7.02/7.79 YES 7.02/7.79 7.02/7.79 Problem 1: 7.02/7.79 7.02/7.79 (VAR x y z) 7.02/7.79 (THEORY 7.02/7.79 (AC plus times)) 7.02/7.79 (RULES 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 ) 7.02/7.79 7.02/7.79 Problem 1: 7.02/7.79 7.02/7.79 Dependency Pairs Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) = PLUS(x4,x3) 7.02/7.79 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 I(plus(x,y)) -> I(x) 7.02/7.79 I(plus(x,y)) -> I(y) 7.02/7.79 I(plus(x,y)) -> PLUS(i(x),i(y)) 7.02/7.79 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> I(times(x,y)) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.79 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.79 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.79 TIMES(x,i(y)) -> I(times(x,y)) 7.02/7.79 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.79 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.79 7.02/7.79 Problem 1: 7.02/7.79 7.02/7.79 SCC Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) = PLUS(x4,x3) 7.02/7.79 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 I(plus(x,y)) -> I(x) 7.02/7.79 I(plus(x,y)) -> I(y) 7.02/7.79 I(plus(x,y)) -> PLUS(i(x),i(y)) 7.02/7.79 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> I(times(x,y)) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.79 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.79 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.79 TIMES(x,i(y)) -> I(times(x,y)) 7.02/7.79 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.79 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.79 ->Strongly Connected Components: 7.02/7.79 ->->Cycle: 7.02/7.79 ->->-> Pairs: 7.02/7.79 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 -> FAxioms: 7.02/7.79 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) -> plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) -> times(x4,x3) 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) -> PLUS(x4,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 ->->-> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 ->->Cycle: 7.02/7.79 ->->-> Pairs: 7.02/7.79 I(plus(x,y)) -> I(x) 7.02/7.79 I(plus(x,y)) -> I(y) 7.02/7.79 -> FAxioms: 7.02/7.79 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) -> plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) -> times(x4,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 ->->-> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 Empty 7.02/7.79 ->->Cycle: 7.02/7.79 ->->-> Pairs: 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.79 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.79 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.79 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.79 -> FAxioms: 7.02/7.79 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) -> plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) -> times(x4,x3) 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 ->->-> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.79 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.79 7.02/7.79 7.02/7.79 The problem is decomposed in 3 subproblems. 7.02/7.79 7.02/7.79 Problem 1.1: 7.02/7.79 7.02/7.79 Reduction Pairs Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) = PLUS(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 PLUS(plus(x,i(x)),x3) -> PLUS(0,x3) 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Usable Equations: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> Usable Rules: 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 ->Interpretation type: 7.02/7.79 Linear 7.02/7.79 ->Coefficients: 7.02/7.79 Natural Numbers 7.02/7.79 ->Dimension: 7.02/7.79 1 7.02/7.79 ->Bound: 7.02/7.79 2 7.02/7.79 ->Interpretation: 7.02/7.79 7.02/7.79 [i](X) = 2 7.02/7.79 [plus](X1,X2) = X1 + X2 7.02/7.79 [times](X1,X2) = 0 7.02/7.79 [0] = 0 7.02/7.79 [1] = 0 7.02/7.79 [I](X) = 0 7.02/7.79 [PLUS](X1,X2) = 2.X1 + 2.X2 7.02/7.79 [TIMES](X1,X2) = 0 7.02/7.79 7.02/7.79 Problem 1.1: 7.02/7.79 7.02/7.79 SCC Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) = PLUS(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 ->Strongly Connected Components: 7.02/7.79 ->->Cycle: 7.02/7.79 ->->-> Pairs: 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 -> FAxioms: 7.02/7.79 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) -> plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) -> times(x4,x3) 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) -> PLUS(x4,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 ->->-> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 7.02/7.79 Problem 1.1: 7.02/7.79 7.02/7.79 Reduction Pairs Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) = PLUS(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 PLUS(plus(x,0),x3) -> PLUS(x,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Usable Equations: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> Usable Rules: 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 ->Interpretation type: 7.02/7.79 Linear 7.02/7.79 ->Coefficients: 7.02/7.79 Natural Numbers 7.02/7.79 ->Dimension: 7.02/7.79 1 7.02/7.79 ->Bound: 7.02/7.79 2 7.02/7.79 ->Interpretation: 7.02/7.79 7.02/7.79 [i](X) = 2 7.02/7.79 [plus](X1,X2) = X1 + X2 7.02/7.79 [times](X1,X2) = 0 7.02/7.79 [0] = 2 7.02/7.79 [1] = 0 7.02/7.79 [I](X) = 0 7.02/7.79 [PLUS](X1,X2) = 2.X1 + 2.X2 7.02/7.79 [TIMES](X1,X2) = 0 7.02/7.79 7.02/7.79 Problem 1.1: 7.02/7.79 7.02/7.79 SCC Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 7.02/7.79 PLUS(x3,x4) = PLUS(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 Empty 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 7.02/7.79 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 7.02/7.79 ->Strongly Connected Components: 7.02/7.79 There is no strongly connected component 7.02/7.79 7.02/7.79 The problem is finite. 