The rewrite relation of the following TRS is considered.
minus(n__0,Y) | → | 0 | (1) |
minus(n__s(X),n__s(Y)) | → | minus(activate(X),activate(Y)) | (2) |
geq(X,n__0) | → | true | (3) |
geq(n__0,n__s(Y)) | → | false | (4) |
geq(n__s(X),n__s(Y)) | → | geq(activate(X),activate(Y)) | (5) |
div(0,n__s(Y)) | → | 0 | (6) |
div(s(X),n__s(Y)) | → | if(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),n__0) | (7) |
if(true,X,Y) | → | activate(X) | (8) |
if(false,X,Y) | → | activate(Y) | (9) |
0 | → | n__0 | (10) |
s(X) | → | n__s(X) | (11) |
activate(n__0) | → | 0 | (12) |
activate(n__s(X)) | → | s(X) | (13) |
activate(X) | → | X | (14) |
minus#(n__0,Y) | → | 0# | (15) |
minus#(n__s(X),n__s(Y)) | → | minus#(activate(X),activate(Y)) | (16) |
minus#(n__s(X),n__s(Y)) | → | activate#(X) | (17) |
minus#(n__s(X),n__s(Y)) | → | activate#(Y) | (18) |
geq#(n__s(X),n__s(Y)) | → | geq#(activate(X),activate(Y)) | (19) |
geq#(n__s(X),n__s(Y)) | → | activate#(X) | (20) |
geq#(n__s(X),n__s(Y)) | → | activate#(Y) | (21) |
div#(s(X),n__s(Y)) | → | if#(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),n__0) | (22) |
div#(s(X),n__s(Y)) | → | geq#(X,activate(Y)) | (23) |
div#(s(X),n__s(Y)) | → | activate#(Y) | (24) |
div#(s(X),n__s(Y)) | → | div#(minus(X,activate(Y)),n__s(activate(Y))) | (25) |
div#(s(X),n__s(Y)) | → | minus#(X,activate(Y)) | (26) |
if#(true,X,Y) | → | activate#(X) | (27) |
if#(false,X,Y) | → | activate#(Y) | (28) |
activate#(n__0) | → | 0# | (29) |
activate#(n__s(X)) | → | s#(X) | (30) |
The dependency pairs are split into 3 components.
div#(s(X),n__s(Y)) | → | div#(minus(X,activate(Y)),n__s(activate(Y))) | (25) |
prec(s) | = | 3 | weight(s) | = | 1 | ||||
prec(minus) | = | 2 | weight(minus) | = | 1 | ||||
prec(0) | = | 1 | weight(0) | = | 1 | ||||
prec(n__0) | = | 0 | weight(n__0) | = | 1 |
π(div#) | = | 1 |
π(s) | = | [] |
π(minus) | = | [] |
π(0) | = | [] |
π(n__0) | = | [] |
minus(n__0,Y) | → | 0 | (1) |
minus(n__s(X),n__s(Y)) | → | minus(activate(X),activate(Y)) | (2) |
0 | → | n__0 | (10) |
div#(s(X),n__s(Y)) | → | div#(minus(X,activate(Y)),n__s(activate(Y))) | (25) |
There are no pairs anymore.
minus#(n__s(X),n__s(Y)) | → | minus#(activate(X),activate(Y)) | (16) |
[activate(x1)] | = | 1 · x1 |
[n__0] | = | 0 |
[0] | = | 0 |
[n__s(x1)] | = | 1 · x1 |
[s(x1)] | = | 1 · x1 |
[minus#(x1, x2)] | = | 1 · x1 + 1 · x2 |
activate(n__0) | → | 0 | (12) |
activate(n__s(X)) | → | s(X) | (13) |
activate(X) | → | X | (14) |
s(X) | → | n__s(X) | (11) |
0 | → | n__0 | (10) |
20
Hence, it suffices to show innermost termination in the following.prec(n__0) | = | 0 | weight(n__0) | = | 1 | ||||
prec(0) | = | 2 | weight(0) | = | 1 | ||||
prec(activate) | = | 1 | weight(activate) | = | 1 | ||||
prec(n__s) | = | 3 | weight(n__s) | = | 1 | ||||
prec(s) | = | 4 | weight(s) | = | 1 | ||||
prec(minus#) | = | 5 | weight(minus#) | = | 0 |
minus#(n__s(X),n__s(Y)) | → | minus#(activate(X),activate(Y)) | (16) |
activate(n__0) | → | 0 | (12) |
activate(n__s(X)) | → | s(X) | (13) |
activate(X) | → | X | (14) |
s(X) | → | n__s(X) | (11) |
0 | → | n__0 | (10) |
There are no pairs anymore.
geq#(n__s(X),n__s(Y)) | → | geq#(activate(X),activate(Y)) | (19) |
[activate(x1)] | = | 1 · x1 |
[n__0] | = | 0 |
[0] | = | 0 |
[n__s(x1)] | = | 1 · x1 |
[s(x1)] | = | 1 · x1 |
[geq#(x1, x2)] | = | 1 · x1 + 1 · x2 |
activate(n__0) | → | 0 | (12) |
activate(n__s(X)) | → | s(X) | (13) |
activate(X) | → | X | (14) |
s(X) | → | n__s(X) | (11) |
0 | → | n__0 | (10) |
20
Hence, it suffices to show innermost termination in the following.prec(n__0) | = | 0 | weight(n__0) | = | 1 | ||||
prec(0) | = | 2 | weight(0) | = | 1 | ||||
prec(activate) | = | 1 | weight(activate) | = | 1 | ||||
prec(n__s) | = | 3 | weight(n__s) | = | 1 | ||||
prec(s) | = | 4 | weight(s) | = | 1 | ||||
prec(geq#) | = | 5 | weight(geq#) | = | 0 |
geq#(n__s(X),n__s(Y)) | → | geq#(activate(X),activate(Y)) | (19) |
activate(n__0) | → | 0 | (12) |
activate(n__s(X)) | → | s(X) | (13) |
activate(X) | → | X | (14) |
s(X) | → | n__s(X) | (11) |
0 | → | n__0 | (10) |
There are no pairs anymore.