4.75/4.89 YES 4.75/4.89 4.75/4.89 Problem 1: 4.75/4.89 4.75/4.89 (VAR x y) 4.75/4.89 (THEORY 4.75/4.89 (AC plus times)) 4.75/4.89 (RULES 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 ) 4.75/4.89 4.75/4.89 Problem 1: 4.75/4.89 4.75/4.89 Dependency Pairs Processor: 4.75/4.89 -> FAxioms: 4.75/4.89 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.89 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.89 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.89 -> Pairs: 4.75/4.89 PLUS(0(x),0(y)) -> 0#(plus(x,y)) 4.75/4.89 PLUS(0(x),0(y)) -> PLUS(x,y) 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y)) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> 0#(1(plus(plus(x,y),S))) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> 0#(1(plus(plus(x,y),S))) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 TIMES(times(x,0(y)),x2) -> 0#(times(x,y)) 4.75/4.89 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.89 TIMES(times(x,0(y)),x2) -> TIMES(x,y) 4.75/4.89 TIMES(times(x,1(y)),x2) -> 0#(times(x,y)) 4.75/4.89 TIMES(times(x,1(y)),x2) -> PLUS(x,0(times(x,y))) 4.75/4.89 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.89 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.89 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.89 TIMES(x,0(y)) -> 0#(times(x,y)) 4.75/4.89 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.89 TIMES(x,1(y)) -> 0#(times(x,y)) 4.75/4.89 TIMES(x,1(y)) -> PLUS(x,0(times(x,y))) 4.75/4.89 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 -> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.89 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.89 4.75/4.89 Problem 1: 4.75/4.89 4.75/4.89 SCC Processor: 4.75/4.89 -> FAxioms: 4.75/4.89 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.89 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.89 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.89 -> Pairs: 4.75/4.89 PLUS(0(x),0(y)) -> 0#(plus(x,y)) 4.75/4.89 PLUS(0(x),0(y)) -> PLUS(x,y) 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y)) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> 0#(1(plus(plus(x,y),S))) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> 0#(1(plus(plus(x,y),S))) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 TIMES(times(x,0(y)),x2) -> 0#(times(x,y)) 4.75/4.89 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.89 TIMES(times(x,0(y)),x2) -> TIMES(x,y) 4.75/4.89 TIMES(times(x,1(y)),x2) -> 0#(times(x,y)) 4.75/4.89 TIMES(times(x,1(y)),x2) -> PLUS(x,0(times(x,y))) 4.75/4.89 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.89 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.89 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.89 TIMES(x,0(y)) -> 0#(times(x,y)) 4.75/4.89 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.89 TIMES(x,1(y)) -> 0#(times(x,y)) 4.75/4.89 TIMES(x,1(y)) -> PLUS(x,0(times(x,y))) 4.75/4.89 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 -> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.89 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.89 ->Strongly Connected Components: 4.75/4.89 ->->Cycle: 4.75/4.89 ->->-> Pairs: 4.75/4.89 PLUS(0(x),0(y)) -> PLUS(x,y) 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> FAxioms: 4.75/4.89 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) -> plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) -> times(x3,x2) 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 ->->-> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 ->->Cycle: 4.75/4.89 ->->-> Pairs: 4.75/4.89 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.89 TIMES(times(x,0(y)),x2) -> TIMES(x,y) 4.75/4.89 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.89 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.89 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.89 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.89 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.89 -> FAxioms: 4.75/4.89 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) -> plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) -> times(x3,x2) 4.75/4.89 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.89 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 ->->-> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.89 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.89 4.75/4.89 4.75/4.89 The problem is decomposed in 2 subproblems. 4.75/4.89 4.75/4.89 Problem 1.1: 4.75/4.89 4.75/4.89 Reduction Pairs Processor: 4.75/4.89 -> FAxioms: 4.75/4.89 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.89 -> Pairs: 4.75/4.89 PLUS(0(x),0(y)) -> PLUS(x,y) 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 -> Usable Equations: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 -> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> Usable Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 ->Interpretation type: 4.75/4.89 Linear 4.75/4.89 ->Coefficients: 4.75/4.89 Natural Numbers 4.75/4.89 ->Dimension: 4.75/4.89 1 4.75/4.89 ->Bound: 4.75/4.89 2 4.75/4.89 ->Interpretation: 4.75/4.89 4.75/4.89 [0](X) = X + 1 4.75/4.89 [plus](X1,X2) = X1 + X2 + 1 4.75/4.89 [times](X1,X2) = 0 4.75/4.89 [1](X) = X + 2 4.75/4.89 [S] = 0 4.75/4.89 [0#](X) = 0 4.75/4.89 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.89 [TIMES](X1,X2) = 0 4.75/4.89 4.75/4.89 Problem 1.1: 4.75/4.89 4.75/4.89 SCC Processor: 4.75/4.89 -> FAxioms: 4.75/4.89 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.89 -> Pairs: 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 -> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 ->Strongly Connected Components: 4.75/4.89 ->->Cycle: 4.75/4.89 ->->-> Pairs: 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> FAxioms: 4.75/4.89 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) -> plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) -> times(x3,x2) 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 ->->-> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 4.75/4.89 Problem 1.1: 4.75/4.89 4.75/4.89 Reduction Pairs Processor: 4.75/4.89 -> FAxioms: 4.75/4.89 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.89 -> Pairs: 4.75/4.89 PLUS(0(x),1(y)) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 -> Usable Equations: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 -> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> Usable Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 ->Interpretation type: 4.75/4.89 Linear 4.75/4.89 ->Coefficients: 4.75/4.89 Natural Numbers 4.75/4.89 ->Dimension: 4.75/4.89 1 4.75/4.89 ->Bound: 4.75/4.89 2 4.75/4.89 ->Interpretation: 4.75/4.89 4.75/4.89 [0](X) = X 4.75/4.89 [plus](X1,X2) = X1 + X2 + 2 4.75/4.89 [times](X1,X2) = 0 4.75/4.89 [1](X) = X + 2 4.75/4.89 [S] = 0 4.75/4.89 [0#](X) = 0 4.75/4.89 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.89 [TIMES](X1,X2) = 0 4.75/4.89 4.75/4.89 Problem 1.1: 4.75/4.89 4.75/4.89 SCC Processor: 4.75/4.89 -> FAxioms: 