Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z080)
The rewrite relation of the following TRS is considered.
|
A(b(x1)) |
→ |
b(a(B(A(x1)))) |
(1) |
|
B(a(x1)) |
→ |
a(b(A(B(x1)))) |
(2) |
|
A(a(x1)) |
→ |
x1 |
(3) |
|
B(b(x1)) |
→ |
x1 |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{A(☐), b(☐), a(☐), B(☐)}
We obtain the transformed TRS
|
A(A(b(x1))) |
→ |
A(b(a(B(A(x1))))) |
(5) |
|
b(A(b(x1))) |
→ |
b(b(a(B(A(x1))))) |
(6) |
|
a(A(b(x1))) |
→ |
a(b(a(B(A(x1))))) |
(7) |
|
B(A(b(x1))) |
→ |
B(b(a(B(A(x1))))) |
(8) |
|
A(B(a(x1))) |
→ |
A(a(b(A(B(x1))))) |
(9) |
|
b(B(a(x1))) |
→ |
b(a(b(A(B(x1))))) |
(10) |
|
a(B(a(x1))) |
→ |
a(a(b(A(B(x1))))) |
(11) |
|
B(B(a(x1))) |
→ |
B(a(b(A(B(x1))))) |
(12) |
|
A(A(a(x1))) |
→ |
A(x1) |
(13) |
|
b(A(a(x1))) |
→ |
b(x1) |
(14) |
|
a(A(a(x1))) |
→ |
a(x1) |
(15) |
|
B(A(a(x1))) |
→ |
B(x1) |
(16) |
|
A(B(b(x1))) |
→ |
A(x1) |
(17) |
|
b(B(b(x1))) |
→ |
b(x1) |
(18) |
|
a(B(b(x1))) |
→ |
a(x1) |
(19) |
|
B(B(b(x1))) |
→ |
B(x1) |
(20) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
AA(Ab(bA(x1))) |
→ |
Ab(ba(aB(BA(AA(x1))))) |
(21) |
|
AA(Ab(bb(x1))) |
→ |
Ab(ba(aB(BA(Ab(x1))))) |
(22) |
|
AA(Ab(ba(x1))) |
→ |
Ab(ba(aB(BA(Aa(x1))))) |
(23) |
|
AA(Ab(bB(x1))) |
→ |
Ab(ba(aB(BA(AB(x1))))) |
(24) |
|
bA(Ab(bA(x1))) |
→ |
bb(ba(aB(BA(AA(x1))))) |
(25) |
|
bA(Ab(bb(x1))) |
→ |
bb(ba(aB(BA(Ab(x1))))) |
(26) |
|
bA(Ab(ba(x1))) |
→ |
bb(ba(aB(BA(Aa(x1))))) |
(27) |
|
bA(Ab(bB(x1))) |
→ |
bb(ba(aB(BA(AB(x1))))) |
(28) |
|
aA(Ab(bA(x1))) |
→ |
ab(ba(aB(BA(AA(x1))))) |
(29) |
|
aA(Ab(bb(x1))) |
→ |
ab(ba(aB(BA(Ab(x1))))) |
(30) |
|
aA(Ab(ba(x1))) |
→ |
ab(ba(aB(BA(Aa(x1))))) |
(31) |
|
aA(Ab(bB(x1))) |
→ |
ab(ba(aB(BA(AB(x1))))) |
(32) |
|
BA(Ab(bA(x1))) |
→ |
Bb(ba(aB(BA(AA(x1))))) |
(33) |
|
BA(Ab(bb(x1))) |
→ |
Bb(ba(aB(BA(Ab(x1))))) |
(34) |
|
BA(Ab(ba(x1))) |
→ |
Bb(ba(aB(BA(Aa(x1))))) |
