:S),0,M:S) -> cons(0,filter(Y:S,M:S,M:S)) 0.002/0.002 filter(cons(X:S,Y:S),s(N:S),M:S) -> cons(X:S,filter(Y:S,N:S,M:S)) 0.002/0.002 nats(N:S) -> cons(N:S,nats(s(N:S))) 0.002/0.002 sieve(cons(0,Y:S)) -> cons(0,sieve(Y:S)) 0.002/0.002 sieve(cons(s(N:S),Y:S)) -> cons(s(N:S),sieve(filter(Y:S,N:S,N:S))) 0.002/0.002 zprimes -> sieve(nats(s(s(0)))) 0.002/0.002 -> Vars: 0.002/0.002 M, X, Y, M, N, X, Y, N, Y, N, Y 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [M, X], UV-RActive: [], UV-LFrozen: [Y], UV-RFrozen: [M, Y]) 0.002/0.002 (UV-RuleId: 2, UV-LActive: [M, N, X], UV-RActive: [X], UV-LFrozen: [Y], UV-RFrozen: [M, N, Y]) 0.002/0.002 (UV-RuleId: 3, UV-LActive: [N], UV-RActive: [N], UV-LFrozen: [], UV-RFrozen: [N]) 0.002/0.002 (UV-RuleId: 4, UV-LActive: [], UV-RActive: [], UV-LFrozen: [Y], UV-RFrozen: [Y]) 0.002/0.002 (UV-RuleId: 5, UV-LActive: [N], UV-RActive: [N], UV-LFrozen: [Y], UV-RFrozen: [N, Y]) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 -> FVars: 0.002/0.002 x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15 0.002/0.002 -> PVars: 0.002/0.002 M: [x5, x8], X: [x6, x10], Y: [x7, x11, x13, x15], N: [x9, x12, x14] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: filter(cons(x6:S,x7:S),0,x5:S) -> cons(0,filter(x7:S,x5:S,x5:S)), id: 1, possubterms: filter(cons(x6:S,x7:S),0,x5:S)->[], cons(x6:S,x7:S)->[1], 0->[2]) 0.002/0.002 (rule: filter(cons(x10:S,x11:S),s(x9:S),x8:S) -> cons(x10:S,filter(x11:S,x9:S,x8:S)), id: 2, possubterms: filter(cons(x10:S,x11:S),s(x9:S),x8:S)->[], cons(x10:S,x11:S)->[1], s(x9:S)->[2]) 0.002/0.002 (rule: nats(x12:S) -> cons(x12:S,nats(s(x12:S))), id: 3, possubterms: nats(x12:S)->[]) 0.002/0.002 (rule: sieve(cons(0,x13:S)) -> cons(0,sieve(x13:S)), id: 4, possubterms: sieve(cons(0,x13:S))->[], cons(0,x13:S)->[1], 0->[1, 1]) 0.002/0.002 (rule: sieve(cons(s(x14:S),x15:S)) -> cons(s(x14:S),sieve(filter(x15:S,x14:S,x14:S))), id: 5, possubterms: sieve(cons(s(x14:S),x15:S))->[], cons(s(x14:S),x15:S)->[1], s(x14:S)->[1, 1]) 0.002/0.002 (rule: zprimes -> sieve(nats(s(s(0)))), id: 6, possubterms: zprimes->[]) 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, NW1, N2 0.002/0.002 => Not trivial, Not overlay, NW1, 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, 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 => Not trivial, Not overlay, NW1, N10 0.002/0.002 => Not trivial, Not overlay, NW1, 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, NW1, N14 0.002/0.002 => Not trivial, Not overlay, NW0, N15 0.002/0.002 => Not trivial, Not overlay, NW1, N16 0.002/0.002 => Not trivial, Not overlay, NW1, 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(0,filter(Y:S,filter(cons(X:S,Y:S),s(N:S),M:S),filter(cons(X:S,Y:S),s(N:S),M:S))) 0.002/0.002 Nodes: [0] 0.002/0.002 Edges: [] 0.002/0.002 ID: 0 => ('cons(0,filter(Y:S,filter(cons(X:S,Y:S),s(N:S),M:S),filter(cons(X:S,Y:S),s(N:S),M:S)))', D0) 0.002/0.002 t: filter(cons(X:S,Y:S),0,cons(X:S,filter(Y:S,N:S,M:S))) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('filter(cons(X:S,Y:S),0,cons(X:S,filter(Y:S,N:S,M:S)))', D0) 0.002/0.002 ID: 1 => ('cons(0,filter(Y:S,cons(X:S,filter(Y:S,N:S,M:S)),cons(X:S,filter(Y:S,N:S,M:S))))', D1, R1, P[], S{x5:S -> cons(X:S,filter(Y:S,N:S,M:S)), x6:S -> X:S, x7:S -> Y:S}), NR: 'cons(0,filter(Y:S,cons(X:S,filter(Y:S,N:S,M:S)),cons(X:S,filter(Y:S,N:S,M:S))))' 0.002/0.002 cons(0,filter(Y:S,filter(cons(X:S,Y:S),s(N:S),M:S),filter(cons(X:S,Y:S),s(N:S),M:S))) ->* no union *<- filter(cons(X:S,Y:S),0,cons(X:S,filter(Y:S,N:S,M: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 69%CPU (0avgtext+0avgdata 11440maxresident)k 0.002/0.002 0inputs+0outputs (0major+1197minor)pagefaults 0swaps