Certification Problem
Input (TPDB TRS_Innermost/AG01_innermost/#4.31)
The rewrite relation of the following TRS is considered.
|
a(d(x)) |
→ |
d(c(b(a(x)))) |
(1) |
|
b(c(x)) |
→ |
c(d(a(b(x)))) |
(2) |
|
a(c(x)) |
→ |
x |
(3) |
|
b(d(x)) |
→ |
x |
(4) |
The evaluation strategy is innermost.Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), d(☐), c(☐), b(☐)}
We obtain the transformed TRS
|
a(a(d(x))) |
→ |
a(d(c(b(a(x))))) |
(5) |
|
d(a(d(x))) |
→ |
d(d(c(b(a(x))))) |
(6) |
|
c(a(d(x))) |
→ |
c(d(c(b(a(x))))) |
(7) |
|
b(a(d(x))) |
→ |
b(d(c(b(a(x))))) |
(8) |
|
a(b(c(x))) |
→ |
a(c(d(a(b(x))))) |
(9) |
|
d(b(c(x))) |
→ |
d(c(d(a(b(x))))) |
(10) |
|
c(b(c(x))) |
→ |
c(c(d(a(b(x))))) |
(11) |
|
b(b(c(x))) |
→ |
b(c(d(a(b(x))))) |
(12) |
|
a(a(c(x))) |
→ |
a(x) |
(13) |
|
d(a(c(x))) |
→ |
d(x) |
(14) |
|
c(a(c(x))) |
→ |
c(x) |
(15) |
|
b(a(c(x))) |
→ |
b(x) |
(16) |
|
a(b(d(x))) |
→ |
a(x) |
(17) |
|
d(b(d(x))) |
→ |
d(x) |
(18) |
|
c(b(d(x))) |
→ |
c(x) |
(19) |
|
b(b(d(x))) |
→ |
b(x) |
(20) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(ad(da(x))) |
→ |
ad(dc(cb(ba(aa(x))))) |
(21) |
|
aa(ad(dd(x))) |
→ |
ad(dc(cb(ba(ad(x))))) |
(22) |
|
aa(ad(dc(x))) |
→ |
ad(dc(cb(ba(ac(x))))) |
(23) |
|
aa(ad(db(x))) |
→ |
ad(dc(cb(ba(ab(x))))) |
(24) |
|
da(ad(da(x))) |
→ |
dd(dc(cb(ba(aa(x))))) |
(25) |
|
da(ad(dd(x))) |
→ |
dd(dc(cb(ba(ad(x))))) |
(26) |
|
da(ad(dc(x))) |
→ |
dd(dc(cb(ba(ac(x))))) |
(27) |
|
da(ad(db(x))) |
→ |
dd(dc(cb(ba(ab(x))))) |
(28) |
|
ca(ad(da(x))) |
→ |
cd(dc(cb(ba(aa(x))))) |
(29) |
|
ca(ad(dd(x))) |
→ |
cd(dc(cb(ba(ad(x))))) |
(30) |
|
ca(ad(dc(x))) |
→ |
cd(dc(cb(ba(ac(x))))) |
(31) |
|
ca(ad(db(x))) |
→ |
cd(dc(cb(ba(ab(x))))) |
(32) |
|
ba(ad(da(x))) |
→ |
bd(dc(cb(ba(aa(x))))) |
(33) |
|
ba(ad(dd(x))) |
→ |
bd(dc(cb(ba(ad(x))))) |
(34) |
|
ba(ad(dc(x))) |
→ |
bd(dc(cb(ba(ac(x))))) |
(35) |
|
ba(ad(db(x))) |
→ |
bd(dc(cb(ba(ab(x))))) |
(36) |
|
ab(bc(ca(x))) |
→ |
ac(cd(da(ab(ba(x))))) |
(37) |
|
ab(bc(cd(x))) |
→ |
ac(cd(da(ab(bd(x))))) |
(38) |
|
ab(bc(cc(x))) |
→ |
ac(cd(da(ab(bc(x))))) |
(39) |
|
ab(bc(cb(x))) |
→ |
ac(cd(da(ab(bb(x))))) |
(40) |
|
db(bc(ca(x))) |
→ |
