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) |
mark(terms(X)) | → | a__terms(mark(X)) | (10) |
mark(sqr(X)) | → | a__sqr(mark(X)) | (11) |
mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (12) |
mark(dbl(X)) | → | a__dbl(mark(X)) | (13) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (14) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (15) |
mark(recip(X)) | → | recip(mark(X)) | (16) |
mark(s(X)) | → | s(mark(X)) | (17) |
mark(0) | → | 0 | (18) |
mark(nil) | → | nil | (19) |
a__terms(X) | → | terms(X) | (20) |
a__sqr(X) | → | sqr(X) | (21) |
a__add(X1,X2) | → | add(X1,X2) | (22) |
a__dbl(X) | → | dbl(X) | (23) |
a__first(X1,X2) | → | first(X1,X2) | (24) |
mark#(dbl(X)) | → | a__dbl#(mark(X)) | (25) |
mark#(cons(X1,X2)) | → | mark#(X1) | (26) |
mark#(sqr(X)) | → | mark#(X) | (27) |
a__add#(s(X),Y) | → | a__add#(mark(X),mark(Y)) | (28) |
mark#(add(X1,X2)) | → | mark#(X2) | (29) |
mark#(first(X1,X2)) | → | mark#(X1) | (30) |
a__sqr#(s(X)) | → | mark#(X) | (31) |
a__sqr#(s(X)) | → | a__sqr#(mark(X)) | (32) |
mark#(first(X1,X2)) | → | mark#(X2) | (33) |
a__add#(s(X),Y) | → | mark#(X) | (34) |
a__sqr#(s(X)) | → | a__dbl#(mark(X)) | (35) |
mark#(s(X)) | → | mark#(X) | (36) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (37) |
mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (38) |
mark#(sqr(X)) | → | a__sqr#(mark(X)) | (39) |
a__terms#(N) | → | mark#(N) | (40) |
a__sqr#(s(X)) | → | a__add#(a__sqr(mark(X)),a__dbl(mark(X))) | (41) |
a__terms#(N) | → | a__sqr#(mark(N)) | (42) |
mark#(terms(X)) | → | mark#(X) | (43) |
a__dbl#(s(X)) | → | mark#(X) | (44) |
a__sqr#(s(X)) | → | mark#(X) | (31) |
mark#(recip(X)) | → | mark#(X) | (45) |
mark#(terms(X)) | → | a__terms#(mark(X)) | (46) |
mark#(dbl(X)) | → | mark#(X) | (47) |
a__add#(0,X) | → | mark#(X) | (48) |
mark#(add(X1,X2)) | → | mark#(X1) | (49) |
a__add#(s(X),Y) | → | mark#(Y) | (50) |
a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (51) |
a__dbl#(s(X)) | → | a__dbl#(mark(X)) | (52) |
The dependency pairs are split into 1 component.
a__dbl#(s(X)) | → | a__dbl#(mark(X)) | (52) |
a__sqr#(s(X)) | → | a__dbl#(mark(X)) | (35) |
a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (51) |
a__add#(s(X),Y) | → | mark#(X) | (34) |
a__add#(s(X),Y) | → | mark#(Y) | (50) |
mark#(first(X1,X2)) | → | mark#(X2) | (33) |
mark#(add(X1,X2)) | → | mark#(X1) | (49) |
a__add#(0,X) | → | mark#(X) | (48) |
a__sqr#(s(X)) | → | a__sqr#(mark(X)) | (32) |
mark#(dbl(X)) | → | mark#(X) | (47) |
a__sqr#(s(X)) | → | mark#(X) | (31) |
mark#(terms(X)) | → | a__terms#(mark(X)) | (46) |
mark#(recip(X)) | → | mark#(X) | (45) |
a__sqr#(s(X)) | → | mark#(X) | (31) |
mark#(first(X1,X2)) | → | mark#(X1) | (30) |
mark#(add(X1,X2)) | → | mark#(X2) | (29) |
a__add#(s(X),Y) | → | a__add#(mark(X),mark(Y)) | (28) |
a__dbl#(s(X)) | → | mark#(X) | (44) |
mark#(terms(X)) | → | mark#(X) | (43) |
mark#(sqr(X)) | → | mark#(X) | (27) |
a__terms#(N) | → | a__sqr#(mark(N)) | (42) |
a__sqr#(s(X)) | → | a__add#(a__sqr(mark(X)),a__dbl(mark(X))) | (41) |
a__terms#(N) | → | mark#(N) | (40) |
