The rewrite relation of the following TRS is considered.
terms(N) | → | cons(recip(sqr(N)),n__terms(n__s(N))) | (1) |
sqr(0) | → | 0 | (2) |
sqr(s(X)) | → | s(add(sqr(X),dbl(X))) | (3) |
dbl(0) | → | 0 | (4) |
dbl(s(X)) | → | s(s(dbl(X))) | (5) |
add(0,X) | → | X | (6) |
add(s(X),Y) | → | s(add(X,Y)) | (7) |
first(0,X) | → | nil | (8) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (9) |
half(0) | → | 0 | (10) |
half(s(0)) | → | 0 | (11) |
half(s(s(X))) | → | s(half(X)) | (12) |
half(dbl(X)) | → | X | (13) |
terms(X) | → | n__terms(X) | (14) |
s(X) | → | n__s(X) | (15) |
first(X1,X2) | → | n__first(X1,X2) | (16) |
activate(n__terms(X)) | → | terms(activate(X)) | (17) |
activate(n__s(X)) | → | s(activate(X)) | (18) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (19) |
activate(X) | → | X | (20) |
half#(s(s(X))) | → | s#(half(X)) | (21) |
dbl#(s(X)) | → | s#(s(dbl(X))) | (22) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (23) |
activate#(n__terms(X)) | → | terms#(activate(X)) | (24) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (25) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (26) |
sqr#(s(X)) | → | s#(add(sqr(X),dbl(X))) | (27) |
activate#(n__s(X)) | → | activate#(X) | (28) |
dbl#(s(X)) | → | s#(dbl(X)) | (29) |
activate#(n__terms(X)) | → | activate#(X) | (30) |
add#(s(X),Y) | → | add#(X,Y) | (31) |
sqr#(s(X)) | → | dbl#(X) | (32) |
activate#(n__s(X)) | → | s#(activate(X)) | (33) |
sqr#(s(X)) | → | sqr#(X) | (34) |
half#(s(s(X))) | → | half#(X) | (35) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (36) |
dbl#(s(X)) | → | dbl#(X) | (37) |
sqr#(s(X)) | → | add#(sqr(X),dbl(X)) | (38) |
add#(s(X),Y) | → | s#(add(X,Y)) | (39) |
terms#(N) | → | sqr#(N) | (40) |
The dependency pairs are split into 5 components.
half#(s(s(X))) | → | half#(X) | (35) |
[s(x1)] | = | x1 + 1 |
[n__first(x1, x2)] | = | 0 |
[recip(x1)] | = | 0 |
[activate(x1)] | = | 0 |
[dbl(x1)] | = | 0 |
[dbl#(x1)] | = | 0 |
[terms#(x1)] | = | 0 |
[activate#(x1)] | = | 0 |
[half#(x1)] | = | x1 + 0 |
[half(x1)] | = | 0 |
[n__s(x1)] | = | 0 |
[sqr#(x1)] | = | 0 |
[0] | = | 0 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | 0 |
[nil] | = | 0 |
[first(x1, x2)] | = | 0 |
[n__terms(x1)] | = | 0 |
[cons(x1, x2)] | = | 0 |
[add#(x1, x2)] | = | 0 |
[add(x1, x2)] | = | 0 |
[sqr(x1)] | = | 0 |
[terms(x1)] | = | 0 |
half#(s(s(X))) | → | half#(X) | (35) |
The dependency pairs are split into 0 components.
