The rewrite relation of the following TRS is considered.
and(true,X) | → | activate(X) | (1) |
and(false,Y) | → | false | (2) |
if(true,X,Y) | → | activate(X) | (3) |
if(false,X,Y) | → | activate(Y) | (4) |
add(0,X) | → | activate(X) | (5) |
add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
first(0,X) | → | nil | (7) |
first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
add(X1,X2) | → | n__add(X1,X2) | (10) |
first(X1,X2) | → | n__first(X1,X2) | (11) |
from(X) | → | n__from(X) | (12) |
s(X) | → | n__s(X) | (13) |
activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
activate(n__from(X)) | → | from(X) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
and#(true,X) | → | activate#(X) | (19) |
if#(true,X,Y) | → | activate#(X) | (20) |
if#(false,X,Y) | → | activate#(Y) | (21) |
add#(0,X) | → | activate#(X) | (22) |
add#(s(X),Y) | → | s#(n__add(activate(X),activate(Y))) | (23) |
add#(s(X),Y) | → | activate#(X) | (24) |
add#(s(X),Y) | → | activate#(Y) | (25) |
first#(s(X),cons(Y,Z)) | → | activate#(Y) | (26) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (27) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (28) |
from#(X) | → | activate#(X) | (29) |
activate#(n__add(X1,X2)) | → | add#(activate(X1),X2) | (30) |
activate#(n__add(X1,X2)) | → | activate#(X1) | (31) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (32) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (33) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (34) |
activate#(n__from(X)) | → | from#(X) | (35) |
activate#(n__s(X)) | → | s#(X) | (36) |
The dependency pairs are split into 1 component.
activate#(n__add(X1,X2)) | → | add#(activate(X1),X2) | (30) |
add#(0,X) | → | activate#(X) | (22) |
activate#(n__add(X1,X2)) | → | activate#(X1) | (31) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (32) |
first#(s(X),cons(Y,Z)) | → | activate#(Y) | (26) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (33) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (34) |
activate#(n__from(X)) | → | from#(X) | (35) |
from#(X) | → | activate#(X) | (29) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (27) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (28) |
add#(s(X),Y) | → | activate#(X) | (24) |
add#(s(X),Y) | → | activate#(Y) | (25) |
[activate#(x1)] | = |
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[n__add(x1, x2)] | = |
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[add#(x1, x2)] | = |
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[activate(x1)] | = |
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[0] | = |
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[n__first(x1, x2)] | = |
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[first#(x1, x2)] | = |
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[s(x1)] | = |
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[cons(x1, x2)] | = |
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[n__from(x1)] | = |
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[from#(x1)] | = |
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[add(x1, x2)] | = |
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[first(x1, x2)] | = |
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[from(x1)] | = |
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[n__s(x1)] | = |
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[nil] | = |
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activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
add(0,X) | → | activate(X) | (5) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
activate(n__from(X)) | → | from(X) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
add(X1,X2) | → | n__add(X1,X2) | (10) |
first(0,X) | → | nil | (7) |
first(X1,X2) | → | n__first(X1,X2) | (11) |
from(X) | → | n__from(X) | (12) |
from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
s(X) | → | n__s(X) | (13) |
add#(0,X) | → | activate#(X) | (22) |
[activate#(x1)] | = |
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[n__add(x1, x2)] | = |
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[add#(x1, x2)] | = |
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[activate(x1)] | = |
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[n__first(x1, x2)] | = |
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[first#(x1, x2)] | = |
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[s(x1)] | = |
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[cons(x1, x2)] | = |
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[n__from(x1)] | = |
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[from#(x1)] | = |
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[add(x1, x2)] | = |
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[0] | = |
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[first(x1, x2)] | = |
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[from(x1)] | = |
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[n__s(x1)] | = |
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[nil] | = |
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activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
add(0,X) | → | activate(X) | (5) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
activate(n__from(X)) | → | from(X) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
add(X1,X2) | → | n__add(X1,X2) | (10) |
first(0,X) | → | nil | (7) |
first(X1,X2) | → | n__first(X1,X2) | (11) |
from(X) | → | n__from(X) | (12) |
from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
s(X) | → | n__s(X) | (13) |
activate#(n__add(X1,X2)) | → | activate#(X1) | (31) |
from#(X) | → | activate#(X) | (29) |
add#(s(X),Y) | → | activate#(X) | (24) |
The dependency pairs are split into 1 component.
