YES Problem 1: (VAR x y z) (THEORY (AC plus times)) (RULES i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x ) Problem 1: Dependency Pairs Processor: -> FAxioms: PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) PLUS(x3,x4) = PLUS(x4,x3) TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: I(plus(x,y)) -> I(x) I(plus(x,y)) -> I(y) I(plus(x,y)) -> PLUS(i(x),i(y)) PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) PLUS(plus(x,0),x3) -> PLUS(x,x3) TIMES(times(x,i(y)),x3) -> I(times(x,y)) TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> I(times(x,y)) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1: SCC Processor: -> FAxioms: PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) PLUS(x3,x4) = PLUS(x4,x3) TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: I(plus(x,y)) -> I(x) I(plus(x,y)) -> I(y) I(plus(x,y)) -> PLUS(i(x),i(y)) PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) PLUS(plus(x,0),x3) -> PLUS(x,x3) TIMES(times(x,i(y)),x3) -> I(times(x,y)) TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> I(times(x,y)) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) PLUS(plus(x,0),x3) -> PLUS(x,x3) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) PLUS(x3,x4) -> PLUS(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) ->->Cycle: ->->-> Pairs: I(plus(x,y)) -> I(x) I(plus(x,y)) -> I(y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: Empty ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) The problem is decomposed in 3 subproblems. Problem 1.1: Reduction Pairs Processor: -> FAxioms: PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) PLUS(x3,x4) = PLUS(x4,x3) -> Pairs: PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) PLUS(plus(x,0),x3) -> PLUS(x,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: plus(x,i(x)) -> 0 plus(x,0) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [i](X) = 2 [plus](X1,X2) = X1 + X2 [times](X1,X2) = 0 [0] = 0 [1] = 0 [I](X) = 0 [PLUS](X1,X2) = 2.X1 + 2.X2 [TIMES](X1,X2) = 0 Problem 1.1: SCC Processor: -> FAxioms: PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) PLUS(x3,x4) = PLUS(x4,x3) -> Pairs: PLUS(plus(x,0),x3) -> PLUS(x,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(plus(x,0),x3) -> PLUS(x,x3) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) PLUS(x3,x4) -> PLUS(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) Problem 1.1: Reduction Pairs Processor: -> FAxioms: PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) PLUS(x3,x4) = PLUS(x4,x3) -> Pairs: PLUS(plus(x,0),x3) -> PLUS(x,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: plus(x,i(x)) -> 0 plus(x,0) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [i](X) = 2 [plus](X1,X2) = X1 + X2 [times](X1,X2) = 0 [0] = 2 [1] = 0 [I](X) = 0 [PLUS](X1,X2) = 2.X1 + 2.X2 [TIMES](X1,X2) = 0 Problem 1.1: SCC Processor: -> FAxioms: PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) PLUS(x3,x4) = PLUS(x4,x3) -> Pairs: Empty -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> FAxioms: Empty -> Pairs: I(plus(x,y)) -> I(x) I(plus(x,y)) -> I(y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: Empty ->Projection: pi(I) = [1] Problem 1.2: SCC Processor: -> FAxioms: Empty -> Pairs: Empty -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: Empty ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 1 ->Interpretation: [i](X) = X [plus](X1,X2) = X1 + X2 + 1 [times](X1,X2) = X1.X2 + X1 + X2 [0] = 1 [1] = 1 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 1 ->Interpretation: [i](X) = X [plus](X1,X2) = X1 + X2 + 1 [times](X1,X2) = X1.X2 + X1 + X2 [0] = 1 [1] = 1 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(times(x,1),x3) -> TIMES(x,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 1 ->Interpretation: [i](X) = X [plus](X1,X2) = X1 + X2 + 1 [times](X1,X2) = X1.X2 + X1 + X2 [0] = 1 [1] = 1 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 1 ->Interpretation: [i](X) = X [plus](X1,X2) = X1 + X2 + 1 [times](X1,X2) = X1.X2 + X1 + X2 [0] = 1 [1] = 1 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) TIMES(x,plus(y,z)) -> TIMES(x,z) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 1 ->Interpretation: [i](X) = X [plus](X1,X2) = X1 + X2 + 1 [times](X1,X2) = X1.X2 + X1 + X2 [0] = 1 [1] = 1 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 3 ->Interpretation: [i](X) = 3.X + 3 [plus](X1,X2) = X1 + X2 + 3/2 [times](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2 [0] = 0 [1] = 3/2 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,i(y)),x3) -> TIMES(x,y) TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 3 ->Interpretation: [i](X) = 3.X + 3 [plus](X1,X2) = X1 + X2 + 3/2 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 [0] = 3/2 [1] = 3/2 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 2/3.X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 3 ->Interpretation: [i](X) = 3.X + 3 [plus](X1,X2) = X1 + X2 + 2 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 [0] = 0 [1] = 1 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 1/3.X1.X2 + 1/2.X1 + 1/2.X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(times(x,0),x3) -> TIMES(0,x3) TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 3 ->Interpretation: [i](X) = 3.X + 3 [plus](X1,X2) = X1 + X2 + 3/2 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 [0] = 2 [1] = 2 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 2/3.X1.X2 + X1 + X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(x,i(y)) -> TIMES(x,y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(times(x3,x4),x5) -> TIMES(x3,x4) TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 3 ->Interpretation: [i](X) = X [plus](X1,X2) = X1 + X2 + 3 [times](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 [0] = 2 [1] = 3/2 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TIMES(x,i(y)) -> TIMES(x,y) -> FAxioms: plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) plus(x3,x4) -> plus(x4,x3) times(times(x3,x4),x5) -> times(x3,times(x4,x5)) times(x3,x4) -> times(x4,x3) TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) TIMES(x3,x4) -> TIMES(x4,x3) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) ->->-> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) Problem 1.3: Reduction Pairs Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: TIMES(x,i(y)) -> TIMES(x,y) -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Usable Equations: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> Usable Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 3 ->Interpretation: [i](X) = 2.X + 1 [plus](X1,X2) = X1 + X2 + 1 [times](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [0] = 0 [1] = 3 [I](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2 Problem 1.3: SCC Processor: -> FAxioms: TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) TIMES(x3,x4) = TIMES(x4,x3) -> Pairs: Empty -> EAxioms: plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) plus(x3,x4) = plus(x4,x3) times(times(x3,x4),x5) = times(x3,times(x4,x5)) times(x3,x4) = times(x4,x3) -> Rules: i(i(x)) -> x i(plus(x,y)) -> plus(i(x),i(y)) i(0) -> 0 plus(x,i(x)) -> 0 plus(x,0) -> x times(x,i(y)) -> i(times(x,y)) times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) times(x,0) -> 0 times(x,1) -> x -> SRules: TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) ->Strongly Connected Components: There is no strongly connected component The problem is finite.