activate#(n__first(X1,X2)) | → | activate#(X1) | (26) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (36) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (25) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (23) |
activate#(n__terms(X)) | → | activate#(X) | (30) |
activate#(n__s(X)) | → | activate#(X) | (28) |
[s(x1)] | = | x1 + 0 |
[n__first(x1, x2)] | = | x1 + x2 + 15923 |
[recip(x1)] | = | x1 + 0 |
[activate(x1)] | = | x1 + 0 |
[dbl(x1)] | = | x1 + 20538 |
[dbl#(x1)] | = | 0 |
[terms#(x1)] | = | 0 |
[activate#(x1)] | = | x1 + 0 |
[half#(x1)] | = | 0 |
[half(x1)] | = | 0 |
[n__s(x1)] | = | x1 + 0 |
[sqr#(x1)] | = | 0 |
[0] | = | 0 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | x2 + 15922 |
[nil] | = | 0 |
[first(x1, x2)] | = | x1 + x2 + 15923 |
[n__terms(x1)] | = | x1 + 5854 |
[cons(x1, x2)] | = | x2 + 0 |
[add#(x1, x2)] | = | 0 |
[add(x1, x2)] | = | x2 + 8987 |
[sqr(x1)] | = | 1424 |
[terms(x1)] | = | x1 + 5854 |
activate(n__s(X)) | → | s(activate(X)) | (18) |
dbl(0) | → | 0 | (4) |
s(X) | → | n__s(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(n__s(N))) | (1) |
first(X1,X2) | → | n__first(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (19) |
activate(n__terms(X)) | → | terms(activate(X)) | (17) |
dbl(s(X)) | → | s(s(dbl(X))) | (5) |
activate(X) | → | X | (20) |
terms(X) | → | n__terms(X) | (14) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (9) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (26) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (36) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (25) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (23) |
activate#(n__terms(X)) | → | activate#(X) | (30) |
The dependency pairs are split into 1 component.
activate#(n__s(X)) | → | activate#(X) | (28) |
[s(x1)] | = | x1 + 1 |
[n__first(x1, x2)] | = | x1 + 1 |
[recip(x1)] | = | x1 + 0 |
[activate(x1)] | = | x1 + 1 |
[dbl(x1)] | = | 33602 |
[dbl#(x1)] | = | 0 |
[terms#(x1)] | = | 0 |
[activate#(x1)] | = | x1 + 0 |
[half#(x1)] | = | 0 |
[half(x1)] | = | 0 |
[n__s(x1)] | = | x1 + 1 |
[sqr#(x1)] | = | 0 |
[0] | = | 1 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | x2 + 15922 |
[nil] | = | 0 |
[first(x1, x2)] | = | x1 + 1 |
[n__terms(x1)] | = | 38607 |
[cons(x1, x2)] | = | x2 + 0 |
[add#(x1, x2)] | = | 0 |
[add(x1, x2)] | = | x1 + x2 + 4687 |
[sqr(x1)] | = | 1424 |
[terms(x1)] | = | 38608 |
activate(n__s(X)) | → | s(activate(X)) | (18) |
s(X) | → | n__s(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(n__s(N))) | (1) |
first(X1,X2) | → | n__first(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (19) |
activate(n__terms(X)) | → | terms(activate(X)) | (17) |
activate(X) | → | X | (20) |
terms(X) | → | n__terms(X) | (14) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (9) |
activate#(n__s(X)) | → | activate#(X) | (28) |
The dependency pairs are split into 0 components.
sqr#(s(X)) | → | sqr#(X) | (34) |
[s(x1)] | = | x1 + 1 |
[n__first(x1, x2)] | = | x1 + 1 |
[recip(x1)] | = | x1 + 0 |
[activate(x1)] | = | x1 + 35495 |
[dbl(x1)] | = | 1 |
[dbl#(x1)] | = | 0 |
[terms#(x1)] | = | 0 |
[activate#(x1)] | = | 0 |
[half#(x1)] | = | 0 |
[half(x1)] | = | 0 |
[n__s(x1)] | = | x1 + 1 |
[sqr#(x1)] | = | x1 + 0 |
[0] | = | 1 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | x2 + 15922 |
[nil] | = | 0 |
[first(x1, x2)] | = | x1 + 1 |
[n__terms(x1)] | = | 1 |
[cons(x1, x2)] | = | x2 + 0 |
[add#(x1, x2)] | = | 0 |
[add(x1, x2)] | = | x1 + x2 + 4687 |
[sqr(x1)] | = | 1 |
[terms(x1)] | = | 1 |
activate(n__s(X)) | → | s(activate(X)) | (18) |
s(X) | → | n__s(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(n__s(N))) | (1) |
first(X1,X2) | → | n__first(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (19) |
activate(n__terms(X)) | → | terms(activate(X)) | (17) |
activate(X) | → | X | (20) |
terms(X) | → | n__terms(X) | (14) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (9) |
sqr#(s(X)) | → | sqr#(X) | (34) |
The dependency pairs are split into 0 components.