mark#(sqr(X)) | → | a__sqr#(mark(X)) | (39) |
mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (38) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (37) |
mark#(cons(X1,X2)) | → | mark#(X1) | (26) |
mark#(dbl(X)) | → | a__dbl#(mark(X)) | (25) |
mark#(s(X)) | → | mark#(X) | (36) |
[s(x1)] | = | x1 + 0 |
[a__first#(x1, x2)] | = | max(x2 + 19308, 0) |
[recip(x1)] | = | x1 + 1 |
[dbl(x1)] | = | x1 + 32287 |
[a__add#(x1, x2)] | = | max(x1 + 19306, x2 + 19307, 0) |
[a__add(x1, x2)] | = | max(x1 + 0, x2 + 11799, 0) |
[a__dbl(x1)] | = | x1 + 32287 |
[a__sqr(x1)] | = | x1 + 44086 |
[a__terms#(x1)] | = | x1 + 63393 |
[mark#(x1)] | = | x1 + 19306 |
[0] | = | 8368 |
[nil] | = | 4 |
[a__dbl#(x1)] | = | x1 + 19307 |
[mark(x1)] | = | x1 + 0 |
[first(x1, x2)] | = | max(x1 + 1, x2 + 3, 0) |
[a__first(x1, x2)] | = | max(x1 + 1, x2 + 3, 0) |
[cons(x1, x2)] | = | max(x1 + 4, 0) |
[add(x1, x2)] | = | max(x1 + 0, x2 + 11799, 0) |
[sqr(x1)] | = | x1 + 44086 |
[a__terms(x1)] | = | x1 + 44091 |
[a__sqr#(x1)] | = | x1 + 63392 |
[terms(x1)] | = | x1 + 44091 |
mark(0) | → | 0 | (18) |
a__dbl(0) | → | 0 | (4) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (15) |
a__first(0,X) | → | nil | (8) |
a__terms(N) | → | cons(recip(a__sqr(mark(N))),terms(s(N))) | (1) |
a__sqr(s(X)) | → | s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) | (3) |
mark(recip(X)) | → | recip(mark(X)) | (16) |
a__sqr(X) | → | sqr(X) | (21) |
mark(nil) | → | nil | (19) |
mark(s(X)) | → | s(mark(X)) | (17) |
a__add(X1,X2) | → | add(X1,X2) | (22) |
a__dbl(s(X)) | → | s(s(a__dbl(mark(X)))) | (5) |
mark(terms(X)) | → | a__terms(mark(X)) | (10) |
a__add(s(X),Y) | → | s(a__add(mark(X),mark(Y))) | (7) |
a__terms(X) | → | terms(X) | (20) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (14) |
mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (12) |
a__dbl(X) | → | dbl(X) | (23) |
a__first(X1,X2) | → | first(X1,X2) | (24) |
mark(sqr(X)) | → | a__sqr(mark(X)) | (11) |
a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (9) |
mark(dbl(X)) | → | a__dbl(mark(X)) | (13) |
a__add(0,X) | → | mark(X) | (6) |
a__sqr(0) | → | 0 | (2) |
a__sqr#(s(X)) | → | a__dbl#(mark(X)) | (35) |
a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (51) |
a__add#(s(X),Y) | → | mark#(Y) | (50) |
mark#(first(X1,X2)) | → | mark#(X2) | (33) |
a__add#(0,X) | → | mark#(X) | (48) |
mark#(dbl(X)) | → | mark#(X) | (47) |
a__sqr#(s(X)) | → | mark#(X) | (31) |
mark#(terms(X)) | → | a__terms#(mark(X)) | (46) |
mark#(recip(X)) | → | mark#(X) | (45) |
a__sqr#(s(X)) | → | mark#(X) | (31) |
mark#(first(X1,X2)) | → | mark#(X1) | (30) |
mark#(add(X1,X2)) | → | mark#(X2) | (29) |
a__dbl#(s(X)) | → | mark#(X) | (44) |
mark#(terms(X)) | → | mark#(X) | (43) |
mark#(sqr(X)) | → | mark#(X) | (27) |
a__terms#(N) | → | a__sqr#(mark(N)) | (42) |
a__terms#(N) | → | mark#(N) | (40) |
mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (38) |
mark#(cons(X1,X2)) | → | mark#(X1) | (26) |
mark#(dbl(X)) | → | a__dbl#(mark(X)) | (25) |
The dependency pairs are split into 2 components.
