1.45/2.06 YES 1.45/2.06 1.45/2.06 Problem 1: 1.45/2.06 1.45/2.06 (VAR x y z) 1.45/2.06 (THEORY 1.45/2.06 (AC plus times)) 1.45/2.06 (RULES 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 ) 1.45/2.06 1.45/2.06 Problem 1: 1.45/2.06 1.45/2.06 Dependency Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 DIV(div(x,y),z) -> DIV(x,times(y,z)) 1.45/2.06 DIV(div(x,y),z) -> TIMES(y,z) 1.45/2.06 DIV(x,y) -> QUOT(x,y,y) 1.45/2.06 PLUS(plus(0,y),x3) -> PLUS(y,x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 QUOT(s(x),s(y),z) -> QUOT(x,y,z) 1.45/2.06 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(0),y),x3) -> TIMES(y,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> PLUS(y,times(x,y)) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> PLUS(y,times(x,y)) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 1.45/2.06 Problem 1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 DIV(div(x,y),z) -> DIV(x,times(y,z)) 1.45/2.06 DIV(div(x,y),z) -> TIMES(y,z) 1.45/2.06 DIV(x,y) -> QUOT(x,y,y) 1.45/2.06 PLUS(plus(0,y),x3) -> PLUS(y,x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 QUOT(s(x),s(y),z) -> QUOT(x,y,z) 1.45/2.06 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(0),y),x3) -> TIMES(y,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> PLUS(y,times(x,y)) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> PLUS(y,times(x,y)) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 PLUS(plus(0,y),x3) -> PLUS(y,x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) -> PLUS(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(0),y),x3) -> TIMES(y,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) -> TIMES(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 DIV(div(x,y),z) -> DIV(x,times(y,z)) 1.45/2.06 DIV(x,y) -> QUOT(x,y,y) 1.45/2.06 QUOT(s(x),s(y),z) -> QUOT(x,y,z) 1.45/2.06 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 Empty 1.45/2.06 1.45/2.06 1.45/2.06 The problem is decomposed in 3 subproblems. 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(0,y),x3) -> PLUS(y,x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Linear 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 2 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = 0 1.45/2.06 [0] = 2 1.45/2.06 [s](X) = X 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = X1 + X2 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = 0 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) -> PLUS(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Linear 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 2 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 2 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = 0 1.45/2.06 [0] = 0 1.45/2.06 [s](X) = X + 2 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 2.X1 + 2.X2 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = 0 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) -> PLUS(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(plus(x,0),x3) -> PLUS(x,x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Linear 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 2 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 2 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = 0 1.45/2.06 [0] = 2 1.45/2.06 [s](X) = X + 2 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 2.X1 + 2.X2 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = 0 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) -> PLUS(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 -> SRules: 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Linear 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 2 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 1 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = 0 1.45/2.06 [0] = 0 1.45/2.06 [s](X) = X + 2 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 2.X1 + 2.X2 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = 0 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) -> PLUS(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 PLUS(s(x),y) -> PLUS(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Linear 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 2 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 2 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = 0 1.45/2.06 [0] = 0 1.45/2.06 [s](X) = X + 2 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 2.X1 + 2.X2 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = 0 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) -> PLUS(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Linear 