Quasi Reductive Proof
by ttt2
Input
The rewrite relation of the following conditional TRS is considered.
id(s(x)) |
→ |
id(id(z)) |
| id(x) ≈ y, id(y) ≈ z
|
id(0) |
→ |
0 |
Proof
1 Unraveling
To prove that the CTRS is quasi-reductive, we show termination of the following
unraveled system.
For |
id(s(x))id(id(z))id(x)yid(y)z we get |
|
id(s(x)) |
→ |
U(id(x),x) |
|
U(y,x) |
→ |
W(id(y)) |
|
W(z) |
→ |
id(id(z)) |
For |
id(0)0 we get |
|
id(0) |
→ |
0 |
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the rationals with delta = 1/64
[0] |
= |
|
[W(x1)] |
= |
· x1 +
|
[U(x1, x2)] |
= |
· x1 + · x2 +
|
[id(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
the
rules
U(y,x) |
→ |
W(id(y)) |
W(z) |
→ |
id(id(z)) |
id(0) |
→ |
0 |
remain.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the rationals with delta = 1/64
[0] |
= |
|
[W(x1)] |
= |
· x1 +
|
[U(x1, x2)] |
= |
· x1 + · x2 +
|
[id(x1)] |
= |
· x1 +
|
the
rules
W(z) |
→ |
id(id(z)) |
id(0) |
→ |
0 |
remain.
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the rationals with delta = 1/64
[0] |
= |
|
[W(x1)] |
= |
· x1 +
|
[id(x1)] |
= |
· x1 +
|
the
rule
remains.
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
[W(x1)] |
= |
15 · x1 +
-∞ |
[id(x1)] |
= |
0 · x1 +
-∞ |
all rules could be removed.
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
Tool configuration
ttt2
- version: 1.08
- strategy:
(var | con | (if srs then ((sleep -t 50?;rlab;(((( if srs then arctic -dim
1 -ib 4 -ob 5 else fail || if srs then arctic -dim 2 -ib 2 -ob 3 else fail || if
srs then arctic -dim 3 -ib 1 -ob 2 else fail || if srs then arctic -dim 3 -ib 2
-ob 2 else fail || matrix -dim 1 -ib 5 -ob 8 || matrix -dim 2 -ib 3 -ob 4 ||
matrix -dim 3 -ib 2 -ob 3 || matrix -dim 3 -ib 1 -ob 2 || matrix -dim 4 -ib 1
-ob 2 || matrix -dim 5 -ib 1 -ob 1 || fail)[5]*);(dp;edg[1.0]?;(sccs | (sc ||
sct || ur?;( matrix -dp -ur -dim 1 -ib 3 -ob 5 || matrix -dp -ur -dim 1 -ib 3
-ob 8 -rat 2 -db 1 || matrix -dp -ur -dim 1 -ib 4 -ob 10 -rat 4 -db 1 || matrix
-dp -ur -dim 2 -ib 2 -ob 3 || matrix -dp -ur -dim 2 -ib 3 -ob 4 -rat 2 -db 0 ||
matrix -dp -ur -dim 3 -ib 1 -ob 3 || matrix -dp -ur -dim 3 -ib 2 -ob 3 || matrix
-dp -ur -dim 3 -ib 2 -ob 3 -rat 2 -db 0 || matrix -dp -ur -dim 4 -ib 1 -ob 2 ||
lpo -ur -af || (arctic -dp -ur -dim 1 -ib 4 -ob 3[10] | arctic -dp -ur -dim 2
-ib 2 -ob 2[15] | arctic -dp -ur -dim 3 -ib 1 -ob 1[15] | arctic -dp -ur -dim 4
-ib 1 -ob 1[15] | fail) || (arctic -bz -dp -ur -dim 1 -ib 4 -ob 3[10] | arctic
-bz -dp -ur -dim 2 -ib 2 -ob 2[15] | arctic -bz -dp -ur -dim 3 -ib 1 -ob 1[15] |
