Certification Problem
Input (COPS 370)
The rewrite relation of the following conditional TRS is considered.
| a |
→ |
c |
| a |
→ |
d |
| b |
→ |
c |
| b |
→ |
d |
| c |
→ |
e |
| d |
→ |
e |
| k |
→ |
e |
| l |
→ |
e |
|
s(c) |
→ |
t(k) |
|
s(c) |
→ |
t(l) |
|
s(e) |
→ |
t(e) |
|
g(x,x) |
→ |
h(x,x) |
|
f(x) |
→ |
pair(x,y) |
| s(x) ≈ t(y) |
Property / Task
Prove or disprove confluence.Answer / Result
Yes.Proof (by ConCon @ CoCo 2020)
1 Quasi-reductive SDTRS where all CCPs are joinable
The given strongly deterministic oriented 3-CTRS is quasi-reductive and all CCPs are joinable.
1.1 Quasi-Reductive CTRS
The given CTRS is quasi-reductive
1.1.1 Unraveling
To prove that the CTRS is quasi-reductive, we show termination of the following
unraveled system.
| For |
ke we get |
| For |
s(c)t(k) we get |
| For |
bd we get |
| For |
s(c)t(l) we get |
| For |
ad we get |
| For |
g(x,x)h(x,x) we get |
| For |
bc we get |
| For |
f(x)pair(x,y)s(x)t(y) we get |
| For |
s(e)t(e) we get |
| For |
de we get |
| For |
ce we get |
| For |
ac we get |
| For |
le we get |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [t(x1)] |
= |
4 · x1 + 4 |
| [a] |
= |
0 |
| [e] |
= |
0 |
| [s(x1)] |
= |
16 · x1 + 4 |
| [b] |
= |
0 |
| [h(x1, x2)] |
= |
6 · x1 + 8 · x2 + 0 |
| [k] |
= |
0 |
| [l] |
= |
0 |
| [U1(x1, x2)] |
= |
1 · x1 + 15 · x2 + 21 |
| [f(x1)] |
= |
31 · x1 + 25 |
| [g(x1, x2)] |
= |
10 · x1 + 4 · x2 + 0 |
| [pair(x1, x2)] |
= |
2 · x1 + 4 · x2 + 0 |
| [d] |
= |
0 |
| [c] |
= |
0 |
all of the following rules can be deleted.
|
U1(t(y),x) |
→ |
pair(x,y) |
(9) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [t(x1)] |
= |
1 · x1 + 0 |
| [a] |
= |
0 |
| [e] |
= |
0 |
| [s(x1)] |
= |
6 · x1 + 1 |
| [b] |
= |
0 |
| [h(x1, x2)] |
= |
1 · x1 + 6 · x2 + 0 |
| [k] |
= |
0 |
| [l] |
= |
1 |
| [U1(x1, x2)] |
= |
4 · x1 + 4 · x2 + 12 |
| [f(x1)] |
= |
29 · x1 + 16 |
| [g(x1, x2)] |
= |
1 · x1 + 6 · x2 + 0 |
| [d] |
= |
0 |
| [c] |
= |
0 |
all of the following rules can be deleted.
|
s(c) |
→ |
t(k) |
(2) |
|
s(e) |
→ |
t(e) |
(10) |
| l |
→ |
e |
(14) |
1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [t(x1)] |
= |
4 · x1 + 0 |
| [a] |
= |
0 |
| [e] |
= |
0 |
| [s(x1)] |
= |
1 · x1 + 0 |
| [b] |
= |
0 |
| [h(x1, x2)] |
= |
16 · x1 + 16 · x2 + 0 |
| [k] |
= |
0 |
| [l] |
= |
0 |
| [U1(x1, x2)] |
= |
22 · x1 + 8 · x2 + 1 |
| [f(x1)] |
= |
31 · x1 + 16 |
| [g(x1, x2)] |
= |
12 · x1 + 20 · x2 + 0 |
| [d] |
= |
0 |
| [c] |
= |
0 |
all of the following rules can be deleted.
1.1.1.1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(h) |
= |
0 |
|
weight(h) |
= |
0 |
|
|
|
| prec(g) |
= |
1 |
|
weight(g) |
= |
0 |
|
|
|
| prec(a) |
= |
3 |
|
weight(a) |
= |
1 |
|
|
|
| prec(l) |
= |
0 |
|
weight(l) |
= |
1 |
|
|
|
| prec(d) |
= |
2 |
|
weight(d) |
= |
1 |
|
|
|
| prec(b) |
= |
3 |
|
weight(b) |
= |
1 |
|
|
|
| prec(t) |
= |
0 |
|
weight(t) |
= |
2 |
|
|
|
| prec(s) |
= |
7 |
|
weight(s) |
= |
2 |
|
|
|
| prec(c) |
= |
2 |
|
weight(c) |
= |
1 |
|
|
|
| prec(e) |
= |
0 |
|
weight(e) |
= |
1 |
|
|
|
| prec(k) |
= |
1 |
|
weight(k) |
= |
1 |
|
|
|
all of the following rules can be deleted.
| k |
→ |
e |
(1) |
| b |
→ |
d |
(3) |
|
s(c) |
→ |
t(l) |
(4) |
| a |
→ |
d |
(5) |
|
g(x,x) |
→ |
h(x,x) |
(6) |
| b |
→ |
c |
(7) |
| d |
→ |
e |
(11) |
| c |
→ |
e |
(12) |
| a |
→ |
c |
(13) |
1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2 All CCPs are joinable
A CCP is joinable if it is context-joinable, infeasible, or unfeasible.
-
The
1st
CCP
t(k)
=
t(l)
is context-joinable.
-
The
2nd
CCP
c
=
d
is context-joinable.
-
The
3rd
CCP
d
=
c
is context-joinable.
-
The
4th
CCP
d
=
c
is context-joinable.
-
The
5th
CCP
t(k)
=
s(e)
is context-joinable.
-
The
6th
CCP
c
=
d
is context-joinable.
-
The
7th
CCP
t(l)
=
t(k)
is context-joinable.
-
The
8th
CCP
t(l)
=
s(e)
is context-joinable.