Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-201)
The rewrite relation of the following TRS is considered.
a(x1) |
→ |
x1 |
(1) |
a(b(x1)) |
→ |
b(b(a(c(x1)))) |
(2) |
b(x1) |
→ |
x1 |
(3) |
c(c(x1)) |
→ |
b(a(x1)) |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(x1) |
→ |
x1 |
(1) |
b(a(x1)) |
→ |
c(a(b(b(x1)))) |
(5) |
b(x1) |
→ |
x1 |
(3) |
c(c(x1)) |
→ |
a(b(x1)) |
(6) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), b(☐), c(☐)}
We obtain the transformed TRS
a(a(x1)) |
→ |
a(x1) |
(7) |
b(a(x1)) |
→ |
b(x1) |
(8) |
c(a(x1)) |
→ |
c(x1) |
(9) |
a(b(a(x1))) |
→ |
a(c(a(b(b(x1))))) |
(10) |
b(b(a(x1))) |
→ |
b(c(a(b(b(x1))))) |
(11) |
c(b(a(x1))) |
→ |
c(c(a(b(b(x1))))) |
(12) |
a(b(x1)) |
→ |
a(x1) |
(13) |
b(b(x1)) |
→ |
b(x1) |
(14) |
c(b(x1)) |
→ |
c(x1) |
(15) |
a(c(c(x1))) |
→ |
a(a(b(x1))) |
(16) |
b(c(c(x1))) |
→ |
b(a(b(x1))) |
(17) |
c(c(c(x1))) |
→ |
c(a(b(x1))) |
(18) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
aa(aa(x1)) |
→ |
aa(x1) |
(19) |
aa(ab(x1)) |
→ |
ab(x1) |
(20) |
aa(ac(x1)) |
→ |
ac(x1) |
(21) |
ba(aa(x1)) |
→ |
ba(x1) |
(22) |
ba(ab(x1)) |
→ |
bb(x1) |
(23) |
ba(ac(x1)) |
→ |
bc(x1) |
(24) |
ca(aa(x1)) |
→ |
ca(x1) |
(25) |
ca(ab(x1)) |
→ |
cb(x1) |
(26) |
ca(ac(x1)) |
→ |
cc(x1) |
(27) |
ab(ba(aa(x1))) |
→ |
ac(ca(ab(bb(ba(x1))))) |
(28) |
ab(ba(ab(x1))) |
→ |
ac(ca(ab(bb(bb(x1))))) |
(29) |
ab(ba(ac(x1))) |
→ |
ac(ca(ab(bb(bc(x1))))) |
(30) |
bb(ba(aa(x1))) |
→ |
bc(ca(ab(bb(ba(x1))))) |
(31) |
bb(ba(ab(x1))) |
→ |
bc(ca(ab(bb(bb(x1))))) |
(32) |
bb(ba(ac(x1))) |
→ |
bc(ca(ab(bb(bc(x1))))) |
(33) |
cb(ba(aa(x1))) |
→ |
cc(ca(ab(bb(ba(x1))))) |
(34) |
cb(ba(ab(x1))) |
→ |
cc(ca(ab(bb(bb(x1))))) |
(35) |
cb(ba(ac(x1))) |
→ |
cc(ca(ab(bb(bc(x1))))) |
(36) |
ab(ba(x1)) |
→ |
aa(x1) |
(37) |
ab(bb(x1)) |
→ |
ab(x1) |
(38) |
ab(bc(x1)) |
→ |
ac(x1) |
(39) |
bb(ba(x1)) |
→ |
ba(x1) |
(40) |
bb(bb(x1)) |
→ |
bb(x1) |
(41) |
bb(bc(x1)) |
→ |
bc(x1) |
(42) |
cb(ba(x1)) |
→ |
ca(x1) |
(43) |
cb(bb(x1)) |
→ |
cb(x1) |
(44) |
cb(bc(x1)) |
→ |
cc(x1) |
(45) |
ac(cc(ca(x1))) |
→ |
aa(ab(ba(x1))) |
(46) |
ac(cc(cb(x1))) |
→ |
aa(ab(bb(x1))) |
(47) |
ac(cc(cc(x1))) |
→ |
aa(ab(bc(x1))) |
(48) |
bc(cc(ca(x1))) |
→ |
ba(ab(ba(x1))) |
(49) |
bc(cc(cb(x1))) |
→ |
ba(ab(bb(x1))) |
(50) |
bc(cc(cc(x1))) |
→ |
ba(ab(bc(x1))) |
(51) |
cc(cc(ca(x1))) |
→ |
ca(ab(ba(x1))) |
(52) |
cc(cc(cb(x1))) |
→ |
ca(ab(bb(x1))) |
(53) |
cc(cc(cc(x1))) |
→ |
ca(ab(bc(x1))) |
(54) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[aa(x1)] |
= |
1 · x1 + 1 |
[ab(x1)] |
= |
1 · x1
