Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z090)
The rewrite relation of the following TRS is considered.
|
s(b(x1)) |
→ |
b(s(s(s(x1)))) |
(1) |
|
s(b(s(x1))) |
→ |
b(t(x1)) |
(2) |
|
t(b(x1)) |
→ |
b(s(x1)) |
(3) |
|
t(b(s(x1))) |
→ |
u(t(b(x1))) |
(4) |
|
b(u(x1)) |
→ |
b(s(x1)) |
(5) |
|
t(s(x1)) |
→ |
t(t(x1)) |
(6) |
|
t(u(x1)) |
→ |
u(t(x1)) |
(7) |
|
s(u(x1)) |
→ |
s(s(x1)) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{s(☐), b(☐), t(☐), u(☐)}
We obtain the transformed TRS
|
b(u(x1)) |
→ |
b(s(x1)) |
(5) |
|
t(s(x1)) |
→ |
t(t(x1)) |
(6) |
|
s(u(x1)) |
→ |
s(s(x1)) |
(8) |
|
s(s(b(x1))) |
→ |
s(b(s(s(s(x1))))) |
(9) |
|
b(s(b(x1))) |
→ |
b(b(s(s(s(x1))))) |
(10) |
|
t(s(b(x1))) |
→ |
t(b(s(s(s(x1))))) |
(11) |
|
u(s(b(x1))) |
→ |
u(b(s(s(s(x1))))) |
(12) |
|
s(s(b(s(x1)))) |
→ |
s(b(t(x1))) |
(13) |
|
b(s(b(s(x1)))) |
→ |
b(b(t(x1))) |
(14) |
|
t(s(b(s(x1)))) |
→ |
t(b(t(x1))) |
(15) |
|
u(s(b(s(x1)))) |
→ |
u(b(t(x1))) |
(16) |
|
s(t(b(x1))) |
→ |
s(b(s(x1))) |
(17) |
|
b(t(b(x1))) |
→ |
b(b(s(x1))) |
(18) |
|
t(t(b(x1))) |
→ |
t(b(s(x1))) |
(19) |
|
u(t(b(x1))) |
→ |
u(b(s(x1))) |
(20) |
|
s(t(b(s(x1)))) |
→ |
s(u(t(b(x1)))) |
(21) |
|
b(t(b(s(x1)))) |
→ |
b(u(t(b(x1)))) |
(22) |
|
t(t(b(s(x1)))) |
→ |
t(u(t(b(x1)))) |
(23) |
|
u(t(b(s(x1)))) |
→ |
u(u(t(b(x1)))) |
(24) |
|
s(t(u(x1))) |
→ |
s(u(t(x1))) |
(25) |
|
b(t(u(x1))) |
→ |
b(u(t(x1))) |
(26) |
|
t(t(u(x1))) |
→ |
t(u(t(x1))) |
(27) |
|
u(t(u(x1))) |
→ |
u(u(t(x1))) |
(28) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
bu(ub(x1)) |
→ |
bs(sb(x1)) |
(29) |
|
bu(uu(x1)) |
→ |
bs(su(x1)) |
(30) |
|
bu(us(x1)) |
→ |
bs(ss(x1)) |
(31) |
|
bu(ut(x1)) |
→ |
bs(st(x1)) |
(32) |
|
ts(sb(x1)) |
→ |
tt(tb(x1)) |
(33) |
|
ts(su(x1)) |
→ |
tt(tu(x1)) |
(34) |
|
ts(ss(x1)) |
→ |
tt(ts(x1)) |
(35) |
|
ts(st(x1)) |
→ |
tt(tt(x1)) |
(36) |
|
su(ub(x1)) |
→ |
ss(sb(x1)) |
(37) |
|
su(uu(x1)) |
→ |
ss(su(x1)) |
(38) |
|
su(us(x1)) |
→ |
ss(ss(x1)) |
(39) |
|
su(ut(x1)) |
→ |
ss(st(x1)) |
(40) |
|
