1.40/1.55 YES 1.40/1.55 1.40/1.55 Problem 1: 1.40/1.55 1.40/1.55 (VAR x y) 1.40/1.55 (THEORY 1.40/1.55 (AC plus times)) 1.40/1.55 (RULES 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 ) 1.40/1.55 1.40/1.55 Problem 1: 1.40/1.55 1.40/1.55 Dependency Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.55 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(0,x),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> PLUS(times(x,y),x) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(plus(times(x,y),x),x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.55 TIMES(x,s(y)) -> PLUS(times(x,y),x) 1.40/1.55 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.55 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.55 1.40/1.55 Problem 1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.55 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(0,x),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> PLUS(times(x,y),x) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(plus(times(x,y),x),x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.55 TIMES(x,s(y)) -> PLUS(times(x,y),x) 1.40/1.55 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.55 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(0,x),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(plus(times(x,y),x),x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.55 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 1.40/1.55 TIMES(x2,x3) -> TIMES(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.55 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.55 1.40/1.55 1.40/1.55 The problem is decomposed in 2 subproblems. 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(0,x),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 1 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 2 1.40/1.55 [s](X) = X 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(x,y) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = X + 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,0),x2) -> PLUS(x,x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 2 1.40/1.55 [s](X) = X + 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(x,y) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 1 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = X + 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = X + 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(s(x),y) -> PLUS(x,y) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = X + 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(x,s(y)) -> PLUS(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = X + 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(s(x),y),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 ->->Cycle: 1.40/1.55 ->->-> Pairs: 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 -> FAxioms: 1.40/1.55 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) -> plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) -> times(x3,x2) 1.40/1.55 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) -> PLUS(x3,x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 ->->-> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 PLUS(plus(x,s(y)),x2) -> PLUS(s(plus(x,y)),x2) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Linear 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 2 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 + 2 1.40/1.55 [times](X1,X2) = 0 1.40/1.55 [0] = 0 1.40/1.55 [s](X) = 2 1.40/1.55 [PLUS](X1,X2) = 2.X1 + 2.X2 1.40/1.55 [TIMES](X1,X2) = 0 1.40/1.55 1.40/1.55 Problem 1.1: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 1.40/1.55 PLUS(x2,x3) = PLUS(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 Empty 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 1.40/1.55 ->Strongly Connected Components: 1.40/1.55 There is no strongly connected component 1.40/1.55 1.40/1.55 The problem is finite. 1.40/1.55 1.40/1.55 Problem 1.2: 1.40/1.55 1.40/1.55 Reduction Pairs Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.55 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.55 -> Pairs: 1.40/1.55 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(plus(times(x,y),x),x2) 1.40/1.55 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.55 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.55 -> EAxioms: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Usable Equations: 1.40/1.55 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.55 plus(x2,x3) = plus(x3,x2) 1.40/1.55 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.55 times(x2,x3) = times(x3,x2) 1.40/1.55 -> Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> Usable Rules: 1.40/1.55 plus(0,x) -> x 1.40/1.55 plus(s(x),y) -> s(plus(x,y)) 1.40/1.55 plus(x,0) -> x 1.40/1.55 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.55 times(x,0) -> 0 1.40/1.55 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.55 -> SRules: 1.40/1.55 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.55 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.55 ->Interpretation type: 1.40/1.55 Simple mixed 1.40/1.55 ->Coefficients: 1.40/1.55 Natural Numbers 1.40/1.55 ->Dimension: 1.40/1.55 1 1.40/1.55 ->Bound: 1.40/1.55 1 1.40/1.55 ->Interpretation: 1.40/1.55 1.40/1.55 [plus](X1,X2) = X1 + X2 1.40/1.55 [times](X1,X2) = X1.X2 + X1 + X2 1.40/1.55 [0] = 1 1.40/1.55 [s](X) = X + 1 1.40/1.55 [PLUS](X1,X2) = 0 1.40/1.55 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.40/1.55 