Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-561)
The rewrite relation of the following TRS is considered.
a(b(x1)) |
→ |
x1 |
(1) |
a(c(x1)) |
→ |
c(c(x1)) |
(2) |
b(c(x1)) |
→ |
a(b(a(b(x1)))) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
c(a(b(x1))) |
→ |
c(x1) |
(4) |
c(a(c(x1))) |
→ |
c(c(c(x1))) |
(5) |
c(b(c(x1))) |
→ |
c(a(b(a(b(x1))))) |
(6) |
b(a(b(x1))) |
→ |
b(x1) |
(7) |
b(a(c(x1))) |
→ |
b(c(c(x1))) |
(8) |
b(b(c(x1))) |
→ |
b(a(b(a(b(x1))))) |
(9) |
a(a(b(x1))) |
→ |
a(x1) |
(10) |
a(a(c(x1))) |
→ |
a(c(c(x1))) |
(11) |
a(b(c(x1))) |
→ |
a(a(b(a(b(x1))))) |
(12) |
1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,1,2}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 3):
[c(x1)] |
= |
3x1 + 0 |
[b(x1)] |
= |
3x1 + 1 |
[a(x1)] |
= |
3x1 + 2 |
We obtain the labeled TRS
a2(a1(b2(x1))) |
→ |
a2(x1) |
(13) |
a2(a1(b1(x1))) |
→ |
a1(x1) |
(14) |
a2(a1(b0(x1))) |
→ |
a0(x1) |
(15) |
b2(a1(b2(x1))) |
→ |
b2(x1) |
(16) |
b2(a1(b1(x1))) |
→ |
b1(x1) |
(17) |
b2(a1(b0(x1))) |
→ |
b0(x1) |
(18) |
c2(a1(b2(x1))) |
→ |
c2(x1) |
(19) |
c2(a1(b1(x1))) |
→ |
c1(x1) |
(20) |
c2(a1(b0(x1))) |
→ |
c0(x1) |
(21) |
a2(a0(c2(x1))) |
→ |
a0(c0(c2(x1))) |
(22) |
a2(a0(c1(x1))) |
→ |
a0(c0(c1(x1))) |
(23) |
a2(a0(c0(x1))) |
→ |
a0(c0(c0(x1))) |
(24) |
b2(a0(c2(x1))) |
→ |
b0(c0(c2(x1))) |
(25) |
b2(a0(c1(x1))) |
→ |
b0(c0(c1(x1))) |
(26) |
b2(a0(c0(x1))) |
→ |
b0(c0(c0(x1))) |
(27) |
c2(a0(c2(x1))) |
→ |
c0(c0(c2(x1))) |
(28) |
c2(a0(c1(x1))) |
→ |
c0(c0(c1(x1))) |
(29) |
c2(a0(c0(x1))) |
→ |
c0(c0(c0(x1))) |
(30) |
a1(b0(c2(x1))) |
→ |
a2(a1(b2(a1(b2(x1))))) |
(31) |
a1(b0(c1(x1))) |
→ |
a2(a1(b2(a1(b1(x1))))) |
(32) |
a1(b0(c0(x1))) |
→ |
a2(a1(b2(a1(b0(x1))))) |
(33) |
b1(b0(c2(x1))) |
→ |
b2(a1(b2(a1(b2(x1))))) |
(34) |
b1(b0(c1(x1))) |
→ |
b2(a1(b2(a1(b1(x1))))) |
(35) |
b1(b0(c0(x1))) |
→ |
b2(a1(b2(a1(b0(x1))))) |
(36) |
c1(b0(c2(x1))) |
→ |
c2(a1(b2(a1(b2(x1))))) |
(37) |
c1(b0(c1(x1))) |
→ |
c2(a1(b2(a1(b1(x1))))) |
(38) |
c1(b0(c0(x1))) |
→ |
c2(a1(b2(a1(b0(x1))))) |
(39) |
1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] |
= |
x1 +
|
[c1(x1)] |
= |
x1 +
|
[c2(x1)] |
= |
x1 +
|
[b0(x1)] |
= |
x1 +
|
[b1(x1)] |
= |
x1 +
|
[b2(x1)] |
= |
x1 +
|
[a0(x1)] |
= |
x1 +
|
[a1(x1)] |
= |
x1 +
|