dc(cd(da(ab(ba(x))))) |
(41) |
|
db(bc(cd(x))) |
→ |
dc(cd(da(ab(bd(x))))) |
(42) |
|
db(bc(cc(x))) |
→ |
dc(cd(da(ab(bc(x))))) |
(43) |
|
db(bc(cb(x))) |
→ |
dc(cd(da(ab(bb(x))))) |
(44) |
|
cb(bc(ca(x))) |
→ |
cc(cd(da(ab(ba(x))))) |
(45) |
|
cb(bc(cd(x))) |
→ |
cc(cd(da(ab(bd(x))))) |
(46) |
|
cb(bc(cc(x))) |
→ |
cc(cd(da(ab(bc(x))))) |
(47) |
|
cb(bc(cb(x))) |
→ |
cc(cd(da(ab(bb(x))))) |
(48) |
|
bb(bc(ca(x))) |
→ |
bc(cd(da(ab(ba(x))))) |
(49) |
|
bb(bc(cd(x))) |
→ |
bc(cd(da(ab(bd(x))))) |
(50) |
|
bb(bc(cc(x))) |
→ |
bc(cd(da(ab(bc(x))))) |
(51) |
|
bb(bc(cb(x))) |
→ |
bc(cd(da(ab(bb(x))))) |
(52) |
|
aa(ac(ca(x))) |
→ |
aa(x) |
(53) |
|
aa(ac(cd(x))) |
→ |
ad(x) |
(54) |
|
aa(ac(cc(x))) |
→ |
ac(x) |
(55) |
|
aa(ac(cb(x))) |
→ |
ab(x) |
(56) |
|
da(ac(ca(x))) |
→ |
da(x) |
(57) |
|
da(ac(cd(x))) |
→ |
dd(x) |
(58) |
|
da(ac(cc(x))) |
→ |
dc(x) |
(59) |
|
da(ac(cb(x))) |
→ |
db(x) |
(60) |
|
ca(ac(ca(x))) |
→ |
ca(x) |
(61) |
|
ca(ac(cd(x))) |
→ |
cd(x) |
(62) |
|
ca(ac(cc(x))) |
→ |
cc(x) |
(63) |
|
ca(ac(cb(x))) |
→ |
cb(x) |
(64) |
|
ba(ac(ca(x))) |
→ |
ba(x) |
(65) |
|
ba(ac(cd(x))) |
→ |
bd(x) |
(66) |
|
ba(ac(cc(x))) |
→ |
bc(x) |
(67) |
|
ba(ac(cb(x))) |
→ |
bb(x) |
(68) |
|
ab(bd(da(x))) |
→ |
aa(x) |
(69) |
|
ab(bd(dd(x))) |
→ |
ad(x) |
(70) |
|
ab(bd(dc(x))) |
→ |
ac(x) |
(71) |
|
ab(bd(db(x))) |
→ |
ab(x) |
(72) |
|
db(bd(da(x))) |
→ |
da(x) |
(73) |
|
db(bd(dd(x))) |
→ |
dd(x) |
(74) |
|
db(bd(dc(x))) |
→ |
dc(x) |
(75) |
|
db(bd(db(x))) |
→ |
db(x) |
(76) |
|
cb(bd(da(x))) |
→ |
ca(x) |
(77) |
|
cb(bd(dd(x))) |
→ |
cd(x) |
(78) |
|
cb(bd(dc(x))) |
→ |
cc(x) |
(79) |
|
cb(bd(db(x))) |
→ |
cb(x) |
(80) |
|
bb(bd(da(x))) |
→ |
ba(x) |
(81) |
|
bb(bd(dd(x))) |
→ |
bd(x) |
(82) |
|
bb(bd(dc(x))) |
→ |
bc(x) |
(83) |
|
bb(bd(db(x))) |
→ |
bb(x) |
(84) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 · x1 + 1 |
| [ad(x1)] |
= |
1 · x1 + 1 |
| [da(x1)] |
= |
1 · x1 + 1 |
| [dc(x1)] |
= |
1 · x1
|
| [cb(x1)] |
= |
1 · x1 + 1 |
| [ba(x1)] |
= |
1 · x1
|
| [dd(x1)] |
= |
1 · x1 + 1 |
| [ac(x1)] |
= |
1 · x1
|
| [db(x1)] |
= |
1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [ca(x1)] |
= |
