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 X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (first 1 2) 0.001/0.001 (from 1) 0.001/0.001 (sel 1 2) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 first(0,Z:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z: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 X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (first 1 2) 0.001/0.001 (from 1) 0.001/0.001 (sel 1 2) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 first(0,Z:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z: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 X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (first 1 2) 0.001/0.001 (from 1) 0.001/0.001 (sel 1 2) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 first(0,Z:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z: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(x5:S),cons(x6:S,x7:S)) -> cons(x6:S,first(x5:S,x7:S)) 0.001/0.001 Rule 2 (l' :-> r') => first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 Var => x5:S 0.001/0.001 Pos x5:S in l => [1,1] 0.001/0.001 Sigma => {x5:S -> first(s(X:S),cons(Y:S,Z:S))} 0.001/0.001 s => cons(x6:S,first(first(s(X:S),cons(Y:S,Z:S)),x7:S)) 0.001/0.001 t => first(s(cons(Y:S,first(X:S,Z:S))),cons(x6:S,x7: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(x5:S),cons(x6:S,x7:S)) -> cons(x6:S,first(x5:S,x7:S)) 0.001/0.001 Rule 3 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 Var => x5:S 0.001/0.001 Pos x5:S in l => [1,1] 0.001/0.001 Sigma => {x5:S -> from(X:S)} 0.001/0.001 s => cons(x6:S,first(from(X:S),x7:S)) 0.001/0.001 t => first(s(cons(X:S,from(s(X:S)))),cons(x6:S,x7: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(x5:S),cons(x6:S,x7:S)) -> cons(x6:S,first(x5:S,x7:S)) 0.001/0.001 Rule 4 (l' :-> r') => sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 Var => x5:S 0.001/0.001 Pos x5:S in l => [1,1] 0.001/0.001 Sigma => {x5:S -> sel(0,cons(X:S,Z:S))} 0.001/0.001 s => cons(x6:S,first(sel(0,cons(X:S,Z:S)),x7:S)) 0.001/0.001 t => first(s(X:S),cons(x6:S,x7: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(x5:S),cons(x6:S,x7:S)) -> cons(x6:S,first(x5:S,x7:S)) 0.001/0.001 Rule 5 (l' :-> r') => sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z:S) 0.001/0.001 Var => x5:S 0.001/0.001 Pos x5:S in l => [1,1] 0.001/0.001 Sigma => {x5:S -> sel(s(X:S),cons(Y:S,Z:S))} 0.001/0.001 s => cons(x6:S,first(sel(s(X:S),cons(Y:S,Z:S)),x7:S)) 0.001/0.001 t => first(s(sel(X:S,Z:S)),cons(x6:S,x7: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) => from(x8:S) -> cons(x8:S,from(s(x8:S))) 0.001/0.001 Rule 3 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 Var => x8:S 0.001/0.001 Pos x8:S in l => [1] 0.001/0.001 Sigma => {x8:S -> from(X:S)} 0.001/0.001 s => cons(from(X:S),from(s(from(X:S)))) 0.001/0.001 t => from(cons(X:S,from(s(X: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) => from(x8:S) -> cons(x8:S,from(s(x8:S))) 0.001/0.001 Rule 4 (l' :-> r') => sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 Var => x8:S 0.001/0.001 Pos x8:S in l => [1] 0.001/0.001 Sigma => {x8:S -> sel(0,cons(X:S,Z:S))} 0.001/0.001 s => cons(sel(0,cons(X:S,Z:S)),from(s(sel(0,cons(X:S,Z:S))))) 0.001/0.001 t => from(X: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) => from(x8:S) -> cons(x8:S,from(s(x8:S))) 0.001/0.001 Rule 5 (l' :-> r') => sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z:S) 0.001/0.001 Var => x8:S 0.001/0.001 Pos x8:S in l => [1] 