add#(s(X),Y) | → | add#(X,Y) | (31) |
[s(x1)] | = | x1 + 1 |
[n__first(x1, x2)] | = | x1 + 1 |
[recip(x1)] | = | x1 + 0 |
[activate(x1)] | = | x1 + 1 |
[dbl(x1)] | = | 13485 |
[dbl#(x1)] | = | 0 |
[terms#(x1)] | = | 0 |
[activate#(x1)] | = | 0 |
[half#(x1)] | = | 0 |
[half(x1)] | = | 0 |
[n__s(x1)] | = | x1 + 1 |
[sqr#(x1)] | = | 0 |
[0] | = | 1 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | x2 + 15922 |
[nil] | = | 0 |
[first(x1, x2)] | = | x1 + 1 |
[n__terms(x1)] | = | 1 |
[cons(x1, x2)] | = | x2 + 0 |
[add#(x1, x2)] | = | x1 + 0 |
[add(x1, x2)] | = | x1 + x2 + 41990 |
[sqr(x1)] | = | 1 |
[terms(x1)] | = | 1 |
activate(n__s(X)) | → | s(activate(X)) | (18) |
s(X) | → | n__s(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(n__s(N))) | (1) |
first(X1,X2) | → | n__first(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (19) |
activate(n__terms(X)) | → | terms(activate(X)) | (17) |
activate(X) | → | X | (20) |
terms(X) | → | n__terms(X) | (14) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (9) |
add#(s(X),Y) | → | add#(X,Y) | (31) |
The dependency pairs are split into 0 components.
dbl#(s(X)) | → | dbl#(X) | (37) |
[s(x1)] | = | x1 + 1 |
[n__first(x1, x2)] | = | x1 + 1 |
[recip(x1)] | = | x1 + 0 |
[activate(x1)] | = | x1 + 1 |
[dbl(x1)] | = | 1 |
[dbl#(x1)] | = | x1 + 0 |
[terms#(x1)] | = | 0 |
[activate#(x1)] | = | 0 |
[half#(x1)] | = | 0 |
[half(x1)] | = | 0 |
[n__s(x1)] | = | x1 + 1 |
[sqr#(x1)] | = | 0 |
[0] | = | 1 |
[s#(x1)] | = | 0 |
[first#(x1, x2)] | = | x2 + 15922 |
[nil] | = | 0 |
[first(x1, x2)] | = | x1 + 1 |
[n__terms(x1)] | = | 1 |
[cons(x1, x2)] | = | x2 + 0 |
[add#(x1, x2)] | = | 0 |
[add(x1, x2)] | = | x1 + x2 + 1 |
[sqr(x1)] | = | 1 |
[terms(x1)] | = | 1 |
activate(n__s(X)) | → | s(activate(X)) | (18) |
s(X) | → | n__s(X) | (15) |
first(0,X) | → | nil | (8) |
terms(N) | → | cons(recip(sqr(N)),n__terms(n__s(N))) | (1) |
first(X1,X2) | → | n__first(X1,X2) | (16) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (19) |
activate(n__terms(X)) | → | terms(activate(X)) | (17) |
activate(X) | → | X | (20) |
terms(X) | → | n__terms(X) | (14) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (9) |
dbl#(s(X)) | → | dbl#(X) | (37) |
The dependency pairs are split into 0 components.