0.53/0.62 YES 0.53/0.62 0.53/0.62 Problem 1: 0.53/0.62 0.53/0.62 (VAR k l x y z) 0.53/0.62 (THEORY 0.53/0.62 (AC plus)) 0.53/0.62 (RULES 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 ) 0.53/0.62 0.53/0.62 Problem 1: 0.53/0.62 0.53/0.62 Dependency Pairs Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 APP(cons(x,l),k) -> APP(l,k) 0.53/0.62 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z)) 0.53/0.62 MINUS(minus(x,y),z) -> PLUS(y,z) 0.53/0.62 MINUS(s(x),s(y)) -> MINUS(x,y) 0.53/0.62 PLUS(plus(0,y),x5) -> PLUS(y,x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 QUOT(s(x),s(y)) -> MINUS(x,y) 0.53/0.62 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y)) 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> APP(l,sum(cons(x,cons(y,k)))) 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> SUM(cons(x,cons(y,k))) 0.53/0.62 SUM(cons(x,cons(y,l))) -> PLUS(x,y) 0.53/0.62 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l)) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 0.53/0.62 Problem 1: 0.53/0.62 0.53/0.62 SCC Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 APP(cons(x,l),k) -> APP(l,k) 0.53/0.62 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z)) 0.53/0.62 MINUS(minus(x,y),z) -> PLUS(y,z) 0.53/0.62 MINUS(s(x),s(y)) -> MINUS(x,y) 0.53/0.62 PLUS(plus(0,y),x5) -> PLUS(y,x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 QUOT(s(x),s(y)) -> MINUS(x,y) 0.53/0.62 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y)) 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> APP(l,sum(cons(x,cons(y,k)))) 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> SUM(cons(x,cons(y,k))) 0.53/0.62 SUM(cons(x,cons(y,l))) -> PLUS(x,y) 0.53/0.62 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l)) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Strongly Connected Components: 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 PLUS(plus(0,y),x5) -> PLUS(y,x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) -> PLUS(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l)) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 Empty 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z)) 0.53/0.62 MINUS(s(x),s(y)) -> MINUS(x,y) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 Empty 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y)) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 Empty 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 APP(cons(x,l),k) -> APP(l,k) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 Empty 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 Empty 0.53/0.62 0.53/0.62 0.53/0.62 The problem is decomposed in 6 subproblems. 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 Reduction Pairs Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(0,y),x5) -> PLUS(y,x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Usable Equations: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> Usable Rules: 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Interpretation type: 0.53/0.62 Linear 0.53/0.62 ->Coefficients: 0.53/0.62 Natural Numbers 0.53/0.62 ->Dimension: 0.53/0.62 1 0.53/0.62 ->Bound: 0.53/0.62 2 0.53/0.62 ->Interpretation: 0.53/0.62 0.53/0.62 [app](X1,X2) = 0 0.53/0.62 [minus](X1,X2) = 0 0.53/0.62 [plus](X1,X2) = X1 + X2 0.53/0.62 [quot](X1,X2) = 0 0.53/0.62 [sum](X) = 0 0.53/0.62 [0] = 2 0.53/0.62 [cons](X1,X2) = 0 0.53/0.62 [nil] = 0 0.53/0.62 [s](X) = X 0.53/0.62 [APP](X1,X2) = 0 0.53/0.62 [MINUS](X1,X2) = 0 0.53/0.62 [PLUS](X1,X2) = X1 + X2 0.53/0.62 [QUOT](X1,X2) = 0 0.53/0.62 [SUM](X) = 0 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 SCC Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Strongly Connected Components: 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) -> PLUS(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 