(35) |
|
BA(Ab(bB(x1))) |
→ |
Bb(ba(aB(BA(AB(x1))))) |
(36) |
|
AB(Ba(aA(x1))) |
→ |
Aa(ab(bA(AB(BA(x1))))) |
(37) |
|
AB(Ba(ab(x1))) |
→ |
Aa(ab(bA(AB(Bb(x1))))) |
(38) |
|
AB(Ba(aa(x1))) |
→ |
Aa(ab(bA(AB(Ba(x1))))) |
(39) |
|
AB(Ba(aB(x1))) |
→ |
Aa(ab(bA(AB(BB(x1))))) |
(40) |
|
bB(Ba(aA(x1))) |
→ |
ba(ab(bA(AB(BA(x1))))) |
(41) |
|
bB(Ba(ab(x1))) |
→ |
ba(ab(bA(AB(Bb(x1))))) |
(42) |
|
bB(Ba(aa(x1))) |
→ |
ba(ab(bA(AB(Ba(x1))))) |
(43) |
|
bB(Ba(aB(x1))) |
→ |
ba(ab(bA(AB(BB(x1))))) |
(44) |
|
aB(Ba(aA(x1))) |
→ |
aa(ab(bA(AB(BA(x1))))) |
(45) |
|
aB(Ba(ab(x1))) |
→ |
aa(ab(bA(AB(Bb(x1))))) |
(46) |
|
aB(Ba(aa(x1))) |
→ |
aa(ab(bA(AB(Ba(x1))))) |
(47) |
|
aB(Ba(aB(x1))) |
→ |
aa(ab(bA(AB(BB(x1))))) |
(48) |
|
BB(Ba(aA(x1))) |
→ |
Ba(ab(bA(AB(BA(x1))))) |
(49) |
|
BB(Ba(ab(x1))) |
→ |
Ba(ab(bA(AB(Bb(x1))))) |
(50) |
|
BB(Ba(aa(x1))) |
→ |
Ba(ab(bA(AB(Ba(x1))))) |
(51) |
|
BB(Ba(aB(x1))) |
→ |
Ba(ab(bA(AB(BB(x1))))) |
(52) |
|
AA(Aa(aA(x1))) |
→ |
AA(x1) |
(53) |
|
AA(Aa(ab(x1))) |
→ |
Ab(x1) |
(54) |
|
AA(Aa(aa(x1))) |
→ |
Aa(x1) |
(55) |
|
AA(Aa(aB(x1))) |
→ |
AB(x1) |
(56) |
|
bA(Aa(aA(x1))) |
→ |
bA(x1) |
(57) |
|
bA(Aa(ab(x1))) |
→ |
bb(x1) |
(58) |
|
bA(Aa(aa(x1))) |
→ |
ba(x1) |
(59) |
|
bA(Aa(aB(x1))) |
→ |
bB(x1) |
(60) |
|
aA(Aa(aA(x1))) |
→ |
aA(x1) |
(61) |
|
aA(Aa(ab(x1))) |
→ |
ab(x1) |
(62) |
|
aA(Aa(aa(x1))) |
→ |
aa(x1) |
(63) |
|
aA(Aa(aB(x1))) |
→ |
aB(x1) |
(64) |
|
BA(Aa(aA(x1))) |
→ |
BA(x1) |
(65) |
|
BA(Aa(ab(x1))) |
→ |
Bb(x1) |
(66) |
|
BA(Aa(aa(x1))) |
→ |
Ba(x1) |
(67) |
|
BA(Aa(aB(x1))) |
→ |
BB(x1) |
(68) |
|
AB(Bb(bA(x1))) |
→ |
AA(x1) |
(69) |
|
AB(Bb(bb(x1))) |
→ |
Ab(x1) |
(70) |
|
AB(Bb(ba(x1))) |
→ |
Aa(x1) |
(71) |
|
AB(Bb(bB(x1))) |
→ |
AB(x1) |
(72) |
|
bB(Bb(bA(x1))) |
→ |
bA(x1) |
(73) |
|
bB(Bb(bb(x1))) |
→ |
bb(x1) |
(74) |
|
bB(Bb(ba(x1))) |
→ |
ba(x1) |
(75) |
|