1 · x1 + 1 |
| [cd(x1)] |
= |
1 · x1
|
| [bd(x1)] |
= |
1 · x1
|
| [bc(x1)] |
= |
1 · x1 + 1 |
| [cc(x1)] |
= |
1 · x1 + 1 |
| [bb(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
|
ca(ad(da(x))) |
→ |
cd(dc(cb(ba(aa(x))))) |
(29) |
|
ca(ad(dd(x))) |
→ |
cd(dc(cb(ba(ad(x))))) |
(30) |
|
ca(ad(dc(x))) |
→ |
cd(dc(cb(ba(ac(x))))) |
(31) |
|
ca(ad(db(x))) |
→ |
cd(dc(cb(ba(ab(x))))) |
(32) |
|
ab(bc(ca(x))) |
→ |
ac(cd(da(ab(ba(x))))) |
(37) |
|
db(bc(ca(x))) |
→ |
dc(cd(da(ab(ba(x))))) |
(41) |
|
cb(bc(ca(x))) |
→ |
cc(cd(da(ab(ba(x))))) |
(45) |
|
bb(bc(ca(x))) |
→ |
bc(cd(da(ab(ba(x))))) |
(49) |
|
aa(ac(ca(x))) |
→ |
aa(x) |
(53) |
|
aa(ac(cc(x))) |
→ |
ac(x) |
(55) |
|
aa(ac(cb(x))) |
→ |
ab(x) |
(56) |
|
da(ac(ca(x))) |
→ |
da(x) |
(57) |
|
da(ac(cc(x))) |
→ |
dc(x) |
(59) |
|
da(ac(cb(x))) |
→ |
db(x) |
(60) |
|
ca(ac(ca(x))) |
→ |
ca(x) |
(61) |
|
ca(ac(cd(x))) |
→ |
cd(x) |
(62) |
|
ca(ac(cc(x))) |
→ |
cc(x) |
(63) |
|
ca(ac(cb(x))) |
→ |
cb(x) |
(64) |
|
ba(ac(ca(x))) |
→ |
ba(x) |
(65) |
|
cb(bd(da(x))) |
→ |
ca(x) |
(77) |
|
cb(bd(dd(x))) |
→ |
cd(x) |
(78) |
|
bb(bd(da(x))) |
→ |
ba(x) |
(81) |
|
bb(bd(dd(x))) |
→ |
bd(x) |
(82) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 · x1
|
| [ad(x1)] |
= |
1 · x1
|
| [da(x1)] |
= |
1 · x1
|
| [dc(x1)] |
= |
1 · x1
|
| [cb(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 · x1
|
| [dd(x1)] |
= |
1 · x1
|
| [ac(x1)] |
= |
1 · x1
|
| [db(x1)] |
= |
1 · x1 + 1 |
| [ab(x1)] |
= |
1 · x1
|
| [bd(x1)] |
= |
1 · x1
|
| [bc(x1)] |
= |
1 · x1
|
| [cd(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
aa(ad(db(x))) |
→ |
ad(dc(cb(ba(ab(x))))) |
(24) |
|
da(ad(db(x))) |
→ |
dd(dc(cb(ba(ab(x))))) |
(28) |
|
ba(ad(db(x))) |
→ |
bd(dc(cb(ba(ab(x))))) |
(36) |
|
db(bc(cd(x))) |
→ |
dc(cd(da(ab(bd(x))))) |
(42) |
|
db(bc(cc(x))) |
→ |
dc(cd(da(ab(bc(x))))) |
(43) |
|
db(bc(cb(x))) |
→ |
dc(cd(da(ab(bb(x))))) |
(44) |
|
ab(bd(db(x))) |
→ |
ab(x) |
(72) |
|
db(bd(da(x))) |
→ |
da(x) |
(73) |
|
db(bd(dd(x))) |
→ |
dd(x) |
(74) |
|
db(bd(dc(x))) |
→ |
dc(x) |
(75) |
|
db(bd(db(x))) |
→ |
db(x) |
(76) |
|
cb(bd(db(x))) |
→ |
cb(x) |
(80) |
|
bb(bd(db(x))) |
→ |