7.02/7.79 7.02/7.79 Problem 1.2: 7.02/7.79 7.02/7.79 Subterm Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 Empty 7.02/7.79 -> Pairs: 7.02/7.79 I(plus(x,y)) -> I(x) 7.02/7.79 I(plus(x,y)) -> I(y) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 Empty 7.02/7.79 ->Projection: 7.02/7.79 pi(I) = [1] 7.02/7.79 7.02/7.79 Problem 1.2: 7.02/7.79 7.02/7.79 SCC Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 Empty 7.02/7.79 -> Pairs: 7.02/7.79 Empty 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 Empty 7.02/7.79 ->Strongly Connected Components: 7.02/7.79 There is no strongly connected component 7.02/7.79 7.02/7.79 The problem is finite. 7.02/7.79 7.02/7.79 Problem 1.3: 7.02/7.79 7.02/7.79 Reduction Pairs Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.79 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.79 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.79 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Usable Equations: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> Usable Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.79 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.79 ->Interpretation type: 7.02/7.79 Simple mixed 7.02/7.79 ->Coefficients: 7.02/7.79 Natural Numbers 7.02/7.79 ->Dimension: 7.02/7.79 1 7.02/7.79 ->Bound: 7.02/7.79 1 7.02/7.79 ->Interpretation: 7.02/7.79 7.02/7.79 [i](X) = X 7.02/7.79 [plus](X1,X2) = X1 + X2 + 1 7.02/7.79 [times](X1,X2) = X1.X2 + X1 + X2 7.02/7.79 [0] = 1 7.02/7.79 [1] = 1 7.02/7.79 [I](X) = 0 7.02/7.79 [PLUS](X1,X2) = 0 7.02/7.79 [TIMES](X1,X2) = X1.X2 + X1 + X2 7.02/7.79 7.02/7.79 Problem 1.3: 7.02/7.79 7.02/7.79 SCC Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.79 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.79 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.79 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 -> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.79 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.79 ->Strongly Connected Components: 7.02/7.79 ->->Cycle: 7.02/7.79 ->->-> Pairs: 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.79 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.79 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.79 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.79 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.79 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.79 -> FAxioms: 7.02/7.79 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) -> plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) -> times(x4,x3) 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.79 -> EAxioms: 7.02/7.79 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.79 plus(x3,x4) = plus(x4,x3) 7.02/7.79 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.79 times(x3,x4) = times(x4,x3) 7.02/7.79 ->->-> Rules: 7.02/7.79 i(i(x)) -> x 7.02/7.79 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.79 i(0) -> 0 7.02/7.79 plus(x,i(x)) -> 0 7.02/7.79 plus(x,0) -> x 7.02/7.79 times(x,i(y)) -> i(times(x,y)) 7.02/7.79 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.79 times(x,0) -> 0 7.02/7.79 times(x,1) -> x 7.02/7.79 -> SRules: 7.02/7.79 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.79 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.79 7.02/7.79 Problem 1.3: 7.02/7.79 7.02/7.79 Reduction Pairs Processor: 7.02/7.79 -> FAxioms: 7.02/7.79 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.79 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.79 -> Pairs: 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.79 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 Natural Numbers 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 1 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = X 7.02/7.80 [plus](X1,X2) = X1 + X2 + 1 7.02/7.80 [times](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 [0] = 1 7.02/7.80 [1] = 1 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(times(x,1),x3) -> TIMES(x,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 Natural Numbers 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 1 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = X 7.02/7.80 [plus](X1,X2) = X1 + X2 + 1 7.02/7.80 [times](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 [0] = 1 7.02/7.80 [1] = 1 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 Natural Numbers 