bB(Bb(bB(x1))) |
→ |
bB(x1) |
(76) |
|
aB(Bb(bA(x1))) |
→ |
aA(x1) |
(77) |
|
aB(Bb(bb(x1))) |
→ |
ab(x1) |
(78) |
|
aB(Bb(ba(x1))) |
→ |
aa(x1) |
(79) |
|
aB(Bb(bB(x1))) |
→ |
aB(x1) |
(80) |
|
BB(Bb(bA(x1))) |
→ |
BA(x1) |
(81) |
|
BB(Bb(bb(x1))) |
→ |
Bb(x1) |
(82) |
|
BB(Bb(ba(x1))) |
→ |
Ba(x1) |
(83) |
|
BB(Bb(bB(x1))) |
→ |
BB(x1) |
(84) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [AA(x1)] |
= |
1 · x1 + 1 |
| [Ab(x1)] |
= |
1 · x1 + 1 |
| [bA(x1)] |
= |
1 · x1 + 1 |
| [ba(x1)] |
= |
1 · x1
|
| [aB(x1)] |
= |
1 · x1 + 1 |
| [BA(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1 + 1 |
| [Aa(x1)] |
= |
1 · x1
|
| [bB(x1)] |
= |
1 · x1
|
| [AB(x1)] |
= |
1 · x1
|
| [aA(x1)] |
= |
1 · x1 + 1 |
| [ab(x1)] |
= |
1 · x1
|
| [Bb(x1)] |
= |
1 · x1
|
| [Ba(x1)] |
= |
1 · x1 + 1 |
| [aa(x1)] |
= |
1 · x1 + 1 |
| [BB(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
|
aA(Ab(bA(x1))) |
→ |
ab(ba(aB(BA(AA(x1))))) |
(29) |
|
aA(Ab(bb(x1))) |
→ |
ab(ba(aB(BA(Ab(x1))))) |
(30) |
|
aA(Ab(ba(x1))) |
→ |
ab(ba(aB(BA(Aa(x1))))) |
(31) |
|
aA(Ab(bB(x1))) |
→ |
ab(ba(aB(BA(AB(x1))))) |
(32) |
|
AB(Ba(aA(x1))) |
→ |
Aa(ab(bA(AB(BA(x1))))) |
(37) |
|
bB(Ba(aA(x1))) |
→ |
ba(ab(bA(AB(BA(x1))))) |
(41) |
|
aB(Ba(aA(x1))) |
→ |
aa(ab(bA(AB(BA(x1))))) |
(45) |
|
BB(Ba(aA(x1))) |
→ |
Ba(ab(bA(AB(BA(x1))))) |
(49) |
|
AA(Aa(aA(x1))) |
→ |
AA(x1) |
(53) |
|
AA(Aa(aa(x1))) |
→ |
Aa(x1) |
(55) |
|
AA(Aa(aB(x1))) |
→ |
AB(x1) |
(56) |
|
bA(Aa(aA(x1))) |
→ |
bA(x1) |
(57) |
|
bA(Aa(aa(x1))) |
→ |
ba(x1) |
(59) |
|
bA(Aa(aB(x1))) |
→ |
bB(x1) |
(60) |
|
aA(Aa(aA(x1))) |
→ |
aA(x1) |
(61) |
|
aA(Aa(ab(x1))) |
→ |
ab(x1) |
(62) |
|
aA(Aa(aa(x1))) |
→ |
aa(x1) |
(63) |
|
aA(Aa(aB(x1))) |
→ |
aB(x1) |
(64) |
|
BA(Aa(aA(x1))) |
→ |
BA(x1) |
(65) |
|
aB(Bb(bA(x1))) |
→ |
aA(x1) |
(77) |
|
aB(Bb(bb(x1))) |
→ |
ab(x1) |
(78) |
|
BB(Bb(bA(x1))) |
→ |
BA(x1) |