4.75/4.89 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.89 -> Pairs: 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 -> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.89 ->Strongly Connected Components: 4.75/4.89 ->->Cycle: 4.75/4.89 ->->-> Pairs: 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.89 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.89 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.89 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.89 -> FAxioms: 4.75/4.89 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) -> plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) -> times(x3,x2) 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.89 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.89 -> EAxioms: 4.75/4.89 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.89 plus(x2,x3) = plus(x3,x2) 4.75/4.89 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.89 times(x2,x3) = times(x3,x2) 4.75/4.89 ->->-> Rules: 4.75/4.89 0(S) -> S 4.75/4.89 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.89 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.89 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.89 plus(x,S) -> x 4.75/4.89 times(x,0(y)) -> 0(times(x,y)) 4.75/4.89 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.89 times(x,S) -> S 4.75/4.89 -> SRules: 4.75/4.89 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.89 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X + 1 4.75/4.90 [plus](X1,X2) = X1 + X2 + 1 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X + 1 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 1 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(1(plus(plus(x,y),S))),x2) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 1 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 2 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y) 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(plus(x,S),x2) -> PLUS(x,x2) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 2 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(plus(x,y),S) 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 2 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) -> PLUS(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 PLUS(1(x),1(y)) -> PLUS(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 1 4.75/4.90 [times](X1,X2) = 0 4.75/4.90 [1](X) = X + 1 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 2.X1 + 2.X2 4.75/4.90 [TIMES](X1,X2) = 0 4.75/4.90 4.75/4.90 Problem 1.1: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 4.75/4.90 PLUS(x2,x3) = PLUS(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 Empty 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 There is no strongly connected component 4.75/4.90 4.75/4.90 The problem is finite. 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Simple mixed 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 1 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X + 1 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 [1](X) = X + 1 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(plus(x,0(times(x,y))),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Simple mixed 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 1 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 [1](X) = X + 1 4.75/4.90 [S] = 1 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,1(y)),x2) -> TIMES(x,y) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Simple mixed 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 1 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X + 1 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 [1](X) = X + 1 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,0(y)) -> TIMES(x,y) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Simple mixed 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 1 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X + 1 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 [1](X) = X + 1 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 TIMES(x,1(y)) -> TIMES(x,y) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Simple mixed 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 1 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = X 4.75/4.90 [plus](X1,X2) = X1 + X2 + 1 4.75/4.90 [times](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 [1](X) = X + 1 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,0(y)),x2) -> TIMES(0(times(x,y)),x2) 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = 0 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = X1 + X2 + 1 4.75/4.90 [1](X) = 1 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = X1 + X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 ->->Cycle: 4.75/4.90 ->->-> Pairs: 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 -> FAxioms: 4.75/4.90 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) -> plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) -> times(x3,x2) 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) -> TIMES(x3,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 ->->-> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 Reduction Pairs Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 TIMES(times(x,S),x2) -> TIMES(S,x2) 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Usable Equations: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> Usable Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Interpretation type: 4.75/4.90 Linear 4.75/4.90 ->Coefficients: 4.75/4.90 Natural Numbers 4.75/4.90 ->Dimension: 4.75/4.90 1 4.75/4.90 ->Bound: 4.75/4.90 2 4.75/4.90 ->Interpretation: 4.75/4.90 4.75/4.90 [0](X) = 2 4.75/4.90 [plus](X1,X2) = X1 + X2 4.75/4.90 [times](X1,X2) = X1 + X2 + 2 4.75/4.90 [1](X) = 2 4.75/4.90 [S] = 0 4.75/4.90 [0#](X) = 0 4.75/4.90 [PLUS](X1,X2) = 0 4.75/4.90 [TIMES](X1,X2) = 2.X1 + 2.X2 4.75/4.90 4.75/4.90 Problem 1.2: 4.75/4.90 4.75/4.90 SCC Processor: 4.75/4.90 -> FAxioms: 4.75/4.90 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 4.75/4.90 TIMES(x2,x3) = TIMES(x3,x2) 4.75/4.90 -> Pairs: 4.75/4.90 Empty 4.75/4.90 -> EAxioms: 4.75/4.90 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 4.75/4.90 plus(x2,x3) = plus(x3,x2) 4.75/4.90 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 4.75/4.90 times(x2,x3) = times(x3,x2) 4.75/4.90 -> Rules: 4.75/4.90 0(S) -> S 4.75/4.90 plus(0(x),0(y)) -> 0(plus(x,y)) 4.75/4.90 plus(0(x),1(y)) -> 1(plus(x,y)) 4.75/4.90 plus(1(x),1(y)) -> 0(1(plus(plus(x,y),S))) 4.75/4.90 plus(x,S) -> x 4.75/4.90 times(x,0(y)) -> 0(times(x,y)) 4.75/4.90 times(x,1(y)) -> plus(x,0(times(x,y))) 4.75/4.90 times(x,S) -> S 4.75/4.90 -> SRules: 4.75/4.90 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 4.75/4.90 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 4.75/4.90 ->Strongly Connected Components: 4.75/4.90 There is no strongly connected component 4.75/4.90 4.75/4.90 The problem is finite. 4.75/4.90 EOF