YES 3 decompositions #1 ----------- 1: from(x1) -> :(x1,from(s(x1))) #2 ----------- 2: +(+(x1,x2),x3) -> +(x1,+(x2,x3)) 3: +(x1,+(x2,x3)) -> +(+(x1,x2),x3) 4: +(0(),x1) -> x1 5: +(x1,0()) -> x1 6: f(c(x1)) -> x1 7: f(+(c(x1),x2)) -> x1 #3 ----------- 2: +(+(x1,x2),x3) -> +(x1,+(x2,x3)) 3: +(x1,+(x2,x3)) -> +(+(x1,x2),x3) 4: +(0(),x1) -> x1 5: +(x1,0()) -> x1 8: *(0(),x2) -> 0() 9: *(s(x1),x2) -> +(x2,*(x1,x2)) @Development Closedness --- R 1: from(x1) -> :(x1,from(s(x1))) --- S 1: from(x1) -> :(x1,from(s(x1))) @Jouannaud and Kirchner's criterion --- R 2: +(+(x1,x2),x3) -> +(x1,+(x2,x3)) 3: +(x1,+(x2,x3)) -> +(+(x1,x2),x3) 4: +(0(),x1) -> x1 5: +(x1,0()) -> x1 6: f(c(x1)) -> x1 7: f(+(c(x1),x2)) -> x1 --- S 2: +(+(x1,x2),x3) -> +(x1,+(x2,x3)) 3: +(x1,+(x2,x3)) -> +(+(x1,x2),x3) 4: +(0(),x1) -> x1 5: +(x1,0()) -> x1 6: f(c(x1)) -> x1 7: f(+(c(x1),x2)) -> x1 @Jouannaud and Kirchner's criterion --- R 2: +(+(x1,x2),x3) -> +(x1,+(x2,x3)) 3: +(x1,+(x2,x3)) -> +(+(x1,x2),x3) 4: +(0(),x1) -> x1 5: +(x1,0()) -> x1 8: *(0(),x2) -> 0() 9: *(s(x1),x2) -> +(x2,*(x1,x2)) --- S 2: +(+(x1,x2),x3) -> +(x1,+(x2,x3)) 3: +(x1,+(x2,x3)) -> +(+(x1,x2),x3) 4: +(0(),x1) -> x1 5: +(x1,0()) -> x1 8: *(0(),x2) -> 0() 9: *(s(x1),x2) -> +(x2,*(x1,x2))