[a2(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
a2(a1(b1(x1))) |
→ |
a1(x1) |
(14) |
c2(a1(b1(x1))) |
→ |
c1(x1) |
(20) |
c2(a1(b0(x1))) |
→ |
c0(x1) |
(21) |
c2(a0(c2(x1))) |
→ |
c0(c0(c2(x1))) |
(28) |
c2(a0(c1(x1))) |
→ |
c0(c0(c1(x1))) |
(29) |
c2(a0(c0(x1))) |
→ |
c0(c0(c0(x1))) |
(30) |
a1(b0(c2(x1))) |
→ |
a2(a1(b2(a1(b2(x1))))) |
(31) |
a1(b0(c1(x1))) |
→ |
a2(a1(b2(a1(b1(x1))))) |
(32) |
b1(b0(c2(x1))) |
→ |
b2(a1(b2(a1(b2(x1))))) |
(34) |
b1(b0(c1(x1))) |
→ |
b2(a1(b2(a1(b1(x1))))) |
(35) |
b1(b0(c0(x1))) |
→ |
b2(a1(b2(a1(b0(x1))))) |
(36) |
c1(b0(c2(x1))) |
→ |
c2(a1(b2(a1(b2(x1))))) |
(37) |
c1(b0(c1(x1))) |
→ |
c2(a1(b2(a1(b1(x1))))) |
(38) |
c1(b0(c0(x1))) |
→ |
c2(a1(b2(a1(b0(x1))))) |
(39) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
c2#(a1(b2(x1))) |
→ |
c2#(x1) |
(40) |
a1#(b0(c0(x1))) |
→ |
b2#(a1(b0(x1))) |
(41) |
a1#(b0(c0(x1))) |
→ |
a1#(b0(x1)) |
(42) |
a1#(b0(c0(x1))) |
→ |
a1#(b2(a1(b0(x1)))) |
(43) |
a1#(b0(c0(x1))) |
→ |
a2#(a1(b2(a1(b0(x1))))) |
(44) |
a2#(a1(b2(x1))) |
→ |
a2#(x1) |
(45) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] |
= |
x1 +
|
[c1(x1)] |
= |
x1 +
|
[c2(x1)] |
= |
x1 +
|
[b0(x1)] |
= |
x1 +
|
[b1(x1)] |
= |
x1 +
|
[b2(x1)] |
= |
x1 +
|
[a0(x1)] |
= |
x1 +
|
[a1(x1)] |
= |
x1 +
|
[a2(x1)] |
= |
x1 +
|
[c2#(x1)] |
= |
x1 +
|
[b2#(x1)] |
= |
x1 +
|
[a1#(x1)] |
= |
x1 +
|
[a2#(x1)] |
= |
x1 +
|
together with the usable
rules
a2(a1(b2(x1))) |
→ |
a2(x1) |
(13) |
a2(a1(b0(x1))) |
→ |
a0(x1) |
(15) |
b2(a1(b2(x1))) |
→ |
b2(x1) |
(16) |
b2(a1(b1(x1))) |
→ |
b1(x1) |
(17) |
b2(a1(b0(x1))) |
→ |
b0(x1) |
(18) |
c2(a1(b2(x1))) |
→ |
c2(x1) |
(19) |
a2(a0(c2(x1))) |
→ |
a0(c0(c2(x1))) |
(22) |
a2(a0(c1(x1))) |
→ |
a0(c0(c1(x1))) |
(23) |
a2(a0(c0(x1))) |
→ |
a0(c0(c0(x1))) |
(24) |
b2(a0(c2(x1))) |
→ |
b0(c0(c2(x1))) |
(25) |
b2(a0(c1(x1))) |
→ |
b0(c0(c1(x1))) |
(26) |
b2(a0(c0(x1))) |
→ |
b0(c0(c0(x1))) |
(27) |
a1(b0(c0(x1))) |
→ |
a2(a1(b2(a1(b0(x1))))) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
a1#(b0(c0(x1))) |
→ |
b2#(a1(b0(x1))) |
(41) |
a1#(b0(c0(x1))) |
→ |
a1#(b0(x1)) |
(42) |
a1#(b0(c0(x1))) |
→ |
a1#(b2(a1(b0(x1)))) |
(43) |
a1#(b0(c0(x1))) |
→ |
a2#(a1(b2(a1(b0(x1))))) |
(44) |
and
no rules
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.