0.001/0.001 Sigma => {x8:S -> sel(s(X:S),cons(Y:S,Z:S))} 0.001/0.001 s => cons(sel(s(X:S),cons(Y:S,Z:S)),from(s(sel(s(X:S),cons(Y:S,Z:S))))) 0.001/0.001 t => from(sel(X:S,Z: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 X:S Y:S Z:S) 0.001/0.001 (STRATEGY CONTEXTSENSITIVE 0.001/0.001 (first 1 2) 0.001/0.001 (from 1) 0.001/0.001 (sel 1 2) 0.001/0.001 (0) 0.001/0.001 (cons 1) 0.001/0.001 (fSNonEmpty) 0.001/0.001 (nil) 0.001/0.001 (s 1) 0.001/0.001 ) 0.001/0.001 (RULES 0.001/0.001 first(0,Z:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z: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, NW0, N5 0.001/0.001 => Not trivial, Not overlay, NW0, N6 0.001/0.001 => Not trivial, Not overlay, NW0, N7 0.001/0.001 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.001/0.001 0.001/0.001 Huet Levy Processor: 0.001/0.001 -> Rules: 0.001/0.001 first(0,Z:S) -> nil 0.001/0.001 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) 0.001/0.001 from(X:S) -> cons(X:S,from(s(X:S))) 0.001/0.001 sel(0,cons(X:S,Z:S)) -> X:S 0.001/0.001 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,Z:S) 0.001/0.001 -> Vars: 0.001/0.001 Z, X, Y, Z, X, X, Z, X, Y, Z 0.001/0.001 -> UVars: 0.001/0.001 (UV-RuleId: 1, UV-LActive: [Z], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 2, UV-LActive: [X, Y], UV-RActive: [Y], UV-LFrozen: [Z], UV-RFrozen: [X, Z]) 0.001/0.001 (UV-RuleId: 3, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: [X]) 0.001/0.001 (UV-RuleId: 4, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [Z], UV-RFrozen: []) 0.001/0.001 (UV-RuleId: 5, UV-LActive: [X, Y], UV-RActive: [X, Z], UV-LFrozen: [Z], UV-RFrozen: []) 0.001/0.001 -> FVars: 0.001/0.001 x4, x5, x6, x7, x8, x9, x10, x11, x12, x13 0.001/0.001 -> PVars: 0.001/0.001 Z: [x4, x7, x10, x13], X: [x5, x8, x9, x11], Y: [x6, x12] 0.001/0.001 0.001/0.001 -> Rlps: 0.001/0.001 (rule: first(0,x4:S) -> nil, id: 1, possubterms: first(0,x4:S)->[], 0->[1]) 0.001/0.001 (rule: first(s(x5:S),cons(x6:S,x7:S)) -> cons(x6:S,first(x5:S,x7:S)), id: 2, possubterms: first(s(x5:S),cons(x6:S,x7:S))->[], s(x5:S)->[1], cons(x6:S,x7:S)->[2]) 0.001/0.001 (rule: from(x8:S) -> cons(x8:S,from(s(x8:S))), id: 3, possubterms: from(x8:S)->[]) 0.001/0.001 (rule: sel(0,cons(x9:S,x10:S)) -> x9:S, id: 4, possubterms: sel(0,cons(x9:S,x10:S))->[], 0->[1], cons(x9:S,x10:S)->[2]) 0.001/0.001 (rule: sel(s(x11:S),cons(x12:S,x13:S)) -> sel(x11:S,x13:S), id: 5, possubterms: sel(s(x11:S),cons(x12:S,x13:S))->[], s(x11:S)->[1], cons(x12:S,x13:S)->[2]) 0.001/0.001 0.001/0.001 -> Unifications: 0.001/0.001 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, NW1, 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, NW0, N6 0.001/0.001 => Not trivial, Not overlay, NW0, N7 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,first(sel(0,cons(X:S,Z:S)),Z:S)) 0.001/0.001 Nodes: [0] 0.001/0.001 Edges: [] 0.001/0.001 ID: 0 => ('cons(Y:S,first(sel(0,cons(X:S,Z:S)),Z:S))', D0) 0.001/0.001 t: first(s(X:S),cons(Y:S,Z:S)) 0.001/0.001 Nodes: [0,1] 0.001/0.001 Edges: [(0,1)] 0.001/0.001 ID: 0 => ('first(s(X:S),cons(Y:S,Z:S))', D0) 0.001/0.001 ID: 1 => ('cons(Y:S,first(X:S,Z:S))', D1, R2, P[], S{x5:S -> X:S, x6:S -> Y:S, x7:S -> Z:S}), NR: 'cons(Y:S,first(X:S,Z:S))' 0.001/0.001 cons(Y:S,first(sel(0,cons(X:S,Z:S)),Z:S)) ->* no union *<- first(s(X:S),cons(Y:S,Z: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 84%CPU (0avgtext+0avgdata 10428maxresident)k 0.001/0.001 0inputs+0outputs (0major+923minor)pagefaults 0swaps