Reduction Pairs Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(x,y) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Usable Equations: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> Usable Rules: 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Interpretation type: 0.53/0.62 Linear 0.53/0.62 ->Coefficients: 0.53/0.62 Natural Numbers 0.53/0.62 ->Dimension: 0.53/0.62 1 0.53/0.62 ->Bound: 0.53/0.62 2 0.53/0.62 ->Interpretation: 0.53/0.62 0.53/0.62 [app](X1,X2) = 0 0.53/0.62 [minus](X1,X2) = 0 0.53/0.62 [plus](X1,X2) = X1 + X2 + 1 0.53/0.62 [quot](X1,X2) = 0 0.53/0.62 [sum](X) = 0 0.53/0.62 [0] = 0 0.53/0.62 [cons](X1,X2) = 0 0.53/0.62 [nil] = 0 0.53/0.62 [s](X) = X + 2 0.53/0.62 [APP](X1,X2) = 0 0.53/0.62 [MINUS](X1,X2) = 0 0.53/0.62 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.62 [QUOT](X1,X2) = 0 0.53/0.62 [SUM](X) = 0 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 SCC Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Strongly Connected Components: 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) -> PLUS(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 Reduction Pairs Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Usable Equations: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> Usable Rules: 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,x6) 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Interpretation type: 0.53/0.62 Linear 0.53/0.62 ->Coefficients: 0.53/0.62 Natural Numbers 0.53/0.62 ->Dimension: 0.53/0.62 1 0.53/0.62 ->Bound: 0.53/0.62 2 0.53/0.62 ->Interpretation: 0.53/0.62 0.53/0.62 [app](X1,X2) = 0 0.53/0.62 [minus](X1,X2) = 0 0.53/0.62 [plus](X1,X2) = X1 + X2 + 1 0.53/0.62 [quot](X1,X2) = 0 0.53/0.62 [sum](X) = 0 0.53/0.62 [0] = 0 0.53/0.62 [cons](X1,X2) = 0 0.53/0.62 [nil] = 0 0.53/0.62 [s](X) = X + 2 0.53/0.62 [APP](X1,X2) = 0 0.53/0.62 [MINUS](X1,X2) = 0 0.53/0.62 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.62 [QUOT](X1,X2) = 0 0.53/0.62 [SUM](X) = 0 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 SCC Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Strongly Connected Components: 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) -> PLUS(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 Reduction Pairs Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 PLUS(s(x),y) -> PLUS(x,y) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Usable Equations: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> Usable Rules: 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Interpretation type: 0.53/0.62 Linear 0.53/0.62 ->Coefficients: 0.53/0.62 Natural Numbers 0.53/0.62 ->Dimension: 0.53/0.62 1 0.53/0.62 ->Bound: 0.53/0.62 2 0.53/0.62 ->Interpretation: 0.53/0.62 0.53/0.62 [app](X1,X2) = 0 0.53/0.62 [minus](X1,X2) = 0 0.53/0.62 [plus](X1,X2) = X1 + X2 + 2 0.53/0.62 [quot](X1,X2) = 0 0.53/0.62 [sum](X) = 0 0.53/0.62 [0] = 0 0.53/0.62 [cons](X1,X2) = 0 0.53/0.62 [nil] = 0 0.53/0.62 [s](X) = X + 2 0.53/0.62 [APP](X1,X2) = 0 0.53/0.62 [MINUS](X1,X2) = 0 0.53/0.62 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.62 [QUOT](X1,X2) = 0 0.53/0.62 [SUM](X) = 0 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 SCC Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 ->Strongly Connected Components: 0.53/0.62 ->->Cycle: 0.53/0.62 ->->-> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 -> FAxioms: 0.53/0.62 plus(plus(x5,x6),x7) -> plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) -> plus(x6,x5) 0.53/0.62 PLUS(plus(x5,x6),x7) -> PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) -> PLUS(x6,x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 ->->-> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.62 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.62 