arctic -bz -dp -ur -dim 4 -ib 1 -ob 1[15] | fail) || fail)) )*[59])) || ((
unfold || fail)*[7])!)[59])! || ((((( if srs then arctic -dim 1 -ib 4 -ob 5 else
fail || if srs then arctic -dim 2 -ib 2 -ob 3 else fail || if srs then arctic
-dim 3 -ib 1 -ob 2 else fail || if srs then arctic -dim 3 -ib 2 -ob 2 else fail
|| matrix -dim 1 -ib 5 -ob 8 || matrix -dim 2 -ib 3 -ob 4 || matrix -dim 3 -ib 2
-ob 3 || matrix -dim 3 -ib 1 -ob 2 || matrix -dim 4 -ib 1 -ob 2 || matrix -dim 5
-ib 1 -ob 1 || fail)[5]*);(dp;edg[1.0]?;(sccs | (sc || sct || ur?;( matrix -dp
-ur -dim 1 -ib 3 -ob 5 || matrix -dp -ur -dim 1 -ib 3 -ob 8 -rat 2 -db 1 ||
matrix -dp -ur -dim 1 -ib 4 -ob 10 -rat 4 -db 1 || matrix -dp -ur -dim 2 -ib 2
-ob 3 || matrix -dp -ur -dim 2 -ib 3 -ob 4 -rat 2 -db 0 || matrix -dp -ur -dim 3
-ib 1 -ob 3 || matrix -dp -ur -dim 3 -ib 2 -ob 3 || matrix -dp -ur -dim 3 -ib 2
-ob 3 -rat 2 -db 0 || matrix -dp -ur -dim 4 -ib 1 -ob 2 || lpo -ur -af ||
(arctic -dp -ur -dim 1 -ib 4 -ob 3[10] | arctic -dp -ur -dim 2 -ib 2 -ob 2[15] |
arctic -dp -ur -dim 3 -ib 1 -ob 1[15] | arctic -dp -ur -dim 4 -ib 1 -ob 1[15] |
fail) || (arctic -bz -dp -ur -dim 1 -ib 4 -ob 3[10] | arctic -bz -dp -ur -dim 2
-ib 2 -ob 2[15] | arctic -bz -dp -ur -dim 3 -ib 1 -ob 1[15] | arctic -bz -dp -ur
-dim 4 -ib 1 -ob 1[15] | fail) || fail)) )*[59])) || (( unfold ||
fail)*[7])!)[59])! ) else (((( if srs then arctic -dim 1 -ib 4 -ob 5 else fail
|| if srs then arctic -dim 2 -ib 2 -ob 3 else fail || if srs then arctic -dim 3
-ib 1 -ob 2 else fail || if srs then arctic -dim 3 -ib 2 -ob 2 else fail ||
matrix -dim 1 -ib 5 -ob 8 || matrix -dim 2 -ib 3 -ob 4 || matrix -dim 3 -ib 2
-ob 3 || matrix -dim 3 -ib 1 -ob 2 || matrix -dim 4 -ib 1 -ob 2 || matrix -dim 5
-ib 1 -ob 1 || fail)[5]*);(dp;edg[1.0]?;(sccs | (sc || sct || ur?;( matrix -dp
-ur -dim 1 -ib 3 -ob 5 || matrix -dp -ur -dim 1 -ib 3 -ob 8 -rat 2 -db 1 ||
matrix -dp -ur -dim 1 -ib 4 -ob 10 -rat 4 -db 1 || matrix -dp -ur -dim 2 -ib 2
-ob 3 || matrix -dp -ur -dim 2 -ib 3 -ob 4 -rat 2 -db 0 || matrix -dp -ur -dim 3
-ib 1 -ob 3 || matrix -dp -ur -dim 3 -ib 2 -ob 3 || matrix -dp -ur -dim 3 -ib 2
-ob 3 -rat 2 -db 0 || matrix -dp -ur -dim 4 -ib 1 -ob 2 || lpo -ur -af ||
(arctic -dp -ur -dim 1 -ib 4 -ob 3[10] | arctic -dp -ur -dim 2 -ib 2 -ob 2[15] |
arctic -dp -ur -dim 3 -ib 1 -ob 1[15] | arctic -dp -ur -dim 4 -ib 1 -ob 1[15] |
fail) || (arctic -bz -dp -ur -dim 1 -ib 4 -ob 3[10] | arctic -bz -dp -ur -dim 2
-ib 2 -ob 2[15] | arctic -bz -dp -ur -dim 3 -ib 1 -ob 1[15] | arctic -bz -dp -ur
-dim 4 -ib 1 -ob 1[15] | fail) || fail)) )*[59])) || (( unfold ||
fail)*[7])!)[59]))