ss(sb(bb(x1))) |
→ |
sb(bs(ss(ss(sb(x1))))) |
(41) |
|
ss(sb(bu(x1))) |
→ |
sb(bs(ss(ss(su(x1))))) |
(42) |
|
ss(sb(bs(x1))) |
→ |
sb(bs(ss(ss(ss(x1))))) |
(43) |
|
ss(sb(bt(x1))) |
→ |
sb(bs(ss(ss(st(x1))))) |
(44) |
|
bs(sb(bb(x1))) |
→ |
bb(bs(ss(ss(sb(x1))))) |
(45) |
|
bs(sb(bu(x1))) |
→ |
bb(bs(ss(ss(su(x1))))) |
(46) |
|
bs(sb(bs(x1))) |
→ |
bb(bs(ss(ss(ss(x1))))) |
(47) |
|
bs(sb(bt(x1))) |
→ |
bb(bs(ss(ss(st(x1))))) |
(48) |
|
ts(sb(bb(x1))) |
→ |
tb(bs(ss(ss(sb(x1))))) |
(49) |
|
ts(sb(bu(x1))) |
→ |
tb(bs(ss(ss(su(x1))))) |
(50) |
|
ts(sb(bs(x1))) |
→ |
tb(bs(ss(ss(ss(x1))))) |
(51) |
|
ts(sb(bt(x1))) |
→ |
tb(bs(ss(ss(st(x1))))) |
(52) |
|
us(sb(bb(x1))) |
→ |
ub(bs(ss(ss(sb(x1))))) |
(53) |
|
us(sb(bu(x1))) |
→ |
ub(bs(ss(ss(su(x1))))) |
(54) |
|
us(sb(bs(x1))) |
→ |
ub(bs(ss(ss(ss(x1))))) |
(55) |
|
us(sb(bt(x1))) |
→ |
ub(bs(ss(ss(st(x1))))) |
(56) |
|
ss(sb(bs(sb(x1)))) |
→ |
sb(bt(tb(x1))) |
(57) |
|
ss(sb(bs(su(x1)))) |
→ |
sb(bt(tu(x1))) |
(58) |
|
ss(sb(bs(ss(x1)))) |
→ |
sb(bt(ts(x1))) |
(59) |
|
ss(sb(bs(st(x1)))) |
→ |
sb(bt(tt(x1))) |
(60) |
|
bs(sb(bs(sb(x1)))) |
→ |
bb(bt(tb(x1))) |
(61) |
|
bs(sb(bs(su(x1)))) |
→ |
bb(bt(tu(x1))) |
(62) |
|
bs(sb(bs(ss(x1)))) |
→ |
bb(bt(ts(x1))) |
(63) |
|
bs(sb(bs(st(x1)))) |
→ |
bb(bt(tt(x1))) |
(64) |
|
ts(sb(bs(sb(x1)))) |
→ |
tb(bt(tb(x1))) |
(65) |
|
ts(sb(bs(su(x1)))) |
→ |
tb(bt(tu(x1))) |
(66) |
|
ts(sb(bs(ss(x1)))) |
→ |
tb(bt(ts(x1))) |
(67) |
|
ts(sb(bs(st(x1)))) |
→ |
tb(bt(tt(x1))) |
(68) |
|
us(sb(bs(sb(x1)))) |
→ |
ub(bt(tb(x1))) |
(69) |
|
us(sb(bs(su(x1)))) |
→ |
ub(bt(tu(x1))) |
(70) |
|
us(sb(bs(ss(x1)))) |
→ |
ub(bt(ts(x1))) |
(71) |
|
us(sb(bs(st(x1)))) |
→ |
ub(bt(tt(x1))) |
(72) |
|
st(tb(bb(x1))) |
→ |
sb(bs(sb(x1))) |
(73) |
|
st(tb(bu(x1))) |
→ |
sb(bs(su(x1))) |
(74) |
|
st(tb(bs(x1))) |
→ |
sb(bs(ss(x1))) |
(75) |
|
st(tb(bt(x1))) |
→ |
sb(bs(st(x1))) |
(76) |
|
bt(tb(bb(x1))) |
→ |
bb(bs(sb(x1))) |
(77) |
|
bt(tb(bu(x1))) |
→ |
bb(bs(su(x1))) |
(78) |
|
bt(tb(bs(x1))) |
→ |
