The rewrite relation of the following TRS is considered.
a__minus(0,Y) | → | 0 | (1) |
a__minus(s(X),s(Y)) | → | a__minus(X,Y) | (2) |
a__geq(X,0) | → | true | (3) |
a__geq(0,s(Y)) | → | false | (4) |
a__geq(s(X),s(Y)) | → | a__geq(X,Y) | (5) |
a__div(0,s(Y)) | → | 0 | (6) |
a__div(s(X),s(Y)) | → | a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0) | (7) |
a__if(true,X,Y) | → | mark(X) | (8) |
a__if(false,X,Y) | → | mark(Y) | (9) |
mark(minus(X1,X2)) | → | a__minus(X1,X2) | (10) |
mark(geq(X1,X2)) | → | a__geq(X1,X2) | (11) |
mark(div(X1,X2)) | → | a__div(mark(X1),X2) | (12) |
mark(if(X1,X2,X3)) | → | a__if(mark(X1),X2,X3) | (13) |
mark(0) | → | 0 | (14) |
mark(s(X)) | → | s(mark(X)) | (15) |
mark(true) | → | true | (16) |
mark(false) | → | false | (17) |
a__minus(X1,X2) | → | minus(X1,X2) | (18) |
a__geq(X1,X2) | → | geq(X1,X2) | (19) |
a__div(X1,X2) | → | div(X1,X2) | (20) |
a__if(X1,X2,X3) | → | if(X1,X2,X3) | (21) |
a__minus#(s(X),s(Y)) | → | a__minus#(X,Y) | (22) |
a__geq#(s(X),s(Y)) | → | a__geq#(X,Y) | (23) |
a__div#(s(X),s(Y)) | → | a__geq#(X,Y) | (24) |
a__div#(s(X),s(Y)) | → | a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0) | (25) |
a__if#(true,X,Y) | → | mark#(X) | (26) |
a__if#(false,X,Y) | → | mark#(Y) | (27) |
mark#(minus(X1,X2)) | → | a__minus#(X1,X2) | (28) |
mark#(geq(X1,X2)) | → | a__geq#(X1,X2) | (29) |
mark#(div(X1,X2)) | → | mark#(X1) | (30) |
mark#(div(X1,X2)) | → | a__div#(mark(X1),X2) | (31) |
mark#(if(X1,X2,X3)) | → | mark#(X1) | (32) |
mark#(if(X1,X2,X3)) | → | a__if#(mark(X1),X2,X3) | (33) |
mark#(s(X)) | → | mark#(X) | (34) |
The dependency pairs are split into 3 components.
mark#(if(X1,X2,X3)) | → | mark#(X1) | (32) |
mark#(div(X1,X2)) | → | mark#(X1) | (30) |
mark#(div(X1,X2)) | → | a__div#(mark(X1),X2) | (31) |
a__div#(s(X),s(Y)) | → | a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0) | (25) |
a__if#(true,X,Y) | → | mark#(X) | (26) |
mark#(if(X1,X2,X3)) | → | a__if#(mark(X1),X2,X3) | (33) |
a__if#(false,X,Y) | → | mark#(Y) | (27) |
mark#(s(X)) | → | mark#(X) | (34) |
[true] | = | 2 |
[a__div#(x1, x2)] | = | -∞ · x1 + 5 · x2 + 6 |
[div(x1, x2)] | = | 4 · x1 + 5 · x2 + 0 |
[a__minus(x1, x2)] | = | 2 · x1 + 2 · x2 + 0 |
[a__geq(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__if#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 5 · x3 + 6 |
[mark#(x1)] | = | 0 · x1 + 6 |
[false] | = | 0 |
[mark(x1)] | = | 0 · x1 + -∞ |
[0] | = | 0 |
[minus(x1, x2)] | = | -∞ · x1 + -∞ · x2 + 0 |
[if(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 6 · x3 + 7 |
[geq(x1, x2)] | = | 0 · x1 + -∞ · x2 + 6 |
[a__if(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + -∞ · x3 + 8 |
[a__div(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + 0 |
mark#(if(X1,X2,X3)) | → | a__if#(mark(X1),X2,X3) | (33) |
[true] | = | 0 |
[a__div#(x1, x2)] | = | 6 · x1 + 0 · x2 + 7 |
[div(x1, x2)] | = | 3 · x1 + 0 · x2 + 3 |
[a__minus(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__geq(x1, x2)] | = | 0 · x1 + 0 · x2 + 1 |
[a__if#(x1, x2, x3)] | = | 1 · x1 + 2 · x2 + 4 · x3 + 1 |
[mark#(x1)] | = | 2 · x1 + 1 |
[false] | = | 1 |
[mark(x1)] | = | 1 · x1 + 0 |
[0] | = | 0 |
[minus(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[if(x1, x2, x3)] | = | 2 · x1 + 1 · x2 + 1 · x3 + 0 |
[geq(x1, x2)] | = | 0 · x1 + 0 · x2 + 1 |
[a__if(x1, x2, x3)] | = | 2 · x1 + 1 · x2 + 1 · x3 + 0 |
[a__div(x1, x2)] | = | 3 · x1 + 0 · x2 + 3 |
[s(x1)] | = | 2 · x1 + 3 |
a__minus(0,Y) | → | 0 | (1) |
a__minus(s(X),s(Y)) | → | a__minus(X,Y) | (2) |
a__geq(X,0) | → | true | (3) |
a__geq(0,s(Y)) | → | false | (4) |
a__geq(s(X),s(Y)) | → | a__geq(X,Y) | (5) |
a__div(0,s(Y)) | → | 0 | (6) |
a__div(s(X),s(Y)) | → | a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0) | (7) |
a__if(true,X,Y) | → | mark(X) | (8) |
a__if(false,X,Y) | → | mark(Y) | (9) |
mark(minus(X1,X2)) | → | a__minus(X1,X2) | (10) |
mark(geq(X1,X2)) | → | a__geq(X1,X2) | (11) |
mark(div(X1,X2)) | → | a__div(mark(X1),X2) | (12) |
mark(if(X1,X2,X3)) | → | a__if(mark(X1),X2,X3) | (13) |
mark(0) | → | 0 | (14) |
mark(s(X)) | → | s(mark(X)) | (15) |
mark(true) | → | true | (16) |
mark(false) | → | false | (17) |
a__minus(X1,X2) | → | minus(X1,X2) | (18) |
a__geq(X1,X2) | → | geq(X1,X2) | (19) |
a__div(X1,X2) | → | div(X1,X2) | (20) |
a__if(X1,X2,X3) | → | if(X1,X2,X3) | (21) |
mark#(div(X1,X2)) | → | mark#(X1) | (30) |
a__div#(s(X),s(Y)) | → | a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0) | (25) |
a__if#(false,X,Y) | → | mark#(Y) | (27) |
mark#(s(X)) | → | mark#(X) | (34) |
The dependency pairs are split into 1 component.
mark#(if(X1,X2,X3)) | → | mark#(X1) | (32) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(if(X1,X2,X3)) | → | mark#(X1) | (32) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__geq#(s(X),s(Y)) | → | a__geq#(X,Y) | (23) |
π(a__geq#) | = | 1 |
a__geq#(s(X),s(Y)) | → | a__geq#(X,Y) | (23) |
There are no pairs anymore.
a__minus#(s(X),s(Y)) | → | a__minus#(X,Y) | (22) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__minus#(s(X),s(Y)) | → | a__minus#(X,Y) | (22) |
2 | > | 2 | |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.