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 1 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = X 7.02/7.80 [plus](X1,X2) = X1 + X2 + 1 7.02/7.80 [times](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 [0] = 1 7.02/7.80 [1] = 1 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 TIMES(x,plus(y,z)) -> TIMES(x,z) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 Natural Numbers 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 1 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = X 7.02/7.80 [plus](X1,X2) = X1 + X2 + 1 7.02/7.80 [times](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 [0] = 1 7.02/7.80 [1] = 1 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = X1.X2 + X1 + X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(i(times(x,y)),x3) 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 All rationals 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 3 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = 3.X + 3 7.02/7.80 [plus](X1,X2) = X1 + X2 + 3/2 7.02/7.80 [times](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2 7.02/7.80 [0] = 0 7.02/7.80 [1] = 3/2 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,i(y)),x3) -> TIMES(x,y) 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 All rationals 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 3 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = 3.X + 3 7.02/7.80 [plus](X1,X2) = X1 + X2 + 3/2 7.02/7.80 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 7.02/7.80 [0] = 3/2 7.02/7.80 [1] = 3/2 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = 2/3.X1.X2 + X1 + X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 All rationals 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 3 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = 3.X + 3 7.02/7.80 [plus](X1,X2) = X1 + X2 + 2 7.02/7.80 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 7.02/7.80 [0] = 0 7.02/7.80 [1] = 1 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = 1/3.X1.X2 + 1/2.X1 + 1/2.X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(times(x,0),x3) -> TIMES(0,x3) 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 All rationals 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 3 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = 3.X + 3 7.02/7.80 [plus](X1,X2) = X1 + X2 + 3/2 7.02/7.80 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 7.02/7.80 [0] = 2 7.02/7.80 [1] = 2 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = 2/3.X1.X2 + X1 + X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 All rationals 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 3 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = X 7.02/7.80 [plus](X1,X2) = X1 + X2 + 3 7.02/7.80 [times](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 7.02/7.80 [0] = 2 7.02/7.80 [1] = 3/2 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 ->->Cycle: 7.02/7.80 ->->-> Pairs: 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> FAxioms: 7.02/7.80 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) -> plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) -> times(x4,x3) 7.02/7.80 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) -> TIMES(x4,x3) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 ->->-> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 Reduction Pairs Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 TIMES(x,i(y)) -> TIMES(x,y) 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Usable Equations: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> Usable Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Interpretation type: 7.02/7.80 Simple mixed 7.02/7.80 ->Coefficients: 7.02/7.80 All rationals 7.02/7.80 ->Dimension: 7.02/7.80 1 7.02/7.80 ->Bound: 7.02/7.80 3 7.02/7.80 ->Interpretation: 7.02/7.80 7.02/7.80 [i](X) = 2.X + 1 7.02/7.80 [plus](X1,X2) = X1 + X2 + 1 7.02/7.80 [times](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 7.02/7.80 [0] = 0 7.02/7.80 [1] = 3 7.02/7.80 [I](X) = 0 7.02/7.80 [PLUS](X1,X2) = 0 7.02/7.80 [TIMES](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2 7.02/7.80 7.02/7.80 Problem 1.3: 7.02/7.80 7.02/7.80 SCC Processor: 7.02/7.80 -> FAxioms: 7.02/7.80 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 7.02/7.80 TIMES(x3,x4) = TIMES(x4,x3) 7.02/7.80 -> Pairs: 7.02/7.80 Empty 7.02/7.80 -> EAxioms: 7.02/7.80 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 7.02/7.80 plus(x3,x4) = plus(x4,x3) 7.02/7.80 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 7.02/7.80 times(x3,x4) = times(x4,x3) 7.02/7.80 -> Rules: 7.02/7.80 i(i(x)) -> x 7.02/7.80 i(plus(x,y)) -> plus(i(x),i(y)) 7.02/7.80 i(0) -> 0 7.02/7.80 plus(x,i(x)) -> 0 7.02/7.80 plus(x,0) -> x 7.02/7.80 times(x,i(y)) -> i(times(x,y)) 7.02/7.80 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 7.02/7.80 times(x,0) -> 0 7.02/7.80 times(x,1) -> x 7.02/7.80 -> SRules: 7.02/7.80 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 7.02/7.80 ->Strongly Connected Components: 7.02/7.80 There is no strongly connected component 7.02/7.80 7.02/7.80 The problem is finite. 7.02/7.80 EOF