1.40/1.55 Problem 1.2: 1.40/1.55 1.40/1.55 SCC Processor: 1.40/1.55 -> FAxioms: 1.40/1.55 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.55 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.56 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Strongly Connected Components: 1.40/1.56 ->->Cycle: 1.40/1.56 ->->-> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.56 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.56 -> FAxioms: 1.40/1.56 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) -> plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) -> times(x3,x2) 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) -> TIMES(x3,x2) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 ->->-> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 1.40/1.56 Problem 1.2: 1.40/1.56 1.40/1.56 Reduction Pairs Processor: 1.40/1.56 -> FAxioms: 1.40/1.56 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 TIMES(times(x,s(y)),x2) -> TIMES(x,y) 1.40/1.56 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Usable Equations: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> Usable Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Interpretation type: 1.40/1.56 Simple mixed 1.40/1.56 ->Coefficients: 1.40/1.56 Natural Numbers 1.40/1.56 ->Dimension: 1.40/1.56 1 1.40/1.56 ->Bound: 1.40/1.56 1 1.40/1.56 ->Interpretation: 1.40/1.56 1.40/1.56 [plus](X1,X2) = X1 + X2 + 1 1.40/1.56 [times](X1,X2) = X1.X2 + X1 + X2 1.40/1.56 [0] = 1 1.40/1.56 [s](X) = X + 1 1.40/1.56 [PLUS](X1,X2) = 0 1.40/1.56 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.40/1.56 1.40/1.56 Problem 1.2: 1.40/1.56 1.40/1.56 SCC Processor: 1.40/1.56 -> FAxioms: 1.40/1.56 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Strongly Connected Components: 1.40/1.56 ->->Cycle: 1.40/1.56 ->->-> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.56 -> FAxioms: 1.40/1.56 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) -> plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) -> times(x3,x2) 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) -> TIMES(x3,x2) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 ->->-> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 1.40/1.56 Problem 1.2: 1.40/1.56 1.40/1.56 Reduction Pairs Processor: 1.40/1.56 -> FAxioms: 1.40/1.56 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 TIMES(x,s(y)) -> TIMES(x,y) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Usable Equations: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> Usable Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Interpretation type: 1.40/1.56 Simple mixed 1.40/1.56 ->Coefficients: 1.40/1.56 Natural Numbers 1.40/1.56 ->Dimension: 1.40/1.56 1 1.40/1.56 ->Bound: 1.40/1.56 1 1.40/1.56 ->Interpretation: 1.40/1.56 1.40/1.56 [plus](X1,X2) = X1 + X2 + 1 1.40/1.56 [times](X1,X2) = X1.X2 + X1 + X2 1.40/1.56 [0] = 1 1.40/1.56 [s](X) = X + 1 1.40/1.56 [PLUS](X1,X2) = 0 1.40/1.56 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.40/1.56 1.40/1.56 Problem 1.2: 1.40/1.56 1.40/1.56 SCC Processor: 1.40/1.56 -> FAxioms: 1.40/1.56 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Strongly Connected Components: 1.40/1.56 ->->Cycle: 1.40/1.56 ->->-> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 -> FAxioms: 1.40/1.56 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) -> plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) -> times(x3,x2) 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) -> TIMES(x3,x2) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 ->->-> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 1.40/1.56 Problem 1.2: 1.40/1.56 1.40/1.56 Reduction Pairs Processor: 1.40/1.56 -> FAxioms: 1.40/1.56 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 TIMES(times(x,0),x2) -> TIMES(0,x2) 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Usable Equations: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> Usable Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Interpretation type: 1.40/1.56 Simple mixed 1.40/1.56 ->Coefficients: 1.40/1.56 All rationals 1.40/1.56 ->Dimension: 1.40/1.56 1 1.40/1.56 ->Bound: 1.40/1.56 3 1.40/1.56 ->Interpretation: 1.40/1.56 1.40/1.56 [plus](X1,X2) = X1 + X2 + 3 1.40/1.56 [times](X1,X2) = 1/2.X1.X2 + 3/2.X1 + 3/2.X2 + 3/2 1.40/1.56 [0] = 3 1.40/1.56 [s](X) = X + 3 1.40/1.56 [PLUS](X1,X2) = 0 1.40/1.56 [TIMES](X1,X2) = 1/3.X1.X2 + X1 + X2 1.40/1.56 1.40/1.56 Problem 1.2: 1.40/1.56 1.40/1.56 SCC Processor: 1.40/1.56 -> FAxioms: 1.40/1.56 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 1.40/1.56 TIMES(x2,x3) = TIMES(x3,x2) 1.40/1.56 -> Pairs: 1.40/1.56 Empty 1.40/1.56 -> EAxioms: 1.40/1.56 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 1.40/1.56 plus(x2,x3) = plus(x3,x2) 1.40/1.56 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 1.40/1.56 times(x2,x3) = times(x3,x2) 1.40/1.56 -> Rules: 1.40/1.56 plus(0,x) -> x 1.40/1.56 plus(s(x),y) -> s(plus(x,y)) 1.40/1.56 plus(x,0) -> x 1.40/1.56 plus(x,s(y)) -> s(plus(x,y)) 1.40/1.56 times(x,0) -> 0 1.40/1.56 times(x,s(y)) -> plus(times(x,y),x) 1.40/1.56 -> SRules: 1.40/1.56 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 1.40/1.56 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 1.40/1.56 ->Strongly Connected Components: 1.40/1.56 There is no strongly connected component 1.40/1.56 1.40/1.56 The problem is finite. 1.40/1.56 EOF