6.44/7.68 YES 6.44/7.68 6.44/7.68 Problem 1: 6.44/7.68 6.44/7.68 (VAR x y z) 6.44/7.68 (THEORY 6.44/7.68 (AC plus times)) 6.44/7.68 (RULES 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 ) 6.44/7.68 6.44/7.68 Problem 1: 6.44/7.68 6.44/7.68 Dependency Pairs Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) = PLUS(x4,x3) 6.44/7.68 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 I(plus(x,y)) -> I(x) 6.44/7.68 I(plus(x,y)) -> I(y) 6.44/7.68 I(plus(x,y)) -> PLUS(i(x),i(y)) 6.44/7.68 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> I(times(x,y)) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> I(times(x,y)) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 6.44/7.68 Problem 1: 6.44/7.68 6.44/7.68 SCC Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) = PLUS(x4,x3) 6.44/7.68 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 I(plus(x,y)) -> I(x) 6.44/7.68 I(plus(x,y)) -> I(y) 6.44/7.68 I(plus(x,y)) -> PLUS(i(x),i(y)) 6.44/7.68 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> I(times(x,y)) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> I(times(x,y)) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 ->Strongly Connected Components: 6.44/7.68 ->->Cycle: 6.44/7.68 ->->-> Pairs: 6.44/7.68 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 -> FAxioms: 6.44/7.68 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) -> plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) -> times(x4,x3) 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) -> PLUS(x4,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 ->->-> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 ->->Cycle: 6.44/7.68 ->->-> Pairs: 6.44/7.68 I(plus(x,y)) -> I(x) 6.44/7.68 I(plus(x,y)) -> I(y) 6.44/7.68 -> FAxioms: 6.44/7.68 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) -> plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) -> times(x4,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 ->->-> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 Empty 6.44/7.68 ->->Cycle: 6.44/7.68 ->->-> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> FAxioms: 6.44/7.68 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) -> plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) -> times(x4,x3) 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 ->->-> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 6.44/7.68 6.44/7.68 The problem is decomposed in 3 subproblems. 6.44/7.68 6.44/7.68 Problem 1.1: 6.44/7.68 6.44/7.68 Reduction Pairs Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) = PLUS(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Usable Equations: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> Usable Rules: 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 ->Interpretation type: 6.44/7.68 Linear 6.44/7.68 ->Coefficients: 6.44/7.68 Natural Numbers 6.44/7.68 ->Dimension: 6.44/7.68 1 6.44/7.68 ->Bound: 6.44/7.68 2 6.44/7.68 ->Interpretation: 6.44/7.68 6.44/7.68 [i](X) = 2 6.44/7.68 [plus](X1,X2) = X1 + X2 6.44/7.68 [times](X1,X2) = 0 6.44/7.68 [0] = 0 6.44/7.68 [I](X) = 0 6.44/7.68 [PLUS](X1,X2) = 2.X1 + 2.X2 6.44/7.68 [TIMES](X1,X2) = 0 6.44/7.68 6.44/7.68 Problem 1.1: 6.44/7.68 6.44/7.68 SCC Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) = PLUS(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 ->Strongly Connected Components: 6.44/7.68 ->->Cycle: 6.44/7.68 ->->-> Pairs: 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 -> FAxioms: 6.44/7.68 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) -> plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) -> times(x4,x3) 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) -> PLUS(x4,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 ->->-> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 6.44/7.68 Problem 1.1: 6.44/7.68 6.44/7.68 Reduction Pairs Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) = PLUS(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 PLUS(plus(x,0),x3) -> PLUS(x,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Usable Equations: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> Usable Rules: 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 ->Interpretation type: 6.44/7.68 Linear 6.44/7.68 ->Coefficients: 6.44/7.68 Natural Numbers 6.44/7.68 ->Dimension: 6.44/7.68 1 6.44/7.68 ->Bound: 6.44/7.68 2 6.44/7.68 ->Interpretation: 6.44/7.68 6.44/7.68 [i](X) = 2 6.44/7.68 [plus](X1,X2) = X1 + X2 6.44/7.68 [times](X1,X2) = 0 6.44/7.68 [0] = 2 6.44/7.68 [I](X) = 0 6.44/7.68 [PLUS](X1,X2) = 2.X1 + 2.X2 6.44/7.68 [TIMES](X1,X2) = 0 6.44/7.68 6.44/7.68 Problem 1.1: 6.44/7.68 6.44/7.68 SCC Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 6.44/7.68 PLUS(x3,x4) = PLUS(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 Empty 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 6.44/7.68 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 6.44/7.68 ->Strongly Connected Components: 6.44/7.68 There is no strongly connected component 6.44/7.68 6.44/7.68 The problem is finite. 