Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-30)
The rewrite relation of the following TRS is considered.
a(x1) |
→ |
x1 |
(1) |
a(x1) |
→ |
b(b(x1)) |
(2) |
a(b(x1)) |
→ |
a(c(a(c(x1)))) |
(3) |
c(c(x1)) |
→ |
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) |
a(x1) |
→ |
b(b(x1)) |
(2) |
b(a(x1)) |
→ |
c(a(c(a(x1)))) |
(5) |
c(c(x1)) |
→ |
x1 |
(4) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), b(☐), c(☐)}
We obtain the transformed TRS
a(a(x1)) |
→ |
a(x1) |
(6) |
b(a(x1)) |
→ |
b(x1) |
(7) |
c(a(x1)) |
→ |
c(x1) |
(8) |
a(a(x1)) |
→ |
a(b(b(x1))) |
(9) |
b(a(x1)) |
→ |
b(b(b(x1))) |
(10) |
c(a(x1)) |
→ |
c(b(b(x1))) |
(11) |
a(b(a(x1))) |
→ |
a(c(a(c(a(x1))))) |
(12) |
b(b(a(x1))) |
→ |
b(c(a(c(a(x1))))) |
(13) |
c(b(a(x1))) |
→ |
c(c(a(c(a(x1))))) |
(14) |
a(c(c(x1))) |
→ |
a(x1) |
(15) |
b(c(c(x1))) |
→ |
b(x1) |
(16) |
c(c(c(x1))) |
→ |
c(x1) |
(17) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
aa(aa(x1)) |
→ |
aa(x1) |
(18) |
aa(ab(x1)) |
→ |
ab(x1) |
(19) |
aa(ac(x1)) |
→ |
ac(x1) |
(20) |
ba(aa(x1)) |
→ |
ba(x1) |
(21) |
ba(ab(x1)) |
→ |
bb(x1) |
(22) |
ba(ac(x1)) |
→ |
bc(x1) |
(23) |
ca(aa(x1)) |
→ |
ca(x1) |
(24) |
ca(ab(x1)) |
→ |
cb(x1) |
(25) |
ca(ac(x1)) |
→ |
cc(x1) |
(26) |
aa(aa(x1)) |
→ |
ab(bb(ba(x1))) |
(27) |
aa(ab(x1)) |
→ |
ab(bb(bb(x1))) |
(28) |
aa(ac(x1)) |
→ |
ab(bb(bc(x1))) |
(29) |
ba(aa(x1)) |
→ |
bb(bb(ba(x1))) |
(30) |
ba(ab(x1)) |
→ |
bb(bb(bb(x1))) |
(31) |
ba(ac(x1)) |
→ |
bb(bb(bc(x1))) |
(32) |
ca(aa(x1)) |
→ |
cb(bb(ba(x1))) |
(33) |
ca(ab(x1)) |
→ |
cb(bb(bb(x1))) |
(34) |
ca(ac(x1)) |
→ |
cb(bb(bc(x1))) |
(35) |
ab(ba(aa(x1))) |
→ |
ac(ca(ac(ca(aa(x1))))) |
(36) |
ab(ba(ab(x1))) |
→ |
ac(ca(ac(ca(ab(x1))))) |
(37) |
ab(ba(ac(x1))) |
→ |
ac(ca(ac(ca(ac(x1))))) |
(38) |
bb(ba(aa(x1))) |
→ |
bc(ca(ac(ca(aa(x1))))) |
(39) |
bb(ba(ab(x1))) |
→ |
bc(ca(ac(ca(ab(x1))))) |
(40) |
bb(ba(ac(x1))) |
→ |
bc(ca(ac(ca(ac(x1))))) |
(41) |
cb(ba(aa(x1))) |
→ |
cc(ca(ac(ca(aa(x1))))) |
(42) |
cb(ba(ab(x1))) |
→ |
cc(ca(ac(ca(ab(x1))))) |
(43) |
cb(ba(ac(x1))) |
→ |
cc(ca(ac(ca(ac(x1))))) |
(44) |
ac(cc(ca(x1))) |
→ |
aa(x1) |
(45) |
ac(cc(cb(x1))) |
→ |
ab(x1) |
(46) |
ac(cc(cc(x1))) |
→ |
ac(x1) |
(47) |
bc(cc(ca(x1))) |
→ |
ba(x1) |
(48) |
bc(cc(cb(x1))) |
→ |
bb(x1) |
(49) |
bc(cc(cc(x1))) |
→ |
bc(x1) |
(50) |
cc(cc(ca(x1))) |
→ |
ca(x1) |
(51) |
cc(cc(cb(x1))) |
→ |
cb(x1) |
(52) |
cc(cc(cc(x1))) |
→ |
cc(x1) |
(53) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[aa(x1)] |
= |
1 · x1 + 2 |
[ab(x1)] |
= |
1 · x1 + 3 |
[ac(x1)] |
= |
1 · x1 + 1 |
[ba(x1)] |
= |
1 · x1 + 1 |
[bb(x1)] |
= |
1 · x1
|
[bc(x1)] |
= |
1 · x1
|
[ca(x1)] |
= |
1 · x1
|
[cb(x1)] |
= |
