The rewrite relation of the following TRS is considered.
| le(0,y) | → | true | (1) |
| le(s(x),0) | → | false | (2) |
| le(s(x),s(y)) | → | le(x,y) | (3) |
| pred(s(x)) | → | x | (4) |
| minus(x,0) | → | x | (5) |
| minus(x,s(y)) | → | pred(minus(x,y)) | (6) |
| gcd(0,y) | → | y | (7) |
| gcd(s(x),0) | → | s(x) | (8) |
| gcd(s(x),s(y)) | → | if_gcd(le(y,x),s(x),s(y)) | (9) |
| if_gcd(true,s(x),s(y)) | → | gcd(minus(x,y),s(y)) | (10) |
| if_gcd(false,s(x),s(y)) | → | gcd(minus(y,x),s(x)) | (11) |
| minus#(x,s(y)) | → | minus#(x,y) | (12) |
| if_gcd#(false,s(x),s(y)) | → | minus#(y,x) | (13) |
| le#(s(x),s(y)) | → | le#(x,y) | (14) |
| if_gcd#(true,s(x),s(y)) | → | gcd#(minus(x,y),s(y)) | (15) |
| if_gcd#(false,s(x),s(y)) | → | gcd#(minus(y,x),s(x)) | (16) |
| gcd#(s(x),s(y)) | → | if_gcd#(le(y,x),s(x),s(y)) | (17) |
| minus#(x,s(y)) | → | pred#(minus(x,y)) | (18) |
| if_gcd#(true,s(x),s(y)) | → | minus#(x,y) | (19) |
| gcd#(s(x),s(y)) | → | le#(y,x) | (20) |
The dependency pairs are split into 3 components.
| gcd#(s(x),s(y)) | → | if_gcd#(le(y,x),s(x),s(y)) | (17) |
| if_gcd#(false,s(x),s(y)) | → | gcd#(minus(y,x),s(x)) | (16) |
| if_gcd#(true,s(x),s(y)) | → | gcd#(minus(x,y),s(y)) | (15) |
| [le(x1, x2)] | = | x1 + x2 + 1 |
| [s(x1)] | = | x1 + 3 |
| [le#(x1, x2)] | = | 0 |
| [minus(x1, x2)] | = | x1 + 1 |
| [gcd(x1, x2)] | = | 0 |
| [false] | = | 6 |
| [true] | = | 3 |
| [pred(x1)] | = | x1 + 0 |
| [0] | = | 1 |
| [minus#(x1, x2)] | = | 0 |
| [pred#(x1)] | = | 0 |
| [if_gcd(x1, x2, x3)] | = | 0 |
| [if_gcd#(x1, x2, x3)] | = | x2 + x3 + 1 |
| [gcd#(x1, x2)] | = | x1 + x2 + 2 |
| pred(s(x)) | → | x | (4) |
| minus(x,0) | → | x | (5) |
| minus(x,s(y)) | → | pred(minus(x,y)) | (6) |
| gcd#(s(x),s(y)) | → | if_gcd#(le(y,x),s(x),s(y)) | (17) |
| if_gcd#(false,s(x),s(y)) | → | gcd#(minus(y,x),s(x)) | (16) |
| if_gcd#(true,s(x),s(y)) | → | gcd#(minus(x,y),s(y)) | (15) |
The dependency pairs are split into 0 components.
| le#(s(x),s(y)) | → | le#(x,y) | (14) |
| [le(x1, x2)] | = | x1 + x2 + 1 |
| [s(x1)] | = | x1 + 1 |
| [le#(x1, x2)] | = | x2 + 0 |
| [minus(x1, x2)] | = | x1 + 1 |
| [gcd(x1, x2)] | = | 0 |
| [false] | = | 4 |
| [true] | = | 3 |
| [pred(x1)] | = | x1 + 0 |
| [0] | = | 1 |
| [minus#(x1, x2)] | = | 0 |
| [pred#(x1)] | = | 0 |
| [if_gcd(x1, x2, x3)] | = | 0 |
| [if_gcd#(x1, x2, x3)] | = | 1 |
| [gcd#(x1, x2)] | = | x1 + 2 |
| pred(s(x)) | → | x | (4) |
| minus(x,0) | → | x | (5) |
| minus(x,s(y)) | → | pred(minus(x,y)) | (6) |
| le#(s(x),s(y)) | → | le#(x,y) | (14) |
The dependency pairs are split into 0 components.
| minus#(x,s(y)) | → | minus#(x,y) | (12) |
| [le(x1, x2)] | = | x1 + x2 + 1 |
| [s(x1)] | = | x1 + 1 |
| [le#(x1, x2)] | = | 0 |
| [minus(x1, x2)] | = | x1 + 1 |
| [gcd(x1, x2)] | = | 0 |
| [false] | = | 4 |
| [true] | = | 3 |
| [pred(x1)] | = | x1 + 0 |
| [0] | = | 1 |
| [minus#(x1, x2)] | = | x2 + 0 |
| [pred#(x1)] | = | 0 |
| [if_gcd(x1, x2, x3)] | = | 0 |
| [if_gcd#(x1, x2, x3)] | = | 1 |
| [gcd#(x1, x2)] | = | x1 + 2 |
| pred(s(x)) | → | x | (4) |
| minus(x,0) | → | x | (5) |
| minus(x,s(y)) | → | pred(minus(x,y)) | (6) |
| minus#(x,s(y)) | → | minus#(x,y) | (12) |
The dependency pairs are split into 0 components.