6.44/7.68 6.44/7.68 Problem 1.2: 6.44/7.68 6.44/7.68 Subterm Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 Empty 6.44/7.68 -> Pairs: 6.44/7.68 I(plus(x,y)) -> I(x) 6.44/7.68 I(plus(x,y)) -> I(y) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 Empty 6.44/7.68 ->Projection: 6.44/7.68 pi(I) = [1] 6.44/7.68 6.44/7.68 Problem 1.2: 6.44/7.68 6.44/7.68 SCC Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 Empty 6.44/7.68 -> Pairs: 6.44/7.68 Empty 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 Empty 6.44/7.68 ->Strongly Connected Components: 6.44/7.68 There is no strongly connected component 6.44/7.68 6.44/7.68 The problem is finite. 6.44/7.68 6.44/7.68 Problem 1.3: 6.44/7.68 6.44/7.68 Reduction Pairs Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Usable Equations: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> Usable Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 ->Interpretation type: 6.44/7.68 Simple mixed 6.44/7.68 ->Coefficients: 6.44/7.68 Natural Numbers 6.44/7.68 ->Dimension: 6.44/7.68 1 6.44/7.68 ->Bound: 6.44/7.68 1 6.44/7.68 ->Interpretation: 6.44/7.68 6.44/7.68 [i](X) = X 6.44/7.68 [plus](X1,X2) = X1 + X2 + 1 6.44/7.68 [times](X1,X2) = X1.X2 + X1 + X2 6.44/7.68 [0] = 1 6.44/7.68 [I](X) = 0 6.44/7.68 [PLUS](X1,X2) = 0 6.44/7.68 [TIMES](X1,X2) = X1.X2 + X1 + X2 6.44/7.68 6.44/7.68 Problem 1.3: 6.44/7.68 6.44/7.68 SCC Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 ->Strongly Connected Components: 6.44/7.68 ->->Cycle: 6.44/7.68 ->->-> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> FAxioms: 6.44/7.68 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) -> plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) -> times(x4,x3) 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 ->->-> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 6.44/7.68 Problem 1.3: 6.44/7.68 6.44/7.68 Reduction Pairs Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Usable Equations: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> Usable Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 ->Interpretation type: 6.44/7.68 Simple mixed 6.44/7.68 ->Coefficients: 6.44/7.68 Natural Numbers 6.44/7.68 ->Dimension: 6.44/7.68 1 6.44/7.68 ->Bound: 6.44/7.68 1 6.44/7.68 ->Interpretation: 6.44/7.68 6.44/7.68 [i](X) = X 6.44/7.68 [plus](X1,X2) = X1 + X2 + 1 6.44/7.68 [times](X1,X2) = X1.X2 + X1 + X2 6.44/7.68 [0] = 1 6.44/7.68 [I](X) = 0 6.44/7.68 [PLUS](X1,X2) = 0 6.44/7.68 [TIMES](X1,X2) = X1.X2 + X1 + X2 6.44/7.68 6.44/7.68 Problem 1.3: 6.44/7.68 6.44/7.68 SCC Processor: 6.44/7.68 -> FAxioms: 6.44/7.68 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.68 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.68 -> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.68 -> EAxioms: 6.44/7.68 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.68 plus(x3,x4) = plus(x4,x3) 6.44/7.68 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.68 times(x3,x4) = times(x4,x3) 6.44/7.68 -> Rules: 6.44/7.68 i(i(x)) -> x 6.44/7.68 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.68 i(0) -> 0 6.44/7.68 plus(x,i(x)) -> 0 6.44/7.68 plus(x,0) -> x 6.44/7.68 times(x,i(y)) -> i(times(x,y)) 6.44/7.68 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.68 times(x,0) -> 0 6.44/7.68 -> SRules: 6.44/7.68 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.68 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.68 ->Strongly Connected Components: 6.44/7.68 ->->Cycle: 6.44/7.68 ->->-> Pairs: 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.68 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.68 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.68 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.68 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.68 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 TIMES(x,plus(y,z)) -> TIMES(x,y) 6.44/7.69 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 Natural Numbers 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 1 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = X 6.44/7.69 [plus](X1,X2) = X1 + X2 + 1 