1 · x1 + 1 |
[cc(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
aa(aa(x1)) |
→ |
aa(x1) |
(18) |
aa(ab(x1)) |
→ |
ab(x1) |
(19) |
aa(ac(x1)) |
→ |
ac(x1) |
(20) |
ba(aa(x1)) |
→ |
ba(x1) |
(21) |
ba(ab(x1)) |
→ |
bb(x1) |
(22) |
ba(ac(x1)) |
→ |
bc(x1) |
(23) |
ca(aa(x1)) |
→ |
ca(x1) |
(24) |
ca(ab(x1)) |
→ |
cb(x1) |
(25) |
aa(ab(x1)) |
→ |
ab(bb(bb(x1))) |
(28) |
ba(aa(x1)) |
→ |
bb(bb(ba(x1))) |
(30) |
ba(ab(x1)) |
→ |
bb(bb(bb(x1))) |
(31) |
ba(ac(x1)) |
→ |
bb(bb(bc(x1))) |
(32) |
ca(ab(x1)) |
→ |
cb(bb(bb(x1))) |
(34) |
ab(ba(aa(x1))) |
→ |
ac(ca(ac(ca(aa(x1))))) |
(36) |
ab(ba(ab(x1))) |
→ |
ac(ca(ac(ca(ab(x1))))) |
(37) |
ab(ba(ac(x1))) |
→ |
ac(ca(ac(ca(ac(x1))))) |
(38) |
ac(cc(cc(x1))) |
→ |
ac(x1) |
(47) |
bc(cc(cb(x1))) |
→ |
bb(x1) |
(49) |
bc(cc(cc(x1))) |
→ |
bc(x1) |
(50) |
cc(cc(ca(x1))) |
→ |
ca(x1) |
(51) |
cc(cc(cb(x1))) |
→ |
cb(x1) |
(52) |
cc(cc(cc(x1))) |
→ |
cc(x1) |
(53) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[ca(x1)] |
= |
1 · x1 + 1 |
[ac(x1)] |
= |
1 · x1
|
[cc(x1)] |
= |
1 · x1 + 1 |
[aa(x1)] |
= |
1 · x1 + 2 |
[ab(x1)] |
= |
1 · x1 + 1 |
[bb(x1)] |
= |
1 · x1
|
[ba(x1)] |
= |
1 · x1 + 2 |
[bc(x1)] |
= |
1 · x1
|
[cb(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
aa(aa(x1)) |
→ |
ab(bb(ba(x1))) |
(27) |
aa(ac(x1)) |
→ |
ab(bb(bc(x1))) |
(29) |
ac(cc(cb(x1))) |
→ |
ab(x1) |
(46) |
1.1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
ca#(aa(x1)) |
→ |
cb#(bb(ba(x1))) |
(54) |
ca#(aa(x1)) |
→ |
bb#(ba(x1)) |
(55) |
ca#(ac(x1)) |
→ |
cb#(bb(bc(x1))) |
(56) |
ca#(ac(x1)) |
→ |
bb#(bc(x1)) |
(57) |
ca#(ac(x1)) |
→ |
bc#(x1) |
(58) |
bb#(ba(aa(x1))) |
→ |
bc#(ca(ac(ca(aa(x1))))) |
(59) |
bb#(ba(aa(x1))) |
→ |
ca#(ac(ca(aa(x1)))) |
(60) |
bb#(ba(aa(x1))) |
→ |
ac#(ca(aa(x1))) |
(61) |
bb#(ba(aa(x1))) |
→ |
ca#(aa(x1)) |
(62) |
bb#(ba(ab(x1))) |
→ |
bc#(ca(ac(ca(ab(x1))))) |
(63) |
bb#(ba(ab(x1))) |
→ |
ca#(ac(ca(ab(x1)))) |
(64) |
bb#(ba(ab(x1))) |
→ |
ac#(ca(ab(x1))) |
(65) |
bb#(ba(ab(x1))) |
→ |
ca#(ab(x1)) |
(66) |
bb#(ba(ac(x1))) |
→ |
bc#(ca(ac(ca(ac(x1))))) |
(67) |
bb#(ba(ac(x1))) |
→ |
ca#(ac(ca(ac(x1)))) |
(68) |
bb#(ba(ac(x1))) |
→ |
ac#(ca(ac(x1))) |
(69) |
bb#(ba(ac(x1))) |
→ |
ca#(ac(x1)) |
(70) |
cb#(ba(aa(x1))) |
→ |
ca#(ac(ca(aa(x1)))) |
(71) |
cb#(ba(aa(x1))) |
→ |
ac#(ca(aa(x1))) |
(72) |
cb#(ba(aa(x1))) |
→ |
ca#(aa(x1)) |
(73) |
cb#(ba(ab(x1))) |
→ |
ca#(ac(ca(ab(x1)))) |
(74) |
cb#(ba(ab(x1))) |
→ |
ac#(ca(ab(x1))) |
(75) |
cb#(ba(ab(x1))) |
→ |
ca#(ab(x1)) |
(76) |
cb#(ba(ac(x1))) |
→ |
ca#(ac(ca(ac(x1)))) |
(77) |
cb#(ba(ac(x1))) |
→ |
ac#(ca(ac(x1))) |
(78) |
cb#(ba(ac(x1))) |
→ |
ca#(ac(x1)) |
(79) |
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.