bb(x) |
(84) |
1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
aa#(ad(da(x))) |
→ |
cb#(ba(aa(x))) |
(85) |
|
aa#(ad(da(x))) |
→ |
ba#(aa(x)) |
(86) |
|
aa#(ad(da(x))) |
→ |
aa#(x) |
(87) |
|
aa#(ad(dd(x))) |
→ |
cb#(ba(ad(x))) |
(88) |
|
aa#(ad(dd(x))) |
→ |
ba#(ad(x)) |
(89) |
|
aa#(ad(dc(x))) |
→ |
cb#(ba(ac(x))) |
(90) |
|
aa#(ad(dc(x))) |
→ |
ba#(ac(x)) |
(91) |
|
da#(ad(da(x))) |
→ |
cb#(ba(aa(x))) |
(92) |
|
da#(ad(da(x))) |
→ |
ba#(aa(x)) |
(93) |
|
da#(ad(da(x))) |
→ |
aa#(x) |
(94) |
|
da#(ad(dd(x))) |
→ |
cb#(ba(ad(x))) |
(95) |
|
da#(ad(dd(x))) |
→ |
ba#(ad(x)) |
(96) |
|
da#(ad(dc(x))) |
→ |
cb#(ba(ac(x))) |
(97) |
|
da#(ad(dc(x))) |
→ |
ba#(ac(x)) |
(98) |
|
ba#(ad(da(x))) |
→ |
cb#(ba(aa(x))) |
(99) |
|
ba#(ad(da(x))) |
→ |
ba#(aa(x)) |
(100) |
|
ba#(ad(da(x))) |
→ |
aa#(x) |
(101) |
|
ba#(ad(dd(x))) |
→ |
cb#(ba(ad(x))) |
(102) |
|
ba#(ad(dd(x))) |
→ |
ba#(ad(x)) |
(103) |
|
ba#(ad(dc(x))) |
→ |
cb#(ba(ac(x))) |
(104) |
|
ba#(ad(dc(x))) |
→ |
ba#(ac(x)) |
(105) |
|
ab#(bc(cd(x))) |
→ |
da#(ab(bd(x))) |
(106) |
|
ab#(bc(cd(x))) |
→ |
ab#(bd(x)) |
(107) |
|
ab#(bc(cc(x))) |
→ |
da#(ab(bc(x))) |
(108) |
|
ab#(bc(cc(x))) |
→ |
ab#(bc(x)) |
(109) |
|
ab#(bc(cb(x))) |
→ |
da#(ab(bb(x))) |
(110) |
|
ab#(bc(cb(x))) |
→ |
ab#(bb(x)) |
(111) |
|
ab#(bc(cb(x))) |
→ |
bb#(x) |
(112) |
|
cb#(bc(cd(x))) |
→ |
da#(ab(bd(x))) |
(113) |
|
cb#(bc(cd(x))) |
→ |
ab#(bd(x)) |
(114) |
|
cb#(bc(cc(x))) |
→ |
da#(ab(bc(x))) |
(115) |
|
cb#(bc(cc(x))) |
→ |
ab#(bc(x)) |
(116) |
|
cb#(bc(cb(x))) |
→ |
da#(ab(bb(x))) |
(117) |
|
cb#(bc(cb(x))) |
→ |
ab#(bb(x)) |
(118) |
|
cb#(bc(cb(x))) |
→ |
bb#(x) |
(119) |
|
bb#(bc(cd(x))) |
→ |
da#(ab(bd(x))) |
(120) |
|
bb#(bc(cd(x))) |
→ |
ab#(bd(x)) |
(121) |
|
bb#(bc(cc(x))) |
→ |
da#(ab(bc(x))) |
(122) |
|
bb#(bc(cc(x))) |
→ |
ab#(bc(x)) |
(123) |
|
bb#(bc(cb(x))) |
→ |
da#(ab(bb(x))) |
(124) |
|
bb#(bc(cb(x))) |
→ |
ab#(bb(x)) |
(125) |
|
bb#(bc(cb(x))) |
→ |
bb#(x) |
(126) |
|
ba#(ac(cb(x))) |
→ |
bb#(x) |
(127) |
|
ab#(bd(da(x))) |
→ |
aa#(x) |
(128) |
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [aa#(x1)] |
= |
1 + 1 · x1
|
| [ad(x1)] |
= |
1 + 1 · x1
|