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.62 -> SRules: 0.53/0.62 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.62 0.53/0.62 Problem 1.1: 0.53/0.62 0.53/0.62 Reduction Pairs Processor: 0.53/0.62 -> FAxioms: 0.53/0.62 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.62 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.62 -> Pairs: 0.53/0.62 PLUS(plus(s(x),y),x5) -> PLUS(s(plus(x,y)),x5) 0.53/0.62 -> EAxioms: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Usable Equations: 0.53/0.62 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.62 plus(x5,x6) = plus(x6,x5) 0.53/0.62 -> Rules: 0.53/0.62 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.62 app(nil,k) -> k 0.53/0.62 app(l,nil) -> l 0.53/0.62 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.62 minus(s(x),s(y)) -> minus(x,y) 0.53/0.62 minus(x,0) -> x 0.53/0.62 plus(0,y) -> y 0.53/0.62 plus(s(x),y) -> s(plus(x,y)) 0.53/0.62 quot(0,s(y)) -> 0 0.53/0.62 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.62 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> Usable Rules: 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 -> SRules: 0.53/0.63 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.63 ->Interpretation type: 0.53/0.63 Linear 0.53/0.63 ->Coefficients: 0.53/0.63 Natural Numbers 0.53/0.63 ->Dimension: 0.53/0.63 1 0.53/0.63 ->Bound: 0.53/0.63 2 0.53/0.63 ->Interpretation: 0.53/0.63 0.53/0.63 [app](X1,X2) = 0 0.53/0.63 [minus](X1,X2) = 0 0.53/0.63 [plus](X1,X2) = X1 + X2 + 2 0.53/0.63 [quot](X1,X2) = 0 0.53/0.63 [sum](X) = 0 0.53/0.63 [0] = 0 0.53/0.63 [cons](X1,X2) = 0 0.53/0.63 [nil] = 0 0.53/0.63 [s](X) = 2 0.53/0.63 [APP](X1,X2) = 0 0.53/0.63 [MINUS](X1,X2) = 0 0.53/0.63 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.63 [QUOT](X1,X2) = 0 0.53/0.63 [SUM](X) = 0 0.53/0.63 0.53/0.63 Problem 1.1: 0.53/0.63 0.53/0.63 SCC Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 PLUS(plus(x5,x6),x7) = PLUS(x5,plus(x6,x7)) 0.53/0.63 PLUS(x5,x6) = PLUS(x6,x5) 0.53/0.63 -> Pairs: 0.53/0.63 Empty 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 PLUS(x5,plus(x6,x7)) -> PLUS(x6,x7) 0.53/0.63 ->Strongly Connected Components: 0.53/0.63 There is no strongly connected component 0.53/0.63 0.53/0.63 The problem is finite. 0.53/0.63 0.53/0.63 Problem 1.2: 0.53/0.63 0.53/0.63 Reduction Pairs Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l)) 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Usable Equations: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> Usable Rules: 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Interpretation type: 0.53/0.63 Linear 0.53/0.63 ->Coefficients: 0.53/0.63 Natural Numbers 0.53/0.63 ->Dimension: 0.53/0.63 1 0.53/0.63 ->Bound: 0.53/0.63 2 0.53/0.63 ->Interpretation: 0.53/0.63 0.53/0.63 [app](X1,X2) = 0 0.53/0.63 [minus](X1,X2) = 0 0.53/0.63 [plus](X1,X2) = X1 + X2 0.53/0.63 [quot](X1,X2) = 0 0.53/0.63 [sum](X) = 0 0.53/0.63 [0] = 0 0.53/0.63 [cons](X1,X2) = 2.X1 + X2 + 2 0.53/0.63 [nil] = 0 0.53/0.63 [s](X) = 2 0.53/0.63 [APP](X1,X2) = 0 0.53/0.63 [MINUS](X1,X2) = 0 0.53/0.63 [PLUS](X1,X2) = 0 0.53/0.63 [QUOT](X1,X2) = 0 0.53/0.63 [SUM](X) = X 0.53/0.63 0.53/0.63 Problem 1.2: 0.53/0.63 0.53/0.63 SCC Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 Empty 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Strongly Connected Components: 0.53/0.63 There is no strongly connected component 0.53/0.63 0.53/0.63 The problem is finite. 0.53/0.63 0.53/0.63 Problem 1.3: 0.53/0.63 0.53/0.63 Subterm Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 MINUS(minus(x,y),z) -> MINUS(x,plus(y,z)) 0.53/0.63 MINUS(s(x),s(y)) -> MINUS(x,y) 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Projection: 0.53/0.63 pi(MINUS) = [1] 0.53/0.63 0.53/0.63 Problem 1.3: 0.53/0.63 0.53/0.63 SCC Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 Empty 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Strongly Connected Components: 0.53/0.63 There is no strongly connected component 0.53/0.63 0.53/0.63 The problem is finite. 