YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
a(s(x)) |
→ |
s(a(x)) |
|
b(a(b(s(x)))) |
→ |
a(b(s(a(x)))) |
|
b(a(b(b(x)))) |
→ |
a(b(a(b(x)))) |
|
a(b(a(a(x)))) |
→ |
b(a(b(a(x)))) |
Proof
1 Critical Pair Closing System
Confluence is proven using the following terminating critical-pair-closing-system R:
|
a(s(x)) |
→ |
s(a(x)) |
|
b(a(b(s(x)))) |
→ |
a(b(s(a(x)))) |
|
a(b(a(a(x)))) |
→ |
b(a(b(a(x)))) |
|
b(a(b(b(x)))) |
→ |
a(b(a(b(x)))) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [a(x1)] |
= |
2 · x1 + 0 |
| [s(x1)] |
= |
2 · x1 + 1 |
| [b(x1)] |
= |
2 · x1 + 0 |
the
rules
|
a(b(a(a(x)))) |
→ |
b(a(b(a(x)))) |
|
b(a(b(b(x)))) |
→ |
a(b(a(b(x)))) |
remain.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [a(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
all rules could be removed.
1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
Tool configuration
csi
- version: csi 1.2.5 [hg: unknown]
- strategy:
(cr -cpcs2 -cpcscert; ((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))!