The rewrite relation of the following TRS is considered.
a__terms(N) | → | cons(recip(a__sqr(mark(N))),terms(s(N))) | (1) |
a__sqr(0) | → | 0 | (2) |
a__sqr(s(X)) | → | s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) | (3) |
a__dbl(0) | → | 0 | (4) |
a__dbl(s(X)) | → | s(s(a__dbl(mark(X)))) | (5) |
a__add(0,X) | → | mark(X) | (6) |
a__add(s(X),Y) | → | s(a__add(mark(X),mark(Y))) | (7) |
a__first(0,X) | → | nil | (8) |
a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (9) |
a__half(0) | → | 0 | (10) |
a__half(s(0)) | → | 0 | (11) |
a__half(s(s(X))) | → | s(a__half(mark(X))) | (12) |
a__half(dbl(X)) | → | mark(X) | (13) |
mark(terms(X)) | → | a__terms(mark(X)) | (14) |
mark(sqr(X)) | → | a__sqr(mark(X)) | (15) |
mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (16) |
mark(dbl(X)) | → | a__dbl(mark(X)) | (17) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (18) |
mark(half(X)) | → | a__half(mark(X)) | (19) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (20) |
mark(recip(X)) | → | recip(mark(X)) | (21) |
mark(s(X)) | → | s(mark(X)) | (22) |
mark(0) | → | 0 | (23) |
mark(nil) | → | nil | (24) |
a__terms(X) | → | terms(X) | (25) |
a__sqr(X) | → | sqr(X) | (26) |
a__add(X1,X2) | → | add(X1,X2) | (27) |
a__dbl(X) | → | dbl(X) | (28) |
a__first(X1,X2) | → | first(X1,X2) | (29) |
a__half(X) | → | half(X) | (30) |
a__terms#(N) | → | a__sqr#(mark(N)) | (31) |
a__terms#(N) | → | mark#(N) | (32) |
a__sqr#(s(X)) | → | a__add#(a__sqr(mark(X)),a__dbl(mark(X))) | (33) |
a__sqr#(s(X)) | → | a__sqr#(mark(X)) | (34) |
a__sqr#(s(X)) | → | mark#(X) | (35) |
a__sqr#(s(X)) | → | a__dbl#(mark(X)) | (36) |
a__dbl#(s(X)) | → | a__dbl#(mark(X)) | (37) |
a__dbl#(s(X)) | → | mark#(X) | (38) |
a__add#(0,X) | → | mark#(X) | (39) |
a__add#(s(X),Y) | → | a__add#(mark(X),mark(Y)) | (40) |
a__add#(s(X),Y) | → | mark#(X) | (41) |
a__add#(s(X),Y) | → | mark#(Y) | (42) |
a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (43) |
a__half#(s(s(X))) | → | a__half#(mark(X)) | (44) |
a__half#(s(s(X))) | → | mark#(X) | (45) |
a__half#(dbl(X)) | → | mark#(X) | (46) |
mark#(terms(X)) | → | a__terms#(mark(X)) | (47) |
mark#(terms(X)) | → | mark#(X) | (48) |
mark#(sqr(X)) | → | a__sqr#(mark(X)) | (49) |
mark#(sqr(X)) | → | mark#(X) | (50) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (51) |
mark#(add(X1,X2)) | → | mark#(X1) | (52) |
mark#(add(X1,X2)) | → | mark#(X2) | (53) |
mark#(dbl(X)) | → | a__dbl#(mark(X)) | (54) |
mark#(dbl(X)) | → | mark#(X) | (55) |
mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (56) |
mark#(first(X1,X2)) | → | mark#(X1) | (57) |
mark#(first(X1,X2)) | → | mark#(X2) | (58) |
mark#(half(X)) | → | a__half#(mark(X)) | (59) |
mark#(half(X)) | → | mark#(X) | (60) |
mark#(cons(X1,X2)) | → | mark#(X1) | (61) |
mark#(recip(X)) | → | mark#(X) | (62) |
mark#(s(X)) | → | mark#(X) | (63) |
prec(a__terms#) | = | 6 | stat(a__terms#) | = | lex | |
prec(a__sqr#) | = | 6 | stat(a__sqr#) | = | lex | |
prec(mark#) | = | 0 | stat(mark#) | = | mul | |
prec(s) | = | 1 | stat(s) | = | lex | |