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 2 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 2 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = 0 1.45/2.06 [0] = 0 1.45/2.06 [s](X) = 2 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 2.X1 + 2.X2 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = 0 1.45/2.06 1.45/2.06 Problem 1.1: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 1.45/2.06 PLUS(x3,x4) = PLUS(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 Empty 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 There is no strongly connected component 1.45/2.06 1.45/2.06 The problem is finite. 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(0),y),x3) -> TIMES(y,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Simple mixed 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 1 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 1 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 [0] = 1 1.45/2.06 [s](X) = X + 1 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 0 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) -> TIMES(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(plus(y,times(x,y)),x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Simple mixed 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 1 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 [0] = 1 1.45/2.06 [s](X) = X + 1 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 0 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) -> TIMES(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(times(s(x),y),x3) -> TIMES(x,y) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Simple mixed 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 1 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 1 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 [0] = 1 1.45/2.06 [s](X) = X + 1 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 0 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) -> TIMES(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 Reduction Pairs Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 TIMES(s(x),y) -> TIMES(x,y) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Usable Equations: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> Usable Rules: 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Interpretation type: 1.45/2.06 Simple mixed 1.45/2.06 ->Coefficients: 1.45/2.06 Natural Numbers 1.45/2.06 ->Dimension: 1.45/2.06 1 1.45/2.06 ->Bound: 1.45/2.06 1 1.45/2.06 ->Interpretation: 1.45/2.06 1.45/2.06 [div](X1,X2) = 0 1.45/2.06 [plus](X1,X2) = X1 + X2 + 1 1.45/2.06 [quot](X1,X2,X3) = 0 1.45/2.06 [times](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 [0] = 1 1.45/2.06 [s](X) = X + 1 1.45/2.06 [DIV](X1,X2) = 0 1.45/2.06 [PLUS](X1,X2) = 0 1.45/2.06 [QUOT](X1,X2,X3) = 0 1.45/2.06 [TIMES](X1,X2) = X1.X2 + X1 + X2 1.45/2.06 1.45/2.06 Problem 1.2: 1.45/2.06 1.45/2.06 SCC Processor: 1.45/2.06 -> FAxioms: 1.45/2.06 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.06 -> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 -> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.06 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.06 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.06 times(0,y) -> 0 1.45/2.06 times(s(0),y) -> y 1.45/2.06 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.06 -> SRules: 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.06 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.06 ->Strongly Connected Components: 1.45/2.06 ->->Cycle: 1.45/2.06 ->->-> Pairs: 1.45/2.06 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.06 -> FAxioms: 1.45/2.06 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) -> plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) -> times(x4,x3) 1.45/2.06 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 1.45/2.06 TIMES(x3,x4) -> TIMES(x4,x3) 1.45/2.06 -> EAxioms: 1.45/2.06 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.06 plus(x3,x4) = plus(x4,x3) 1.45/2.06 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.06 times(x3,x4) = times(x4,x3) 1.45/2.06 ->->-> Rules: 1.45/2.06 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.06 div(0,y) -> 0 1.45/2.06 div(x,y) -> quot(x,y,y) 1.45/2.06 plus(0,y) -> y 1.45/2.06 plus(s(x),y) -> s(plus(x,y)) 1.45/2.06 plus(x,0) -> x 1.45/2.06 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.07 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.07 1.45/2.07 Problem 1.2: 1.45/2.07 1.45/2.07 Reduction