bb(bs(ss(x1))) |
(79) |
|
bt(tb(bt(x1))) |
→ |
bb(bs(st(x1))) |
(80) |
|
tt(tb(bb(x1))) |
→ |
tb(bs(sb(x1))) |
(81) |
|
tt(tb(bu(x1))) |
→ |
tb(bs(su(x1))) |
(82) |
|
tt(tb(bs(x1))) |
→ |
tb(bs(ss(x1))) |
(83) |
|
tt(tb(bt(x1))) |
→ |
tb(bs(st(x1))) |
(84) |
|
ut(tb(bb(x1))) |
→ |
ub(bs(sb(x1))) |
(85) |
|
ut(tb(bu(x1))) |
→ |
ub(bs(su(x1))) |
(86) |
|
ut(tb(bs(x1))) |
→ |
ub(bs(ss(x1))) |
(87) |
|
ut(tb(bt(x1))) |
→ |
ub(bs(st(x1))) |
(88) |
|
st(tb(bs(sb(x1)))) |
→ |
su(ut(tb(bb(x1)))) |
(89) |
|
st(tb(bs(su(x1)))) |
→ |
su(ut(tb(bu(x1)))) |
(90) |
|
st(tb(bs(ss(x1)))) |
→ |
su(ut(tb(bs(x1)))) |
(91) |
|
st(tb(bs(st(x1)))) |
→ |
su(ut(tb(bt(x1)))) |
(92) |
|
bt(tb(bs(sb(x1)))) |
→ |
bu(ut(tb(bb(x1)))) |
(93) |
|
bt(tb(bs(su(x1)))) |
→ |
bu(ut(tb(bu(x1)))) |
(94) |
|
bt(tb(bs(ss(x1)))) |
→ |
bu(ut(tb(bs(x1)))) |
(95) |
|
bt(tb(bs(st(x1)))) |
→ |
bu(ut(tb(bt(x1)))) |
(96) |
|
tt(tb(bs(sb(x1)))) |
→ |
tu(ut(tb(bb(x1)))) |
(97) |
|
tt(tb(bs(su(x1)))) |
→ |
tu(ut(tb(bu(x1)))) |
(98) |
|
tt(tb(bs(ss(x1)))) |
→ |
tu(ut(tb(bs(x1)))) |
(99) |
|
tt(tb(bs(st(x1)))) |
→ |
tu(ut(tb(bt(x1)))) |
(100) |
|
ut(tb(bs(sb(x1)))) |
→ |
uu(ut(tb(bb(x1)))) |
(101) |
|
ut(tb(bs(su(x1)))) |
→ |
uu(ut(tb(bu(x1)))) |
(102) |
|
ut(tb(bs(ss(x1)))) |
→ |
uu(ut(tb(bs(x1)))) |
(103) |
|
ut(tb(bs(st(x1)))) |
→ |
uu(ut(tb(bt(x1)))) |
(104) |
|
st(tu(ub(x1))) |
→ |
su(ut(tb(x1))) |
(105) |
|
st(tu(uu(x1))) |
→ |
su(ut(tu(x1))) |
(106) |
|
st(tu(us(x1))) |
→ |
su(ut(ts(x1))) |
(107) |
|
st(tu(ut(x1))) |
→ |
su(ut(tt(x1))) |
(108) |
|
bt(tu(ub(x1))) |
→ |
bu(ut(tb(x1))) |
(109) |
|
bt(tu(uu(x1))) |
→ |
bu(ut(tu(x1))) |
(110) |
|
bt(tu(us(x1))) |
→ |
bu(ut(ts(x1))) |
(111) |
|
bt(tu(ut(x1))) |
→ |
bu(ut(tt(x1))) |
(112) |
|
tt(tu(ub(x1))) |
→ |
tu(ut(tb(x1))) |
(113) |
|
tt(tu(uu(x1))) |
→ |
tu(ut(tu(x1))) |
(114) |
|
tt(tu(us(x1))) |
→ |
tu(ut(ts(x1))) |
(115) |
|
tt(tu(ut(x1))) |
→ |
tu(ut(tt(x1))) |
(116) |
|
ut(tu(ub(x1))) |
→ |
uu(ut(tb(x1))) |
(117) |
|
ut(tu(uu(x1))) |
→ |
uu(ut(tu(x1))) |
(118) |