(81) |
|
BB(Bb(bb(x1))) |
→ |
Bb(x1) |
(82) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [AA(x1)] |
= |
1 · x1
|
| [Ab(x1)] |
= |
1 · x1
|
| [bA(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 · x1
|
| [aB(x1)] |
= |
1 · x1
|
| [BA(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1
|
| [Aa(x1)] |
= |
1 · x1
|
| [bB(x1)] |
= |
1 · x1 + 1 |
| [AB(x1)] |
= |
1 · x1
|
| [Bb(x1)] |
= |
1 · x1
|
| [Ba(x1)] |
= |
1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [aa(x1)] |
= |
1 · x1
|
| [BB(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
AA(Ab(bB(x1))) |
→ |
Ab(ba(aB(BA(AB(x1))))) |
(24) |
|
bA(Ab(bB(x1))) |
→ |
bb(ba(aB(BA(AB(x1))))) |
(28) |
|
BA(Ab(bB(x1))) |
→ |
Bb(ba(aB(BA(AB(x1))))) |
(36) |
|
bB(Ba(ab(x1))) |
→ |
ba(ab(bA(AB(Bb(x1))))) |
(42) |
|
bB(Ba(aa(x1))) |
→ |
ba(ab(bA(AB(Ba(x1))))) |
(43) |
|
bB(Ba(aB(x1))) |
→ |
ba(ab(bA(AB(BB(x1))))) |
(44) |
|
AB(Bb(bB(x1))) |
→ |
AB(x1) |
(72) |
|
bB(Bb(bA(x1))) |
→ |
bA(x1) |
(73) |
|
bB(Bb(bb(x1))) |
→ |
bb(x1) |
(74) |
|
bB(Bb(ba(x1))) |
→ |
ba(x1) |
(75) |
|
bB(Bb(bB(x1))) |
→ |
bB(x1) |
(76) |
|
aB(Bb(bB(x1))) |
→ |
aB(x1) |
(80) |
|
BB(Bb(bB(x1))) |
→ |
BB(x1) |
(84) |
1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
AA#(Ab(bA(x1))) |
→ |
aB#(BA(AA(x1))) |
(85) |
|
AA#(Ab(bA(x1))) |
→ |
BA#(AA(x1)) |
(86) |
|
AA#(Ab(bA(x1))) |
→ |
AA#(x1) |
(87) |
|
AA#(Ab(bb(x1))) |
→ |
aB#(BA(Ab(x1))) |
(88) |
|
AA#(Ab(bb(x1))) |
→ |
BA#(Ab(x1)) |
(89) |
|
AA#(Ab(ba(x1))) |
→ |
aB#(BA(Aa(x1))) |
(90) |
|
AA#(Ab(ba(x1))) |
→ |
BA#(Aa(x1)) |
(91) |
|
bA#(Ab(bA(x1))) |
→ |
aB#(BA(AA(x1))) |
(92) |
|
bA#(Ab(bA(x1))) |
→ |
BA#(AA(x1)) |
(93) |
|
bA#(Ab(bA(x1))) |
→ |
AA#(x1) |
(94) |
|
bA#(Ab(bb(x1))) |
→ |
aB#(BA(Ab(x1))) |
(95) |
|
bA#(Ab(bb(x1))) |
→ |
BA#(Ab(x1)) |
(96) |
|
bA#(Ab(ba(x1))) |
→ |
aB#(BA(Aa(x1))) |
(97) |
|
bA#(Ab(ba(x1))) |
→ |
BA#(Aa(x1)) |
