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 (add 1 2) 0.002/0.002 (from 1) 0.002/0.002 (fst 1 2) 0.002/0.002 (len 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 fst(0,Z:S) -> nil 0.002/0.002 fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z:S)) 0.002/0.002 len(cons(X:S,Z:S)) -> s(len(Z:S)) 0.002/0.002 len(nil) -> 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 X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (add 1 2) 0.002/0.002 (from 1) 0.002/0.002 (fst 1 2) 0.002/0.002 (len 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 fst(0,Z:S) -> nil 0.002/0.002 fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z:S)) 0.002/0.002 len(cons(X:S,Z:S)) -> s(len(Z:S)) 0.002/0.002 len(nil) -> 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 X:S Y:S Z:S) 0.002/0.002 (STRATEGY CONTEXTSENSITIVE 0.002/0.002 (add 1 2) 0.002/0.002 (from 1) 0.002/0.002 (fst 1 2) 0.002/0.002 (len 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 fst(0,Z:S) -> nil 0.002/0.002 fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z:S)) 0.002/0.002 len(cons(X:S,Z:S)) -> s(len(Z:S)) 0.002/0.002 len(nil) -> 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) => add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)) 0.002/0.002 Rule 2 (l' :-> r') => add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 Var => x6:S 0.002/0.002 Pos x6:S in l => [2] 0.002/0.002 Sigma => {x6:S -> add(s(X:S),Y:S)} 0.002/0.002 s => s(add(x5:S,add(s(X:S),Y:S))) 0.002/0.002 t => add(s(x5:S),s(add(X:S,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) => add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)) 0.002/0.002 Rule 3 (l' :-> r') => from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 Var => x6:S 0.002/0.002 Pos x6:S in l => [2] 0.002/0.002 Sigma => {x6:S -> from(X:S)} 0.002/0.002 s => s(add(x5:S,from(X:S))) 0.002/0.002 t => add(s(x5:S),cons(X:S,from(s(X: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) => add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)) 0.002/0.002 Rule 4 (l' :-> r') => fst(0,Z:S) -> nil 0.002/0.002 Var => x6:S 0.002/0.002 Pos x6:S in l => [2] 0.002/0.002 Sigma => {x6:S -> fst(0,Z:S)} 0.002/0.002 s => s(add(x5:S,fst(0,Z:S))) 0.002/0.002 t => add(s(x5:S),nil) 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) => add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)) 0.002/0.002 Rule 5 (l' :-> r') => fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z:S)) 0.002/0.002 Var => x6:S 0.002/0.002 Pos x6:S in l => [2] 0.002/0.002 Sigma => {x6:S -> fst(s(X:S),cons(Y:S,Z:S))} 0.002/0.002 s => s(add(x5:S,fst(s(X:S),cons(Y:S,Z:S)))) 0.002/0.002 t => add(s(x5:S),cons(Y:S,fst(X:S,Z: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) => add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)) 0.002/0.002 Rule 6 (l' :-> r') => len(cons(X:S,Z:S)) -> s(len(Z:S)) 0.002/0.002 Var => x6:S 0.002/0.002 Pos x6:S in l => [2] 0.002/0.002 Sigma => {x6:S -> len(cons(X:S,Z:S))} 0.002/0.002 s => s(add(x5:S,len(cons(X:S,Z:S)))) 0.002/0.002 t => add(s(x5:S),s(len(Z: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) => add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)) 0.002/0.002 Rule 7 (l' :-> r') => len(nil) -> 0 0.002/0.002 Var => x6:S 0.002/0.002 Pos x6:S in l => [2] 0.002/0.002 Sigma => {x6:S -> len(nil)} 0.002/0.002 s => s(add(x5:S,len(nil))) 0.002/0.002 t => add(s(x5: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) => from(x7:S) -> cons(x7:S,from(s(x7:S))) 0.002/0.002 Rule 3 (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 4 (l' :-> r') => fst(0,Z:S) -> nil 0.002/0.002 Var => x7:S 0.002/0.002 Pos x7:S in l => [1] 0.002/0.002 Sigma => {x7:S -> fst(0,Z:S)} 0.002/0.002 s => cons(fst(0,Z:S),from(s(fst(0,Z:S)))) 0.002/0.002 t => from(nil) 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') => fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z: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 -> fst(s(X:S),cons(Y:S,Z:S))} 0.002/0.002 s => cons(fst(s(X:S),cons(Y:S,Z:S)),from(s(fst(s(X:S),cons(Y:S,Z:S))))) 