a__dbl#(s(X)) | → | a__dbl#(mark(X)) | (52) |
π(a__terms#) | = | 1 |
π(a__dbl#) | = | 1 |
π(mark) | = | 1 |
π(a__sqr#) | = | 1 |
prec(s) | = | 0 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(a__first#) | = | 0 | status(a__first#) | = | [2, 1] | list-extension(a__first#) | = | Lex | ||
prec(recip) | = | 0 | status(recip) | = | [1] | list-extension(recip) | = | Lex | ||
prec(dbl) | = | 1 | status(dbl) | = | [1] | list-extension(dbl) | = | Lex | ||
prec(a__add#) | = | 0 | status(a__add#) | = | [1, 2] | list-extension(a__add#) | = | Lex | ||
prec(a__add) | = | 1 | status(a__add) | = | [2, 1] | list-extension(a__add) | = | Lex | ||
prec(a__dbl) | = | 1 | status(a__dbl) | = | [1] | list-extension(a__dbl) | = | Lex | ||
prec(a__sqr) | = | 2 | status(a__sqr) | = | [1] | list-extension(a__sqr) | = | Lex | ||
prec(mark#) | = | 0 | status(mark#) | = | [] | list-extension(mark#) | = | Lex | ||
prec(0) | = | 3 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(nil) | = | 4 | status(nil) | = | [] | list-extension(nil) | = | Lex | ||
prec(first) | = | 4 | status(first) | = | [] | list-extension(first) | = | Lex | ||
prec(a__first) | = | 4 | status(a__first) | = | [] | list-extension(a__first) | = | Lex | ||
prec(cons) | = | 0 | status(cons) | = | [] | list-extension(cons) | = | Lex | ||
prec(add) | = | 1 | status(add) | = | [2, 1] | list-extension(add) | = | Lex | ||
prec(sqr) | = | 2 | status(sqr) | = | [1] | list-extension(sqr) | = | Lex | ||
prec(a__terms) | = | 3 | status(a__terms) | = | [] | list-extension(a__terms) | = | Lex | ||
prec(terms) | = | 3 | status(terms) | = | [] | list-extension(terms) | = | Lex |
[s(x1)] | = | x1 + 0 |
[a__first#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[recip(x1)] | = | x1 + 0 |
[dbl(x1)] | = | x1 + 0 |
[a__add#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[a__add(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[a__dbl(x1)] | = | x1 + 0 |
[a__sqr(x1)] | = | x1 + 0 |
[mark#(x1)] | = | 0 |
[0] | = | 0 |
[nil] | = | 0 |
[first(x1, x2)] | = | max(0) |
[a__first(x1, x2)] | = | max(0) |
[cons(x1, x2)] | = | max(0) |
[add(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[sqr(x1)] | = | x1 + 0 |
[a__terms(x1)] | = | 0 |
[terms(x1)] | = | 0 |
mark(0) | → | 0 | (18) |
a__dbl(0) | → | 0 | (4) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (15) |
a__first(0,X) | → | nil | (8) |
a__terms(N) | → | cons(recip(a__sqr(mark(N))),terms(s(N))) | (1) |
a__sqr(s(X)) | → | s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) | (3) |
mark(recip(X)) | → | recip(mark(X)) | (16) |
a__sqr(X) | → | sqr(X) | (21) |
mark(nil) | → | nil | (19) |
mark(s(X)) | → | s(mark(X)) | (17) |
a__add(X1,X2) | → | add(X1,X2) | (22) |
a__dbl(s(X)) | → | s(s(a__dbl(mark(X)))) | (5) |
mark(terms(X)) | → | a__terms(mark(X)) | (10) |
a__add(s(X),Y) | → | s(a__add(mark(X),mark(Y))) | (7) |
a__terms(X) | → | terms(X) | (20) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (14) |
mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (12) |
a__dbl(X) | → | dbl(X) | (23) |
a__first(X1,X2) | → | first(X1,X2) | (24) |
mark(sqr(X)) | → | a__sqr(mark(X)) | (11) |
a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (9) |
mark(dbl(X)) | → | a__dbl(mark(X)) | (13) |
a__add(0,X) | → | mark(X) | (6) |
a__sqr(0) | → | 0 | (2) |
a__dbl#(s(X)) | → | a__dbl#(mark(X)) | (52) |
The dependency pairs are split into 0 components.
