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) |
terms#(N) | → | s#(N) | (21) |
terms#(N) | → | sqr#(N) | (22) |
sqr#(s(X)) | → | dbl#(activate(X)) | (23) |
sqr#(s(X)) | → | activate#(X) | (24) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
sqr#(s(X)) | → | s#(n__add(sqr(activate(X)),dbl(activate(X)))) | (26) |
dbl#(s(X)) | → | activate#(X) | (27) |
dbl#(s(X)) | → | s#(n__s(n__dbl(activate(X)))) | (28) |
add#(s(X),Y) | → | activate#(X) | (29) |
add#(s(X),Y) | → | s#(n__add(activate(X),Y)) | (30) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (31) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (32) |
activate#(n__terms(X)) | → | terms#(X) | (33) |
activate#(n__add(X1,X2)) | → | add#(X1,X2) | (34) |
activate#(n__s(X)) | → | s#(X) | (35) |
activate#(n__dbl(X)) | → | dbl#(X) | (36) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (37) |
The dependency pairs are split into 1 component.
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (31) |
activate#(n__terms(X)) | → | terms#(X) | (33) |
terms#(N) | → | sqr#(N) | (22) |
sqr#(s(X)) | → | dbl#(activate(X)) | (23) |
dbl#(s(X)) | → | activate#(X) | (27) |
activate#(n__add(X1,X2)) | → | add#(X1,X2) | (34) |
add#(s(X),Y) | → | activate#(X) | (29) |
activate#(n__dbl(X)) | → | dbl#(X) | (36) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (37) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (32) |
sqr#(s(X)) | → | activate#(X) | (24) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
[sqr#(x1)] | = | 0 · x1 + 0 |
[n__terms(x1)] | = | 0 · x1 + 0 |
[n__first(x1, x2)] | = | 7 · x1 + 0 · x2 + 7 |
[dbl(x1)] | = | 0 · x1 + 5 |
[sqr(x1)] | = | 0 · x1 + 4 |
[s(x1)] | = | 0 · x1 + 0 |
[terms#(x1)] | = | 0 · x1 + 0 |
[add#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[dbl#(x1)] | = | 0 · x1 + -∞ |
[first#(x1, x2)] | = | 7 · x1 + 0 · x2 + -∞ |
[cons(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[n__dbl(x1)] | = | 0 · x1 + 5 |
[terms(x1)] | = | 0 · x1 + 0 |
[activate(x1)] | = | 0 · x1 + 0 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[nil] | = | 0 |
[add(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[n__s(x1)] | = | 0 · x1 + -∞ |
[n__add(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[first(x1, x2)] | = | 7 · x1 + 0 · x2 + 7 |
[0] | = | 0 |
[recip(x1)] | = | 0 · x1 + 0 |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
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#(X) | (32) |
[sqr#(x1)] | = | 0 · x1 + 4 |
[n__terms(x1)] | = | 4 · x1 + -∞ |
[n__first(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[dbl(x1)] | = | 1 · x1 + 5 |
[sqr(x1)] | = | 3 · x1 + 0 |
[s(x1)] | = | 0 · x1 + -∞ |
[terms#(x1)] | = | 3 · x1 + 4 |
[add#(x1, x2)] | = | 0 · x1 + -∞ · x2 + 4 |
[dbl#(x1)] | = | 0 · x1 + 4 |
[first#(x1, x2)] | = | 0 · x1 + 0 · x2 + 4 |
[cons(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[n__dbl(x1)] | = | 1 · x1 + 5 |
[terms(x1)] | = | 4 · x1 + -∞ |
[activate(x1)] | = | 0 · x1 + 0 |
[activate#(x1)] | = | 0 · x1 + 4 |
[nil] | = | 0 |
[add(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[n__s(x1)] | = | 0 · x1 + -∞ |
[n__add(x1, x2)] | = | 0 · x1 + 1 · x2 + -∞ |
[first(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[0] | = | 1 |
[recip(x1)] | = | 0 · x1 + 0 |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
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) |
activate#(n__dbl(X)) | → | dbl#(X) | (36) |
[sqr#(x1)] | = | 0 · x1 + -∞ |
[n__terms(x1)] | = | 0 · x1 + 4 |
[n__first(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[dbl(x1)] | = | 0 · x1 + -∞ |
[sqr(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[terms#(x1)] | = | 0 · x1 + -∞ |
[add#(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[dbl#(x1)] | = | 0 · x1 + 0 |
[first#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[cons(x1, x2)] | = | -∞ · x1 + 0 · x2 + 2 |
[n__dbl(x1)] | = | 0 · x1 + -∞ |
[terms(x1)] | = | 0 · x1 + 4 |
[activate(x1)] | = | 0 · x1 + 0 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[nil] | = | 0 |
[add(x1, x2)] | = | 1 · x1 + 0 · x2 + 6 |
[n__s(x1)] | = | 0 · x1 + -∞ |
[n__add(x1, x2)] | = | 1 · x1 + 0 · x2 + 6 |
[first(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[0] | = | 0 |
[recip(x1)] | = | 0 · x1 + 0 |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
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) |
add#(s(X),Y) | → | activate#(X) | (29) |
The dependency pairs are split into 1 component.