0.53/0.63 0.53/0.63 Problem 1.4: 0.53/0.63 0.53/0.63 Reduction Pairs Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y)) 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Usable Equations: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> Usable Rules: 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Interpretation type: 0.53/0.63 Linear 0.53/0.63 ->Coefficients: 0.53/0.63 Natural Numbers 0.53/0.63 ->Dimension: 0.53/0.63 1 0.53/0.63 ->Bound: 0.53/0.63 2 0.53/0.63 ->Interpretation: 0.53/0.63 0.53/0.63 [app](X1,X2) = 0 0.53/0.63 [minus](X1,X2) = X1 0.53/0.63 [plus](X1,X2) = X1 + X2 + 2 0.53/0.63 [quot](X1,X2) = 0 0.53/0.63 [sum](X) = 0 0.53/0.63 [0] = 0 0.53/0.63 [cons](X1,X2) = 0 0.53/0.63 [nil] = 0 0.53/0.63 [s](X) = X + 2 0.53/0.63 [APP](X1,X2) = 0 0.53/0.63 [MINUS](X1,X2) = 0 0.53/0.63 [PLUS](X1,X2) = 0 0.53/0.63 [QUOT](X1,X2) = 2.X1 0.53/0.63 [SUM](X) = 0 0.53/0.63 0.53/0.63 Problem 1.4: 0.53/0.63 0.53/0.63 SCC Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 Empty 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Strongly Connected Components: 0.53/0.63 There is no strongly connected component 0.53/0.63 0.53/0.63 The problem is finite. 0.53/0.63 0.53/0.63 Problem 1.5: 0.53/0.63 0.53/0.63 Subterm Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 APP(cons(x,l),k) -> APP(l,k) 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Projection: 0.53/0.63 pi(APP) = [1] 0.53/0.63 0.53/0.63 Problem 1.5: 0.53/0.63 0.53/0.63 SCC Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 Empty 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Strongly Connected Components: 0.53/0.63 There is no strongly connected component 0.53/0.63 0.53/0.63 The problem is finite. 0.53/0.63 0.53/0.63 Problem 1.6: 0.53/0.63 0.53/0.63 Reduction Pairs Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Usable Equations: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> Usable Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Interpretation type: 0.53/0.63 Linear 0.53/0.63 ->Coefficients: 0.53/0.63 Natural Numbers 0.53/0.63 ->Dimension: 0.53/0.63 1 0.53/0.63 ->Bound: 0.53/0.63 2 0.53/0.63 ->Interpretation: 0.53/0.63 0.53/0.63 [app](X1,X2) = 2.X1 + 2.X2 + 2 0.53/0.63 [minus](X1,X2) = 0 0.53/0.63 [plus](X1,X2) = X1 + X2 + 2 0.53/0.63 [quot](X1,X2) = 0 0.53/0.63 [sum](X) = 2 0.53/0.63 [0] = 0 0.53/0.63 [cons](X1,X2) = X2 + 2 0.53/0.63 [nil] = 0 0.53/0.63 [s](X) = X + 2 0.53/0.63 [APP](X1,X2) = 0 0.53/0.63 [MINUS](X1,X2) = 0 0.53/0.63 [PLUS](X1,X2) = 0 0.53/0.63 [QUOT](X1,X2) = 0 0.53/0.63 [SUM](X) = 2.X 0.53/0.63 0.53/0.63 Problem 1.6: 0.53/0.63 0.53/0.63 SCC Processor: 0.53/0.63 -> FAxioms: 0.53/0.63 Empty 0.53/0.63 -> Pairs: 0.53/0.63 Empty 0.53/0.63 -> EAxioms: 0.53/0.63 plus(plus(x5,x6),x7) = plus(x5,plus(x6,x7)) 0.53/0.63 plus(x5,x6) = plus(x6,x5) 0.53/0.63 -> Rules: 0.53/0.63 app(cons(x,l),k) -> cons(x,app(l,k)) 0.53/0.63 app(nil,k) -> k 0.53/0.63 app(l,nil) -> l 0.53/0.63 minus(minus(x,y),z) -> minus(x,plus(y,z)) 0.53/0.63 minus(s(x),s(y)) -> minus(x,y) 0.53/0.63 minus(x,0) -> x 0.53/0.63 plus(0,y) -> y 0.53/0.63 plus(s(x),y) -> s(plus(x,y)) 0.53/0.63 quot(0,s(y)) -> 0 0.53/0.63 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) 0.53/0.63 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.53/0.63 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.53/0.63 sum(cons(x,nil)) -> cons(x,nil) 0.53/0.63 -> SRules: 0.53/0.63 Empty 0.53/0.63 ->Strongly Connected Components: 0.53/0.63 There is no strongly connected component 0.53/0.63 0.53/0.63 The problem is finite. 0.53/0.63 EOF