Pairs Processor: 1.45/2.07 -> FAxioms: 1.45/2.07 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.07 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.07 -> Pairs: 1.45/2.07 TIMES(times(0,y),x3) -> TIMES(0,x3) 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Usable Equations: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> Usable Rules: 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.07 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.07 ->Interpretation type: 1.45/2.07 Simple mixed 1.45/2.07 ->Coefficients: 1.45/2.07 All rationals 1.45/2.07 ->Dimension: 1.45/2.07 1 1.45/2.07 ->Bound: 1.45/2.07 3 1.45/2.07 ->Interpretation: 1.45/2.07 1.45/2.07 [div](X1,X2) = 0 1.45/2.07 [plus](X1,X2) = X1 + X2 + 3 1.45/2.07 [quot](X1,X2,X3) = 0 1.45/2.07 [times](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2 1.45/2.07 [0] = 3/2 1.45/2.07 [s](X) = X + 2 1.45/2.07 [DIV](X1,X2) = 0 1.45/2.07 [PLUS](X1,X2) = 0 1.45/2.07 [QUOT](X1,X2,X3) = 0 1.45/2.07 [TIMES](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2 1.45/2.07 1.45/2.07 Problem 1.2: 1.45/2.07 1.45/2.07 SCC Processor: 1.45/2.07 -> FAxioms: 1.45/2.07 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 1.45/2.07 TIMES(x3,x4) = TIMES(x4,x3) 1.45/2.07 -> Pairs: 1.45/2.07 Empty 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 1.45/2.07 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 1.45/2.07 ->Strongly Connected Components: 1.45/2.07 There is no strongly connected component 1.45/2.07 1.45/2.07 The problem is finite. 1.45/2.07 1.45/2.07 Problem 1.3: 1.45/2.07 1.45/2.07 Subterm Processor: 1.45/2.07 -> FAxioms: 1.45/2.07 Empty 1.45/2.07 -> Pairs: 1.45/2.07 DIV(div(x,y),z) -> DIV(x,times(y,z)) 1.45/2.07 DIV(x,y) -> QUOT(x,y,y) 1.45/2.07 QUOT(s(x),s(y),z) -> QUOT(x,y,z) 1.45/2.07 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 Empty 1.45/2.07 ->Projection: 1.45/2.07 pi(DIV) = [1] 1.45/2.07 pi(QUOT) = [1] 1.45/2.07 1.45/2.07 Problem 1.3: 1.45/2.07 1.45/2.07 SCC Processor: 1.45/2.07 -> FAxioms: 1.45/2.07 Empty 1.45/2.07 -> Pairs: 1.45/2.07 DIV(x,y) -> QUOT(x,y,y) 1.45/2.07 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 Empty 1.45/2.07 ->Strongly Connected Components: 1.45/2.07 ->->Cycle: 1.45/2.07 ->->-> Pairs: 1.45/2.07 DIV(x,y) -> QUOT(x,y,y) 1.45/2.07 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.07 -> FAxioms: 1.45/2.07 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) -> plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) -> times(x4,x3) 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 ->->-> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 Empty 1.45/2.07 1.45/2.07 Problem 1.3: 1.45/2.07 1.45/2.07 Reduction Pairs Processor: 1.45/2.07 -> FAxioms: 1.45/2.07 Empty 1.45/2.07 -> Pairs: 1.45/2.07 DIV(x,y) -> QUOT(x,y,y) 1.45/2.07 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Usable Equations: 1.45/2.07 Empty 1.45/2.07 -> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> Usable Rules: 1.45/2.07 Empty 1.45/2.07 -> SRules: 1.45/2.07 Empty 1.45/2.07 ->Interpretation type: 1.45/2.07 Linear 1.45/2.07 ->Coefficients: 1.45/2.07 Natural Numbers 1.45/2.07 ->Dimension: 1.45/2.07 1 1.45/2.07 ->Bound: 1.45/2.07 2 1.45/2.07 ->Interpretation: 1.45/2.07 1.45/2.07 [div](X1,X2) = 0 1.45/2.07 [plus](X1,X2) = 0 1.45/2.07 [quot](X1,X2,X3) = 0 1.45/2.07 [times](X1,X2) = 0 1.45/2.07 [0] = 2 1.45/2.07 [s](X) = 0 1.45/2.07 [DIV](X1,X2) = 2.X2 + 1 1.45/2.07 [PLUS](X1,X2) = 0 1.45/2.07 [QUOT](X1,X2,X3) = X2 + X3 1.45/2.07 [TIMES](X1,X2) = 0 1.45/2.07 1.45/2.07 Problem 1.3: 1.45/2.07 1.45/2.07 SCC Processor: 1.45/2.07 -> FAxioms: 1.45/2.07 Empty 1.45/2.07 -> Pairs: 1.45/2.07 QUOT(x,0,s(z)) -> DIV(x,s(z)) 1.45/2.07 -> EAxioms: 1.45/2.07 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 1.45/2.07 plus(x3,x4) = plus(x4,x3) 1.45/2.07 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 1.45/2.07 times(x3,x4) = times(x4,x3) 1.45/2.07 -> Rules: 1.45/2.07 div(div(x,y),z) -> div(x,times(y,z)) 1.45/2.07 div(0,y) -> 0 1.45/2.07 div(x,y) -> quot(x,y,y) 1.45/2.07 plus(0,y) -> y 1.45/2.07 plus(s(x),y) -> s(plus(x,y)) 1.45/2.07 plus(x,0) -> x 1.45/2.07 quot(0,s(y),z) -> 0 1.45/2.07 quot(s(x),s(y),z) -> quot(x,y,z) 1.45/2.07 quot(x,0,s(z)) -> s(div(x,s(z))) 1.45/2.07 times(0,y) -> 0 1.45/2.07 times(s(0),y) -> y 1.45/2.07 times(s(x),y) -> plus(y,times(x,y)) 1.45/2.07 -> SRules: 1.45/2.07 Empty 1.45/2.07 ->Strongly Connected Components: 1.45/2.07 There is no strongly connected component 1.45/2.07 1.45/2.07 The problem is finite. 1.45/2.07 EOF