The rewrite relation of the following TRS is considered.
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
sqr(0) | → | 0 | (2) |
sqr(s(X)) | → | s(n__add(sqr(activate(X)),dbl(activate(X)))) | (3) |
dbl(0) | → | 0 | (4) |
dbl(s(X)) | → | s(n__s(n__dbl(activate(X)))) | (5) |
add(0,X) | → | X | (6) |
add(s(X),Y) | → | s(n__add(activate(X),Y)) | (7) |
first(0,X) | → | nil | (8) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(activate(X),activate(Z))) | (9) |
terms(X) | → | n__terms(X) | (10) |
add(X1,X2) | → | n__add(X1,X2) | (11) |
s(X) | → | n__s(X) | (12) |
dbl(X) | → | n__dbl(X) | (13) |
first(X1,X2) | → | n__first(X1,X2) | (14) |
activate(n__terms(X)) | → | terms(X) | (15) |
activate(n__add(X1,X2)) | → | add(X1,X2) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(n__dbl(X)) | → | dbl(X) | (18) |
activate(n__first(X1,X2)) | → | first(X1,X2) | (19) |
activate(X) | → | X | (20) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (21) |
dbl#(s(X)) | → | activate#(X) | (22) |
sqr#(s(X)) | → | dbl#(activate(X)) | (23) |
activate#(n__add(X1,X2)) | → | add#(X1,X2) | (24) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
sqr#(s(X)) | → | activate#(X) | (26) |
sqr#(s(X)) | → | activate#(X) | (26) |
activate#(n__s(X)) | → | s#(X) | (27) |
sqr#(s(X)) | → | s#(n__add(sqr(activate(X)),dbl(activate(X)))) | (28) |
dbl#(s(X)) | → | s#(n__s(n__dbl(activate(X)))) | (29) |
activate#(n__terms(X)) | → | terms#(X) | (30) |
terms#(N) | → | s#(N) | (31) |
add#(s(X),Y) | → | s#(n__add(activate(X),Y)) | (32) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (33) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (34) |
terms#(N) | → | sqr#(N) | (35) |
add#(s(X),Y) | → | activate#(X) | (36) |
activate#(n__dbl(X)) | → | dbl#(X) | (37) |
The dependency pairs are split into 1 component.
activate#(n__dbl(X)) | → | dbl#(X) | (37) |
add#(s(X),Y) | → | activate#(X) | (36) |
sqr#(s(X)) | → | activate#(X) | (26) |
terms#(N) | → | sqr#(N) | (35) |
sqr#(s(X)) | → | activate#(X) | (26) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (34) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (33) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
activate#(n__add(X1,X2)) | → | add#(X1,X2) | (24) |
sqr#(s(X)) | → | dbl#(activate(X)) | (23) |
activate#(n__terms(X)) | → | terms#(X) | (30) |
dbl#(s(X)) | → | activate#(X) | (22) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (21) |
[s(x1)] | = | x1 + 1 |
[n__first(x1, x2)] | = | x1 + x2 + 1324 |
[recip(x1)] | = | x1 + 0 |
[activate(x1)] | = | x1 + 1 |
[dbl(x1)] | = | x1 + 2242 |
[dbl#(x1)] | = | x1 + 2240 |
[terms#(x1)] | = | x1 + 2242 |
[activate#(x1)] | = | x1 + 0 |
[n__add(x1, x2)] | = | x1 + x2 + 31328 |
[n__s(x1)] | = | x1 + 0 |
[sqr#(x1)] | = | x1 + 2241 |
[n__dbl(x1)] | = | x1 + 2241 |
[0] | = | 1 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | x1 + x2 + 0 |
[nil] | = | 0 |
[first(x1, x2)] | = | x1 + x2 + 1325 |
[n__terms(x1)] | = | x1 + 2243 |
[cons(x1, x2)] | = | x2 + 0 |
[add#(x1, x2)] | = | x1 + 0 |
[add(x1, x2)] | = | x1 + x2 + 31329 |
[sqr(x1)] | = | x1 + 42004 |
[terms(x1)] | = | x1 + 2244 |
activate(n__dbl(X)) | → | dbl(X) | (18) |
dbl(0) | → | 0 | (4) |
activate(n__terms(X)) | → | terms(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
activate(n__add(X1,X2)) | → | add(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(X1,X2) | (19) |
activate(n__s(X)) | → | s(X) | (17) |
dbl(s(X)) | → | s(n__s(n__dbl(activate(X)))) | (5) |
terms(X) | → | n__terms(X) | (10) |
add(s(X),Y) | → | s(n__add(activate(X),Y)) | (7) |
activate(X) | → | X | (20) |
first(X1,X2) | → | n__first(X1,X2) | (14) |
s(X) | → | n__s(X) | (12) |
add(X1,X2) | → | n__add(X1,X2) | (11) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(activate(X),activate(Z))) | (9) |
dbl(X) | → | n__dbl(X) | (13) |
add(0,X) | → | X | (6) |
activate#(n__dbl(X)) | → | dbl#(X) | (37) |
add#(s(X),Y) | → | activate#(X) | (36) |
sqr#(s(X)) | → | activate#(X) | (26) |
terms#(N) | → | sqr#(N) | (35) |
sqr#(s(X)) | → | activate#(X) | (26) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (34) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (33) |
activate#(n__add(X1,X2)) | → | add#(X1,X2) | (24) |
sqr#(s(X)) | → | dbl#(activate(X)) | (23) |
activate#(n__terms(X)) | → | terms#(X) | (30) |
dbl#(s(X)) | → | activate#(X) | (22) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (21) |
The dependency pairs are split into 1 component.