(98) |
|
BA#(Ab(bA(x1))) |
→ |
aB#(BA(AA(x1))) |
(99) |
|
BA#(Ab(bA(x1))) |
→ |
BA#(AA(x1)) |
(100) |
|
BA#(Ab(bA(x1))) |
→ |
AA#(x1) |
(101) |
|
BA#(Ab(bb(x1))) |
→ |
aB#(BA(Ab(x1))) |
(102) |
|
BA#(Ab(bb(x1))) |
→ |
BA#(Ab(x1)) |
(103) |
|
BA#(Ab(ba(x1))) |
→ |
aB#(BA(Aa(x1))) |
(104) |
|
BA#(Ab(ba(x1))) |
→ |
BA#(Aa(x1)) |
(105) |
|
AB#(Ba(ab(x1))) |
→ |
bA#(AB(Bb(x1))) |
(106) |
|
AB#(Ba(ab(x1))) |
→ |
AB#(Bb(x1)) |
(107) |
|
AB#(Ba(aa(x1))) |
→ |
bA#(AB(Ba(x1))) |
(108) |
|
AB#(Ba(aa(x1))) |
→ |
AB#(Ba(x1)) |
(109) |
|
AB#(Ba(aB(x1))) |
→ |
bA#(AB(BB(x1))) |
(110) |
|
AB#(Ba(aB(x1))) |
→ |
AB#(BB(x1)) |
(111) |
|
AB#(Ba(aB(x1))) |
→ |
BB#(x1) |
(112) |
|
aB#(Ba(ab(x1))) |
→ |
bA#(AB(Bb(x1))) |
(113) |
|
aB#(Ba(ab(x1))) |
→ |
AB#(Bb(x1)) |
(114) |
|
aB#(Ba(aa(x1))) |
→ |
bA#(AB(Ba(x1))) |
(115) |
|
aB#(Ba(aa(x1))) |
→ |
AB#(Ba(x1)) |
(116) |
|
aB#(Ba(aB(x1))) |
→ |
bA#(AB(BB(x1))) |
(117) |
|
aB#(Ba(aB(x1))) |
→ |
AB#(BB(x1)) |
(118) |
|
aB#(Ba(aB(x1))) |
→ |
BB#(x1) |
(119) |
|
BB#(Ba(ab(x1))) |
→ |
bA#(AB(Bb(x1))) |
(120) |
|
BB#(Ba(ab(x1))) |
→ |
AB#(Bb(x1)) |
(121) |
|
BB#(Ba(aa(x1))) |
→ |
bA#(AB(Ba(x1))) |
(122) |
|
BB#(Ba(aa(x1))) |
→ |
AB#(Ba(x1)) |
(123) |
|
BB#(Ba(aB(x1))) |
→ |
bA#(AB(BB(x1))) |
(124) |
|
BB#(Ba(aB(x1))) |
→ |
AB#(BB(x1)) |
(125) |
|
BB#(Ba(aB(x1))) |
→ |
BB#(x1) |
(126) |
|
BA#(Aa(aB(x1))) |
→ |
BB#(x1) |
(127) |
|
AB#(Bb(bA(x1))) |
→ |
AA#(x1) |
(128) |
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [AA#(x1)] |
= |
1 · x1
|
| [Ab(x1)] |
= |
1 + 1 · x1
|
| [bA(x1)] |
= |
1 + 1 · x1
|
| [aB#(x1)] |
= |
1 + 1 · x1
|
| [BA(x1)] |
= |
1 · x1
|
| [AA(x1)] |
= |
1 + 1 · x1
|
| [BA#(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 + 1 · x1
|
| [ba(x1)] |
= |
1 · x1
|
| [Aa(x1)] |
= |
1 · x1
|
| [bA#(x1)] |
= |
1 · x1
|
| [AB#(x1)] |
= |
1 · x1
|
| [Ba(x1)] |
= |