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (31) |
activate#(n__terms(X)) | → | terms#(X) | (33) |
terms#(N) | → | sqr#(N) | (22) |
sqr#(s(X)) | → | dbl#(activate(X)) | (23) |
dbl#(s(X)) | → | activate#(X) | (27) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (37) |
sqr#(s(X)) | → | activate#(X) | (24) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
[sqr#(x1)] | = | 0 · x1 + -∞ |
[n__terms(x1)] | = | 6 · x1 + 0 |
[n__first(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[dbl(x1)] | = | 0 · x1 + -∞ |
[sqr(x1)] | = | -∞ · x1 + 1 |
[s(x1)] | = | 0 · x1 + -∞ |
[terms#(x1)] | = | 6 · x1 + -∞ |
[dbl#(x1)] | = | 0 · x1 + -∞ |
[first#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[n__dbl(x1)] | = | 0 · x1 + -∞ |
[terms(x1)] | = | 6 · x1 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
[activate#(x1)] | = | 0 · x1 + -∞ |
[nil] | = | 0 |
[add(x1, x2)] | = | -∞ · x1 + 4 · x2 + -∞ |
[n__s(x1)] | = | 0 · x1 + -∞ |
[n__add(x1, x2)] | = | -∞ · x1 + 4 · x2 + -∞ |
[first(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[0] | = | 0 |
[recip(x1)] | = | -∞ · x1 + 0 |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
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) |
terms#(N) | → | sqr#(N) | (22) |
The dependency pairs are split into 2 components.
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
[sqr#(x1)] | = | 2 · x1 + 1 |
[n__terms(x1)] | = | 0 · x1 + 0 |
[n__first(x1, x2)] | = | 1 · x1 + 4 · x2 + 0 |
[dbl(x1)] | = | 4 · x1 + 6 |
[sqr(x1)] | = | 2 · x1 + 2 |
[s(x1)] | = | 2 · x1 + 2 |
[cons(x1, x2)] | = | 2 · x1 + 0 · x2 + 0 |
[n__dbl(x1)] | = | 2 · x1 + 5 |
[terms(x1)] | = | 0 · x1 + 0 |
[activate(x1)] | = | 2 · x1 + 0 |
[nil] | = | 0 |
[add(x1, x2)] | = | 2 · x1 + 2 · x2 + 0 |
[n__s(x1)] | = | 1 · x1 + 1 |
[n__add(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
[first(x1, x2)] | = | 1 · x1 + 7 · x2 + 0 |
[0] | = | 6 |
[recip(x1)] | = | 0 · x1 + 0 |
terms(N) | → | cons(recip(sqr(N)),n__terms(s(N))) | (1) |
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) |
sqr#(s(X)) | → | sqr#(activate(X)) | (25) |
There are no pairs anymore.
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (31) |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (37) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (31) |
2 | > | 1 | |
activate#(n__first(X1,X2)) | → | first#(X1,X2) | (37) |
1 | > | 2 | |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.