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
π(n__first) | = | 2 |
π(dbl#) | = | 1 |
π(activate#) | = | 1 |
π(sqr#) | = | 1 |
π(s#) | = | 1 |
π(first#) | = | 2 |
π(cons) | = | 2 |
π(add#) | = | 2 |
prec(s) | = | 3 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(recip) | = | 2 | status(recip) | = | [] | list-extension(recip) | = | Lex | ||
prec(activate) | = | 1 | status(activate) | = | [1] | list-extension(activate) | = | Lex | ||
prec(dbl) | = | 4 | status(dbl) | = | [1] | list-extension(dbl) | = | Lex | ||
prec(terms#) | = | 0 | status(terms#) | = | [] | list-extension(terms#) | = | Lex | ||
prec(n__add) | = | 4 | status(n__add) | = | [2, 1] | list-extension(n__add) | = | Lex | ||
prec(n__s) | = | 3 | status(n__s) | = | [1] | list-extension(n__s) | = | Lex | ||
prec(n__dbl) | = | 4 | status(n__dbl) | = | [1] | list-extension(n__dbl) | = | Lex | ||
prec(0) | = | 0 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(nil) | = | 0 | status(nil) | = | [] | list-extension(nil) | = | Lex | ||
prec(first) | = | 1 | status(first) | = | [2] | list-extension(first) | = | Lex | ||
prec(n__terms) | = | 2 | status(n__terms) | = | [] | list-extension(n__terms) | = | Lex | ||
prec(add) | = | 4 | status(add) | = | [2, 1] | list-extension(add) | = | Lex | ||
prec(sqr) | = | 0 | status(sqr) | = | [] | list-extension(sqr) | = | Lex | ||
prec(terms) | = | 2 | status(terms) | = | [] | list-extension(terms) | = | Lex |
[s(x1)] | = | x1 + 0 |
[recip(x1)] | = | 0 |
[activate(x1)] | = | x1 + 0 |
[dbl(x1)] | = | x1 + 0 |
[terms#(x1)] | = | 0 |
[n__add(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[n__s(x1)] | = | x1 + 0 |
[n__dbl(x1)] | = | x1 + 0 |
[0] | = | 0 |
[nil] | = | 0 |
[first(x1, x2)] | = | max(x2 + 0, 0) |
[n__terms(x1)] | = | 0 |
[add(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[sqr(x1)] | = | 0 |
[terms(x1)] | = | 0 |
activate(n__dbl(X)) | → | dbl(X) | (18) |
dbl(0) | → | 0 | (4) |
activate(n__terms(X)) | → | terms(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
activate(n__add(X1,X2)) | → | add(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(X1,X2) | (19) |
activate(n__s(X)) | → | s(X) | (17) |
dbl(s(X)) | → | s(n__s(n__dbl(activate(X)))) | (5) |
terms(X) | → | n__terms(X) | (10) |
add(s(X),Y) | → | s(n__add(activate(X),Y)) | (7) |
activate(X) | → | X | (20) |
first(X1,X2) | → | n__first(X1,X2) | (14) |
s(X) | → | n__s(X) | (12) |
add(X1,X2) | → | n__add(X1,X2) | (11) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(activate(X),activate(Z))) | (9) |
dbl(X) | → | n__dbl(X) | (13) |
add(0,X) | → | X | (6) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
The dependency pairs are split into 0 components.