1 + 1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [AB(x1)] |
= |
1 · x1
|
| [Bb(x1)] |
= |
1 · x1
|
| [aa(x1)] |
= |
1 + 1 · x1
|
| [aB(x1)] |
= |
1 + 1 · x1
|
| [BB(x1)] |
= |
1 + 1 · x1
|
| [BB#(x1)] |
= |
1 · x1
|
the
pairs
|
AA#(Ab(bA(x1))) |
→ |
BA#(AA(x1)) |
(86) |
|
AA#(Ab(bA(x1))) |
→ |
AA#(x1) |
(87) |
|
AA#(Ab(bb(x1))) |
→ |
BA#(Ab(x1)) |
(89) |
|
AA#(Ab(ba(x1))) |
→ |
BA#(Aa(x1)) |
(91) |
|
bA#(Ab(bA(x1))) |
→ |
BA#(AA(x1)) |
(93) |
|
bA#(Ab(bA(x1))) |
→ |
AA#(x1) |
(94) |
|
bA#(Ab(bb(x1))) |
→ |
BA#(Ab(x1)) |
(96) |
|
bA#(Ab(ba(x1))) |
→ |
BA#(Aa(x1)) |
(98) |
|
BA#(Ab(bA(x1))) |
→ |
BA#(AA(x1)) |
(100) |
|
BA#(Ab(bA(x1))) |
→ |
AA#(x1) |
(101) |
|
BA#(Ab(bb(x1))) |
→ |
BA#(Ab(x1)) |
(103) |
|
BA#(Ab(ba(x1))) |
→ |
BA#(Aa(x1)) |
(105) |
|
AB#(Ba(ab(x1))) |
→ |
bA#(AB(Bb(x1))) |
(106) |
|
AB#(Ba(ab(x1))) |
→ |
AB#(Bb(x1)) |
(107) |
|
AB#(Ba(aa(x1))) |
→ |
bA#(AB(Ba(x1))) |
(108) |
|
AB#(Ba(aa(x1))) |
→ |
AB#(Ba(x1)) |
(109) |
|
AB#(Ba(aB(x1))) |
→ |
bA#(AB(BB(x1))) |
(110) |
|
AB#(Ba(aB(x1))) |
→ |
AB#(BB(x1)) |
(111) |
|
AB#(Ba(aB(x1))) |
→ |
BB#(x1) |
(112) |
|
aB#(Ba(ab(x1))) |
→ |
bA#(AB(Bb(x1))) |
(113) |
|
aB#(Ba(ab(x1))) |
→ |
AB#(Bb(x1)) |
(114) |
|
aB#(Ba(aa(x1))) |
→ |
bA#(AB(Ba(x1))) |
(115) |
|
aB#(Ba(aa(x1))) |
→ |
AB#(Ba(x1)) |
(116) |
|
aB#(Ba(aB(x1))) |
→ |
bA#(AB(BB(x1))) |
(117) |
|
aB#(Ba(aB(x1))) |
→ |
AB#(BB(x1)) |
(118) |
|
aB#(Ba(aB(x1))) |
→ |
BB#(x1) |
(119) |
|
BB#(Ba(ab(x1))) |
→ |
bA#(AB(Bb(x1))) |
(120) |
|
BB#(Ba(ab(x1))) |
→ |
AB#(Bb(x1)) |
(121) |
|
BB#(Ba(aa(x1))) |
→ |
bA#(AB(Ba(x1))) |
(122) |
|
BB#(Ba(aa(x1))) |
→ |
AB#(Ba(x1)) |
(123) |
|
BB#(Ba(aB(x1))) |
→ |
bA#(AB(BB(x1))) |
(124) |
|
BB#(Ba(aB(x1))) |
→ |
AB#(BB(x1)) |
(125) |
|
BB#(Ba(aB(x1))) |
→ |
BB#(x1) |
(126) |
|
BA#(Aa(aB(x1))) |
→ |
BB#(x1) |
(127) |
|
AB#(Bb(bA(x1))) |
→ |
AA#(x1) |
(128) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.