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