|
[ac(x1)] |
= |
1 · x1 + 1 |
[ba(x1)] |
= |
1 · x1 + 1 |
[bb(x1)] |
= |
1 · x1
|
[bc(x1)] |
= |
1 · x1 + 1 |
[ca(x1)] |
= |
1 · x1
|
[cb(x1)] |
= |
1 · x1
|
[cc(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
aa(aa(x1)) |
→ |
aa(x1) |
(19) |
aa(ab(x1)) |
→ |
ab(x1) |
(20) |
aa(ac(x1)) |
→ |
ac(x1) |
(21) |
ba(aa(x1)) |
→ |
ba(x1) |
(22) |
ba(ab(x1)) |
→ |
bb(x1) |
(23) |
ba(ac(x1)) |
→ |
bc(x1) |
(24) |
ca(aa(x1)) |
→ |
ca(x1) |
(25) |
cb(ba(x1)) |
→ |
ca(x1) |
(43) |
ac(cc(cb(x1))) |
→ |
aa(ab(bb(x1))) |
(47) |
ac(cc(cc(x1))) |
→ |
aa(ab(bc(x1))) |
(48) |
bc(cc(cb(x1))) |
→ |
ba(ab(bb(x1))) |
(50) |
bc(cc(cc(x1))) |
→ |
ba(ab(bc(x1))) |
(51) |
cc(cc(ca(x1))) |
→ |
ca(ab(ba(x1))) |
(52) |
cc(cc(cb(x1))) |
→ |
ca(ab(bb(x1))) |
(53) |
cc(cc(cc(x1))) |
→ |
ca(ab(bc(x1))) |
(54) |
1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
ca#(ab(x1)) |
→ |
cb#(x1) |
(55) |
ab#(ba(aa(x1))) |
→ |
ac#(ca(ab(bb(ba(x1))))) |
(56) |
ab#(ba(aa(x1))) |
→ |
ca#(ab(bb(ba(x1)))) |
(57) |
ab#(ba(aa(x1))) |
→ |
ab#(bb(ba(x1))) |
(58) |
ab#(ba(aa(x1))) |
→ |
bb#(ba(x1)) |
(59) |
ab#(ba(ab(x1))) |
→ |
ac#(ca(ab(bb(bb(x1))))) |
(60) |
ab#(ba(ab(x1))) |
→ |
ca#(ab(bb(bb(x1)))) |
(61) |
ab#(ba(ab(x1))) |
→ |
ab#(bb(bb(x1))) |
(62) |
ab#(ba(ab(x1))) |
→ |
bb#(bb(x1)) |
(63) |
ab#(ba(ab(x1))) |
→ |
bb#(x1) |
(64) |
ab#(ba(ac(x1))) |
→ |
ac#(ca(ab(bb(bc(x1))))) |
(65) |
ab#(ba(ac(x1))) |
→ |
ca#(ab(bb(bc(x1)))) |
(66) |
ab#(ba(ac(x1))) |
→ |
ab#(bb(bc(x1))) |
(67) |
ab#(ba(ac(x1))) |
→ |
bb#(bc(x1)) |
(68) |
ab#(ba(ac(x1))) |
→ |
bc#(x1) |
(69) |
bb#(ba(aa(x1))) |
→ |
bc#(ca(ab(bb(ba(x1))))) |
(70) |
bb#(ba(aa(x1))) |
→ |
ca#(ab(bb(ba(x1)))) |
(71) |
bb#(ba(aa(x1))) |
→ |
ab#(bb(ba(x1))) |
(72) |
bb#(ba(aa(x1))) |
→ |
bb#(ba(x1)) |
(73) |
bb#(ba(ab(x1))) |
→ |
bc#(ca(ab(bb(bb(x1))))) |
(74) |
bb#(ba(ab(x1))) |
→ |
ca#(ab(bb(bb(x1)))) |
(75) |
bb#(ba(ab(x1))) |
→ |
ab#(bb(bb(x1))) |
(76) |
bb#(ba(ab(x1))) |
→ |
bb#(bb(x1)) |
(77) |
bb#(ba(ab(x1))) |
→ |
bb#(x1) |
(78) |
bb#(ba(ac(x1))) |
→ |
bc#(ca(ab(bb(bc(x1))))) |
(79) |
bb#(ba(ac(x1))) |
→ |
ca#(ab(bb(bc(x1)))) |
(80) |
bb#(ba(ac(x1))) |
→ |
ab#(bb(bc(x1))) |
(81) |
bb#(ba(ac(x1))) |
→ |
bb#(bc(x1)) |
(82) |
bb#(ba(ac(x1))) |
→ |
bc#(x1) |
(83) |
cb#(ba(aa(x1))) |
→ |
ca#(ab(bb(ba(x1)))) |
(84) |
cb#(ba(aa(x1))) |
→ |
ab#(bb(ba(x1))) |
(85) |
cb#(ba(aa(x1))) |
→ |
bb#(ba(x1)) |
(86) |
cb#(ba(ab(x1))) |
→ |
ca#(ab(bb(bb(x1)))) |
(87) |
cb#(ba(ab(x1))) |
→ |
ab#(bb(bb(x1))) |
(88) |
cb#(ba(ab(x1))) |
→ |
bb#(bb(x1)) |
(89) |
cb#(ba(ab(x1))) |
→ |
bb#(x1) |
(90) |