6.44/7.69 [times](X1,X2) = X1.X2 + X1 + X2 6.44/7.69 [0] = 1 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = X1.X2 + X1 + X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 TIMES(x,plus(y,z)) -> TIMES(x,z) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 Natural Numbers 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 1 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = X 6.44/7.69 [plus](X1,X2) = X1 + X2 + 1 6.44/7.69 [times](X1,X2) = X1.X2 + X1 + X2 6.44/7.69 [0] = 1 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = X1.X2 + X1 + X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 All rationals 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 3 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = 3/2.X + 1 6.44/7.69 [plus](X1,X2) = X1 + X2 + 3 6.44/7.69 [times](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 6.44/7.69 [0] = 3/2 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 All rationals 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 3 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = 3.X + 3 6.44/7.69 [plus](X1,X2) = X1 + X2 + 3 6.44/7.69 [times](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 6.44/7.69 [0] = 3 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = X1.X2 + X1 + X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 All rationals 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 3 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = 3/2.X + 1 6.44/7.69 [plus](X1,X2) = X1 + X2 + 3 6.44/7.69 [times](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 6.44/7.69 [0] = 3 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = X1.X2 + 2.X1 + 2.X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(times(x,0),x3) -> TIMES(0,x3) 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 All rationals 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 3 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = X 6.44/7.69 [plus](X1,X2) = X1 + X2 + 3/2 6.44/7.69 [times](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 6.44/7.69 [0] = 1 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 All rationals 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 3 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = 3.X + 2 6.44/7.69 [plus](X1,X2) = X1 + X2 + 1 6.44/7.69 [times](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 6.44/7.69 [0] = 0 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 ->->Cycle: 6.44/7.69 ->->-> Pairs: 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> FAxioms: 6.44/7.69 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) -> plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) -> times(x4,x3) 6.44/7.69 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) -> TIMES(x4,x3) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 ->->-> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 Reduction Pairs Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 TIMES(x,i(y)) -> TIMES(x,y) 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Usable Equations: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> Usable Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Interpretation type: 6.44/7.69 Simple mixed 6.44/7.69 ->Coefficients: 6.44/7.69 All rationals 6.44/7.69 ->Dimension: 6.44/7.69 1 6.44/7.69 ->Bound: 6.44/7.69 3 6.44/7.69 ->Interpretation: 6.44/7.69 6.44/7.69 [i](X) = 2.X + 1 6.44/7.69 [plus](X1,X2) = X1 + X2 + 3/2 6.44/7.69 [times](X1,X2) = 3.X1.X2 + X1 + X2 6.44/7.69 [0] = 0 6.44/7.69 [I](X) = 0 6.44/7.69 [PLUS](X1,X2) = 0 6.44/7.69 [TIMES](X1,X2) = 2.X1.X2 + 2/3.X1 + 2/3.X2 6.44/7.69 6.44/7.69 Problem 1.3: 6.44/7.69 6.44/7.69 SCC Processor: 6.44/7.69 -> FAxioms: 6.44/7.69 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 6.44/7.69 TIMES(x3,x4) = TIMES(x4,x3) 6.44/7.69 -> Pairs: 6.44/7.69 Empty 6.44/7.69 -> EAxioms: 6.44/7.69 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 6.44/7.69 plus(x3,x4) = plus(x4,x3) 6.44/7.69 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 6.44/7.69 times(x3,x4) = times(x4,x3) 6.44/7.69 -> Rules: 6.44/7.69 i(i(x)) -> x 6.44/7.69 i(plus(x,y)) -> plus(i(x),i(y)) 6.44/7.69 i(0) -> 0 6.44/7.69 plus(x,i(x)) -> 0 6.44/7.69 plus(x,0) -> x 6.44/7.69 times(x,i(y)) -> i(times(x,y)) 6.44/7.69 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 6.44/7.69 times(x,0) -> 0 6.44/7.69 -> SRules: 6.44/7.69 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 6.44/7.69 ->Strongly Connected Components: 6.44/7.69 There is no strongly connected component 6.44/7.69 6.44/7.69 The problem is finite. 6.44/7.69 EOF