prec(a__add#) | = | 2 | stat(a__add#) | = | mul | |
prec(a__sqr) | = | 6 | stat(a__sqr) | = | lex | |
prec(a__dbl) | = | 6 | stat(a__dbl) | = | lex | |
prec(a__dbl#) | = | 3 | stat(a__dbl#) | = | lex | |
prec(0) | = | 6 | stat(0) | = | mul | |
prec(a__first#) | = | 0 | stat(a__first#) | = | mul | |
prec(a__half#) | = | 0 | stat(a__half#) | = | mul | |
prec(dbl) | = | 6 | stat(dbl) | = | lex | |
prec(terms) | = | 6 | stat(terms) | = | lex | |
prec(sqr) | = | 6 | stat(sqr) | = | lex | |
prec(add) | = | 4 | stat(add) | = | lex | |
prec(first) | = | 7 | stat(first) | = | lex | |
prec(a__terms) | = | 6 | stat(a__terms) | = | lex | |
prec(a__add) | = | 4 | stat(a__add) | = | lex | |
prec(a__first) | = | 7 | stat(a__first) | = | lex | |
prec(nil) | = | 5 | stat(nil) | = | mul |
π(a__terms#) | = | [1] |
π(a__sqr#) | = | [1] |
π(mark) | = | 1 |
π(mark#) | = | [1] |
π(s) | = | [1] |
π(a__add#) | = | [1,2] |
π(a__sqr) | = | [1] |
π(a__dbl) | = | [1] |
π(a__dbl#) | = | [1] |
π(0) | = | [] |
π(a__first#) | = | [2] |
π(cons) | = | 1 |
π(a__half#) | = | [1] |
π(dbl) | = | [1] |
π(terms) | = | [1] |
π(sqr) | = | [1] |
π(add) | = | [2,1] |
π(first) | = | [2,1] |
π(half) | = | 1 |
π(recip) | = | 1 |
π(a__terms) | = | [1] |
π(a__add) | = | [2,1] |
π(a__half) | = | 1 |
π(a__first) | = | [2,1] |
π(nil) | = | [] |
a__terms#(N) | → | mark#(N) | (32) |
a__sqr#(s(X)) | → | a__add#(a__sqr(mark(X)),a__dbl(mark(X))) | (33) |
a__sqr#(s(X)) | → | a__sqr#(mark(X)) | (34) |
a__sqr#(s(X)) | → | mark#(X) | (35) |
a__sqr#(s(X)) | → | a__dbl#(mark(X)) | (36) |
a__dbl#(s(X)) | → | a__dbl#(mark(X)) | (37) |
a__dbl#(s(X)) | → | mark#(X) | (38) |
a__add#(0,X) | → | mark#(X) | (39) |
a__add#(s(X),Y) | → | a__add#(mark(X),mark(Y)) | (40) |
a__add#(s(X),Y) | → | mark#(X) | (41) |
a__add#(s(X),Y) | → | mark#(Y) | (42) |
a__half#(s(s(X))) | → | a__half#(mark(X)) | (44) |
a__half#(s(s(X))) | → | mark#(X) | (45) |
a__half#(dbl(X)) | → | mark#(X) | (46) |
mark#(terms(X)) | → | a__terms#(mark(X)) | (47) |
mark#(terms(X)) | → | mark#(X) | (48) |
mark#(sqr(X)) | → | a__sqr#(mark(X)) | (49) |
mark#(sqr(X)) | → | mark#(X) | (50) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (51) |
mark#(add(X1,X2)) | → | mark#(X1) | (52) |
mark#(add(X1,X2)) | → | mark#(X2) | (53) |
mark#(dbl(X)) | → | a__dbl#(mark(X)) | (54) |
mark#(dbl(X)) | → | mark#(X) | (55) |
mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (56) |
mark#(first(X1,X2)) | → | mark#(X1) | (57) |
mark#(first(X1,X2)) | → | mark#(X2) | (58) |
mark#(s(X)) | → | mark#(X) | (63) |
The dependency pairs are split into 1 component.
mark#(cons(X1,X2)) | → | mark#(X1) | (61) |
mark#(half(X)) | → | mark#(X) | (60) |
mark#(recip(X)) | → | mark#(X) | (62) |
[cons(x1, x2)] | = | 1 · x1 + 1 · x2 |
[half(x1)] | = | 1 · x1 |
[recip(x1)] | = | 1 · x1 |
[mark#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) | → | mark#(X1) | (61) |
1 | > | 1 | |
mark#(half(X)) | → | mark#(X) | (60) |
1 | > | 1 | |
mark#(recip(X)) | → | mark#(X) | (62) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.