cb#(ba(ac(x1))) |
→ |
ca#(ab(bb(bc(x1)))) |
(91) |
cb#(ba(ac(x1))) |
→ |
ab#(bb(bc(x1))) |
(92) |
cb#(ba(ac(x1))) |
→ |
bb#(bc(x1)) |
(93) |
cb#(ba(ac(x1))) |
→ |
bc#(x1) |
(94) |
ab#(bb(x1)) |
→ |
ab#(x1) |
(95) |
ab#(bc(x1)) |
→ |
ac#(x1) |
(96) |
cb#(bb(x1)) |
→ |
cb#(x1) |
(97) |
ac#(cc(ca(x1))) |
→ |
ab#(ba(x1)) |
(98) |
bc#(cc(ca(x1))) |
→ |
ab#(ba(x1)) |
(99) |
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[ca#(x1)] |
= |
1 · x1
|
[ab(x1)] |
= |
1 + 1 · x1
|
[cb#(x1)] |
= |
1 · x1
|
[ab#(x1)] |
= |
1 · x1
|
[ba(x1)] |
= |
1 · x1
|
[aa(x1)] |
= |
1 + 1 · x1
|
[ac#(x1)] |
= |
1 · x1
|
[ca(x1)] |
= |
1 · x1
|
[bb(x1)] |
= |
1 · x1
|
[bb#(x1)] |
= |
1 · x1
|
[ac(x1)] |
= |
1 + 1 · x1
|
[bc(x1)] |
= |
1 · x1
|
[bc#(x1)] |
= |
1 · x1
|
[cc(x1)] |
= |
1 + 1 · x1
|
[cb(x1)] |
= |
1 + 1 · x1
|
the
pairs
ca#(ab(x1)) |
→ |
cb#(x1) |
(55) |
ab#(ba(aa(x1))) |
→ |
ab#(bb(ba(x1))) |
(58) |
ab#(ba(aa(x1))) |
→ |
bb#(ba(x1)) |
(59) |
ab#(ba(ab(x1))) |
→ |
ab#(bb(bb(x1))) |
(62) |
ab#(ba(ab(x1))) |
→ |
bb#(bb(x1)) |
(63) |
ab#(ba(ab(x1))) |
→ |
bb#(x1) |
(64) |
ab#(ba(ac(x1))) |
→ |
ab#(bb(bc(x1))) |
(67) |
ab#(ba(ac(x1))) |
→ |
bb#(bc(x1)) |
(68) |
ab#(ba(ac(x1))) |
→ |
bc#(x1) |
(69) |
bb#(ba(aa(x1))) |
→ |
ab#(bb(ba(x1))) |
(72) |
bb#(ba(aa(x1))) |
→ |
bb#(ba(x1)) |
(73) |
bb#(ba(ab(x1))) |
→ |
ab#(bb(bb(x1))) |
(76) |
bb#(ba(ab(x1))) |
→ |
bb#(bb(x1)) |
(77) |
bb#(ba(ab(x1))) |
→ |
bb#(x1) |
(78) |
bb#(ba(ac(x1))) |
→ |
ab#(bb(bc(x1))) |
(81) |
bb#(ba(ac(x1))) |
→ |
bb#(bc(x1)) |
(82) |
bb#(ba(ac(x1))) |
→ |
bc#(x1) |
(83) |
cb#(ba(aa(x1))) |
→ |
ab#(bb(ba(x1))) |
(85) |
cb#(ba(aa(x1))) |
→ |
bb#(ba(x1)) |
(86) |
cb#(ba(ab(x1))) |
→ |
ab#(bb(bb(x1))) |
(88) |
cb#(ba(ab(x1))) |
→ |
bb#(bb(x1)) |
(89) |
cb#(ba(ab(x1))) |
→ |
bb#(x1) |
(90) |
cb#(ba(ac(x1))) |
→ |
ab#(bb(bc(x1))) |
(92) |
cb#(ba(ac(x1))) |
→ |
bb#(bc(x1)) |
(93) |
cb#(ba(ac(x1))) |
→ |
bc#(x1) |
(94) |
ac#(cc(ca(x1))) |
→ |
ab#(ba(x1)) |
(98) |
bc#(cc(ca(x1))) |
→ |
ab#(ba(x1)) |
(99) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
cb#(bb(x1)) |
→ |
cb#(x1) |
(97) |
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[bb(x1)] |
= |
1 · x1
|
[cb#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cb#(bb(x1)) |
→ |
cb#(x1) |
(97) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
ab#(bb(x1)) |
→ |
ab#(x1) |
(95) |
1.1.1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[bb(x1)] |
= |
1 · x1
|
[ab#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
ab#(bb(x1)) |
→ |
ab#(x1) |
(95) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.