a__sqr#(s(X)) | → | a__sqr#(mark(X)) | (32) |
a__sqr#(s(X)) | → | a__add#(a__sqr(mark(X)),a__dbl(mark(X))) | (41) |
mark#(s(X)) | → | mark#(X) | (36) |
a__add#(s(X),Y) | → | mark#(X) | (34) |
a__add#(s(X),Y) | → | a__add#(mark(X),mark(Y)) | (28) |
mark#(add(X1,X2)) | → | mark#(X1) | (49) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (37) |
mark#(sqr(X)) | → | a__sqr#(mark(X)) | (39) |
π(a__add#) | = | 1 |
π(a__terms#) | = | 1 |
π(mark#) | = | 1 |
π(mark) | = | 1 |
prec(s) | = | 0 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(a__first#) | = | 0 | status(a__first#) | = | [2, 1] | list-extension(a__first#) | = | Lex | ||
prec(recip) | = | 0 | status(recip) | = | [1] | list-extension(recip) | = | Lex | ||
prec(dbl) | = | 1 | status(dbl) | = | [1] | list-extension(dbl) | = | Lex | ||
prec(a__add) | = | 1 | status(a__add) | = | [2, 1] | list-extension(a__add) | = | Lex | ||
prec(a__dbl) | = | 1 | status(a__dbl) | = | [1] | list-extension(a__dbl) | = | Lex | ||
prec(a__sqr) | = | 2 | status(a__sqr) | = | [1] | list-extension(a__sqr) | = | Lex | ||
prec(0) | = | 3 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(nil) | = | 4 | status(nil) | = | [] | list-extension(nil) | = | Lex | ||
prec(a__dbl#) | = | 0 | status(a__dbl#) | = | [] | list-extension(a__dbl#) | = | Lex | ||
prec(first) | = | 4 | status(first) | = | [] | list-extension(first) | = | Lex | ||
prec(a__first) | = | 4 | status(a__first) | = | [] | list-extension(a__first) | = | Lex | ||
prec(cons) | = | 0 | status(cons) | = | [] | list-extension(cons) | = | Lex | ||
prec(add) | = | 1 | status(add) | = | [2, 1] | list-extension(add) | = | Lex | ||
prec(sqr) | = | 2 | status(sqr) | = | [1] | list-extension(sqr) | = | Lex | ||
prec(a__terms) | = | 3 | status(a__terms) | = | [] | list-extension(a__terms) | = | Lex | ||
prec(a__sqr#) | = | 2 | status(a__sqr#) | = | [1] | list-extension(a__sqr#) | = | Lex | ||
prec(terms) | = | 3 | status(terms) | = | [] | list-extension(terms) | = | Lex |
[s(x1)] | = | x1 + 0 |
[a__first#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[recip(x1)] | = | x1 + 0 |
[dbl(x1)] | = | x1 + 0 |
[a__add(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[a__dbl(x1)] | = | x1 + 0 |
[a__sqr(x1)] | = | x1 + 0 |
[0] | = | 0 |
[nil] | = | 0 |
[a__dbl#(x1)] | = | 0 |
[first(x1, x2)] | = | max(0) |
[a__first(x1, x2)] | = | max(0) |
[cons(x1, x2)] | = | max(0) |
[add(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[sqr(x1)] | = | x1 + 0 |
[a__terms(x1)] | = | 0 |
[a__sqr#(x1)] | = | x1 + 0 |
[terms(x1)] | = | 0 |
mark(0) | → | 0 | (18) |
a__dbl(0) | → | 0 | (4) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (15) |
a__first(0,X) | → | nil | (8) |
a__terms(N) | → | cons(recip(a__sqr(mark(N))),terms(s(N))) | (1) |
a__sqr(s(X)) | → | s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) | (3) |
mark(recip(X)) | → | recip(mark(X)) | (16) |
a__sqr(X) | → | sqr(X) | (21) |
mark(nil) | → | nil | (19) |
mark(s(X)) | → | s(mark(X)) | (17) |
a__add(X1,X2) | → | add(X1,X2) | (22) |
a__dbl(s(X)) | → | s(s(a__dbl(mark(X)))) | (5) |
mark(terms(X)) | → | a__terms(mark(X)) | (10) |
a__add(s(X),Y) | → | s(a__add(mark(X),mark(Y))) | (7) |
a__terms(X) | → | terms(X) | (20) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (14) |
mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (12) |
a__dbl(X) | → | dbl(X) | (23) |
a__first(X1,X2) | → | first(X1,X2) | (24) |
mark(sqr(X)) | → | a__sqr(mark(X)) | (11) |
a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (9) |
mark(dbl(X)) | → | a__dbl(mark(X)) | (13) |
a__add(0,X) | → | mark(X) | (6) |
a__sqr(0) | → | 0 | (2) |
a__sqr#(s(X)) | → | a__sqr#(mark(X)) | (32) |
a__sqr#(s(X)) | → | a__add#(a__sqr(mark(X)),a__dbl(mark(X))) | (41) |
mark#(s(X)) | → | mark#(X) | (36) |
a__add#(s(X),Y) | → | mark#(X) | (34) |
a__add#(s(X),Y) | → | a__add#(mark(X),mark(Y)) | (28) |
mark#(add(X1,X2)) | → | mark#(X1) | (49) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (37) |
The dependency pairs are split into 0 components.