0.002/0.002 t => from(cons(Y:S,fst(X:S,Z: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') => len(cons(X:S,Z:S)) -> s(len(Z: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 -> len(cons(X:S,Z:S))} 0.002/0.002 s => cons(len(cons(X:S,Z:S)),from(s(len(cons(X:S,Z:S))))) 0.002/0.002 t => from(s(len(Z: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 7 (l' :-> r') => len(nil) -> 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 -> len(nil)} 0.002/0.002 s => cons(len(nil),from(s(len(nil)))) 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 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 (add 1 2) 0.002/0.002 (from 1) 0.002/0.002 (fst 1 2) 0.002/0.002 (len 1) 0.002/0.002 (0) 0.002/0.002 (cons 1) 0.002/0.002 (fSNonEmpty) 0.002/0.002 (nil) 0.002/0.002 (s) 0.002/0.002 ) 0.002/0.002 (RULES 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 fst(0,Z:S) -> nil 0.002/0.002 fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z:S)) 0.002/0.002 len(cons(X:S,Z:S)) -> s(len(Z:S)) 0.002/0.002 len(nil) -> 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, 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 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.002/0.002 0.002/0.002 Huet Levy Processor: 0.002/0.002 -> Rules: 0.002/0.002 add(0,X:S) -> X:S 0.002/0.002 add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.002/0.002 from(X:S) -> cons(X:S,from(s(X:S))) 0.002/0.002 fst(0,Z:S) -> nil 0.002/0.002 fst(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,fst(X:S,Z:S)) 0.002/0.002 len(cons(X:S,Z:S)) -> s(len(Z:S)) 0.002/0.002 len(nil) -> 0 0.002/0.002 -> Vars: 0.002/0.002 X, X, Y, X, Z, X, Y, Z, X, Z 0.002/0.002 -> UVars: 0.002/0.002 (UV-RuleId: 1, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 (UV-RuleId: 2, UV-LActive: [Y], UV-RActive: [], UV-LFrozen: [X], UV-RFrozen: [X, Y]) 0.002/0.002 (UV-RuleId: 3, UV-LActive: [X], UV-RActive: [X], UV-LFrozen: [], UV-RFrozen: [X]) 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: [Y], UV-RActive: [Y], UV-LFrozen: [X, Z], UV-RFrozen: [X, Z]) 0.002/0.002 (UV-RuleId: 6, UV-LActive: [X], UV-RActive: [], UV-LFrozen: [Z], UV-RFrozen: [Z]) 0.002/0.002 (UV-RuleId: 7, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: []) 0.002/0.002 -> FVars: 0.002/0.002 x4, x5, x6, x7, x8, x9, x10, x11, x12, x13 0.002/0.002 -> PVars: 0.002/0.002 X: [x4, x5, x7, x9, x12], Y: [x6, x10], Z: [x8, x11, x13] 0.002/0.002 0.002/0.002 -> Rlps: 0.002/0.002 (rule: add(0,x4:S) -> x4:S, id: 1, possubterms: add(0,x4:S)->[], 0->[1]) 0.002/0.002 (rule: add(s(x5:S),x6:S) -> s(add(x5:S,x6:S)), id: 2, possubterms: add(s(x5:S),x6:S)->[], s(x5:S)->[1]) 0.002/0.002 (rule: from(x7:S) -> cons(x7:S,from(s(x7:S))), id: 3, possubterms: from(x7:S)->[]) 0.002/0.002 (rule: fst(0,x8:S) -> nil, id: 4, possubterms: fst(0,x8:S)->[], 0->[1]) 0.002/0.002 (rule: fst(s(x9:S),cons(x10:S,x11:S)) -> cons(x10:S,fst(x9:S,x11:S)), id: 5, possubterms: fst(s(x9:S),cons(x10:S,x11:S))->[], s(x9:S)->[1], cons(x10:S,x11:S)->[2]) 0.002/0.002 (rule: len(cons(x12:S,x13:S)) -> s(len(x13:S)), id: 6, possubterms: len(cons(x12:S,x13:S))->[], cons(x12:S,x13:S)->[1]) 0.002/0.002 (rule: len(nil) -> 0, id: 7, possubterms: len(nil)->[], nil->[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, NW1, 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, NW1, 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, NW0, N11 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: s(add(X:S,len(nil))) 0.002/0.002 Nodes: [0] 0.002/0.002 Edges: [] 0.002/0.002 ID: 0 => ('s(add(X:S,len(nil)))', D0) 0.002/0.002 t: add(s(X:S),0) 0.002/0.002 Nodes: [0,1] 0.002/0.002 Edges: [(0,1)] 0.002/0.002 ID: 0 => ('add(s(X:S),0)', D0) 0.002/0.002 ID: 1 => ('s(add(X:S,0))', D1, R2, P[], S{x5:S -> X:S, x6:S -> 0}), NR: 's(add(X:S,0))' 0.002/0.002 s(add(X:S,len(nil))) ->* no union *<- add(s(X:S),0) 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 56%CPU (0avgtext+0avgdata 11112maxresident)k 0.002/0.002 0inputs+0outputs (0major+1114minor)pagefaults 0swaps