| [da(x1)] |
= |
1 + 1 · x1
|
| [cb#(x1)] |
= |
1 + 1 · x1
|
| [ba(x1)] |
= |
1 · x1
|
| [aa(x1)] |
= |
1 + 1 · x1
|
| [ba#(x1)] |
= |
1 · x1
|
| [dd(x1)] |
= |
1 + 1 · x1
|
| [dc(x1)] |
= |
1 · x1
|
| [ac(x1)] |
= |
1 · x1
|
| [da#(x1)] |
= |
1 · x1
|
| [ab#(x1)] |
= |
1 + 1 · x1
|
| [bc(x1)] |
= |
1 + 1 · x1
|
| [cd(x1)] |
= |
1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [bd(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 + 1 · x1
|
| [cb(x1)] |
= |
1 + 1 · x1
|
| [bb(x1)] |
= |
1 + 1 · x1
|
| [bb#(x1)] |
= |
1 · x1
|
the
pairs
|
aa#(ad(da(x))) |
→ |
cb#(ba(aa(x))) |
(85) |
|
aa#(ad(da(x))) |
→ |
ba#(aa(x)) |
(86) |
|
aa#(ad(da(x))) |
→ |
aa#(x) |
(87) |
|
aa#(ad(dd(x))) |
→ |
cb#(ba(ad(x))) |
(88) |
|
aa#(ad(dd(x))) |
→ |
ba#(ad(x)) |
(89) |
|
aa#(ad(dc(x))) |
→ |
cb#(ba(ac(x))) |
(90) |
|
aa#(ad(dc(x))) |
→ |
ba#(ac(x)) |
(91) |
|
da#(ad(da(x))) |
→ |
ba#(aa(x)) |
(93) |
|
da#(ad(da(x))) |
→ |
aa#(x) |
(94) |
|
da#(ad(dd(x))) |
→ |
ba#(ad(x)) |
(96) |
|
da#(ad(dc(x))) |
→ |
ba#(ac(x)) |
(98) |
|
ba#(ad(da(x))) |
→ |
ba#(aa(x)) |
(100) |
|
ba#(ad(da(x))) |
→ |
aa#(x) |
(101) |
|
ba#(ad(dd(x))) |
→ |
ba#(ad(x)) |
(103) |
|
ba#(ad(dc(x))) |
→ |
ba#(ac(x)) |
(105) |
|
ab#(bc(cd(x))) |
→ |
da#(ab(bd(x))) |
(106) |
|
ab#(bc(cd(x))) |
→ |
ab#(bd(x)) |
(107) |
|
ab#(bc(cc(x))) |
→ |
da#(ab(bc(x))) |
(108) |
|
ab#(bc(cc(x))) |
→ |
ab#(bc(x)) |
(109) |
|
ab#(bc(cb(x))) |
→ |
da#(ab(bb(x))) |
(110) |
|
ab#(bc(cb(x))) |
→ |
ab#(bb(x)) |
(111) |
|
ab#(bc(cb(x))) |
→ |
bb#(x) |
(112) |
|
cb#(bc(cd(x))) |
→ |
da#(ab(bd(x))) |
(113) |
|
cb#(bc(cd(x))) |
→ |
ab#(bd(x)) |
(114) |
|
cb#(bc(cc(x))) |
→ |
da#(ab(bc(x))) |
(115) |
|
cb#(bc(cc(x))) |
→ |
ab#(bc(x)) |
(116) |
|
cb#(bc(cb(x))) |
→ |
da#(ab(bb(x))) |
(117) |
|
cb#(bc(cb(x))) |
→ |
ab#(bb(x)) |
(118) |
|
cb#(bc(cb(x))) |
→ |
bb#(x) |
(119) |
|
bb#(bc(cd(x))) |
→ |
da#(ab(bd(x))) |
(120) |
|
bb#(bc(cc(x))) |
→ |
da#(ab(bc(x))) |
(122) |
|
bb#(bc(cb(x))) |
→ |
da#(ab(bb(x))) |
(124) |
|
bb#(bc(cb(x))) |
→ |
bb#(x) |
(126) |
|
ba#(ac(cb(x))) |
→ |
bb#(x) |
(127) |
|
ab#(bd(da(x))) |
→ |
aa#(x) |
(128) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.