add#(s(X),Y) | → | activate#(Y) | (25) |
activate#(n__add(X1,X2)) | → | add#(activate(X1),X2) | (30) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (32) |
first#(s(X),cons(Y,Z)) | → | activate#(Y) | (26) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (33) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (34) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (27) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (28) |
[add#(x1, x2)] | = | 1/2 + 0 · x1 + 1/2 · x2 |
[s(x1)] | = | 0 + 1/2 · x1 |
[activate#(x1)] | = | 0 + 1/2 · x1 |
[n__add(x1, x2)] | = | 1 + 0 · x1 + 1 · x2 |
[activate(x1)] | = | 0 + 1 · x1 |
[n__first(x1, x2)] | = | 0 + 2 · x1 + 2 · x2 |
[first#(x1, x2)] | = | 0 + 1 · x1 + 1 · x2 |
[cons(x1, x2)] | = | 0 + 1/2 · x1 + 1/2 · x2 |
[add(x1, x2)] | = | 1 + 0 · x1 + 1 · x2 |
[0] | = | 1/4 |
[first(x1, x2)] | = | 0 + 2 · x1 + 2 · x2 |
[n__from(x1)] | = | 1/2 + 4 · x1 |
[from(x1)] | = | 1/2 + 4 · x1 |
[n__s(x1)] | = | 0 + 1/2 · x1 |
[nil] | = | 1/4 |
activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
add(0,X) | → | activate(X) | (5) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
activate(n__from(X)) | → | from(X) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
add(X1,X2) | → | n__add(X1,X2) | (10) |
first(0,X) | → | nil | (7) |
first(X1,X2) | → | n__first(X1,X2) | (11) |
from(X) | → | n__from(X) | (12) |
from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
s(X) | → | n__s(X) | (13) |
add#(s(X),Y) | → | activate#(Y) | (25) |
The dependency pairs are split into 1 component.
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (32) |
first#(s(X),cons(Y,Z)) | → | activate#(Y) | (26) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (33) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (34) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (27) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (28) |
[activate#(x1)] | = | 2 + 1/4 · x1 |
[n__first(x1, x2)] | = | 0 + 4 · x1 + 4 · x2 |
[first#(x1, x2)] | = | 2 + 1 · x1 + 1 · x2 |
[activate(x1)] | = | 0 + 1 · x1 |
[s(x1)] | = | 2 + 1/4 · x1 |
[cons(x1, x2)] | = | 1 + 1/4 · x1 + 1/4 · x2 |
[n__add(x1, x2)] | = | 4 + 0 · x1 + 2 · x2 |
[add(x1, x2)] | = | 4 + 0 · x1 + 2 · x2 |
[0] | = | 4 |
[first(x1, x2)] | = | 0 + 4 · x1 + 4 · x2 |
[n__from(x1)] | = | 4 + 1/2 · x1 |
[from(x1)] | = | 4 + 1/2 · x1 |
[n__s(x1)] | = | 2 + 1/4 · x1 |
[nil] | = | 1 |
activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
add(0,X) | → | activate(X) | (5) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
activate(n__from(X)) | → | from(X) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
add(X1,X2) | → | n__add(X1,X2) | (10) |
first(0,X) | → | nil | (7) |
first(X1,X2) | → | n__first(X1,X2) | (11) |
from(X) | → | n__from(X) | (12) |
from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
s(X) | → | n__s(X) | (13) |
first#(s(X),cons(Y,Z)) | → | activate#(Y) | (26) |
first#(s(X),cons(Y,Z)) | → | activate#(X) | (27) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (28) |
The dependency pairs are split into 1 component.
activate#(n__first(X1,X2)) | → | activate#(X2) | (34) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (33) |
[n__first(x1, x2)] | = | 1 · x1 + 1 · x2 |
[activate#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
activate#(n__first(X1,X2)) | → | activate#(X2) | (34) |
1 | > | 1 | |
activate#(n__first(X1,X2)) | → | activate#(X1) | (33) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.