|
ut(tu(us(x1))) |
→ |
uu(ut(ts(x1))) |
(119) |
|
ut(tu(ut(x1))) |
→ |
uu(ut(tt(x1))) |
(120) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [bu(x1)] |
= |
1 · x1
|
| [ub(x1)] |
= |
1 · x1
|
| [bs(x1)] |
= |
1 · x1
|
| [sb(x1)] |
= |
1 · x1
|
| [uu(x1)] |
= |
1 · x1
|
| [su(x1)] |
= |
1 · x1
|
| [us(x1)] |
= |
1 · x1 + 1 |
| [ss(x1)] |
= |
1 · x1
|
| [ut(x1)] |
= |
1 · x1
|
| [st(x1)] |
= |
1 · x1
|
| [ts(x1)] |
= |
1 · x1
|
| [tt(x1)] |
= |
1 · x1
|
| [tb(x1)] |
= |
1 · x1
|
| [tu(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1
|
| [bt(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
bu(us(x1)) |
→ |
bs(ss(x1)) |
(31) |
|
su(us(x1)) |
→ |
ss(ss(x1)) |
(39) |
|
us(sb(bb(x1))) |
→ |
ub(bs(ss(ss(sb(x1))))) |
(53) |
|
us(sb(bu(x1))) |
→ |
ub(bs(ss(ss(su(x1))))) |
(54) |
|
us(sb(bs(x1))) |
→ |
ub(bs(ss(ss(ss(x1))))) |
(55) |
|
us(sb(bt(x1))) |
→ |
ub(bs(ss(ss(st(x1))))) |
(56) |
|
us(sb(bs(sb(x1)))) |
→ |
ub(bt(tb(x1))) |
(69) |
|
us(sb(bs(su(x1)))) |
→ |
ub(bt(tu(x1))) |
(70) |
|
us(sb(bs(ss(x1)))) |
→ |
ub(bt(ts(x1))) |
(71) |
|
us(sb(bs(st(x1)))) |
→ |
ub(bt(tt(x1))) |
(72) |
|
st(tu(us(x1))) |
→ |
su(ut(ts(x1))) |
(107) |
|
bt(tu(us(x1))) |
→ |
bu(ut(ts(x1))) |
(111) |
|
tt(tu(us(x1))) |
→ |
tu(ut(ts(x1))) |
(115) |
|
ut(tu(us(x1))) |
→ |
uu(ut(ts(x1))) |
(119) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
There are 165 ruless (increase limit for explicit display).
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [bu#(x1)] |
= |
1 + 1 · x1
|
| [ub(x1)] |
= |
1 + 1 · x1
|
| [bs#(x1)] |
= |
1 · x1
|
| [sb(x1)] |
= |
1 + 1 · x1
|
| [uu(x1)] |
= |
1 · x1
|
| [su(x1)] |
= |
1 · x1
|
| [su#(x1)] |
= |
1 + 1 · x1
|
| [ut(x1)] |
= |
1 · x1
|
| [st(x1)] |
= |
1 · x1
|
| [st#(x1)] |
= |
1 + 1 · x1
|
| [ts#(x1)] |
= |
1 + 1 · x1
|
| [tt#(x1)] |
= |
1 · x1
|
| [tb(x1)] |
= |
1 + 1 · x1
|
| [tu(x1)] |
= |
1 · x1
|
| [ss(x1)] |
= |
1 · x1
|
| [ts(x1)] |
= |
1 · x1
|
| [tt(x1)] |
= |
1 · x1
|
| [ss#(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 + 1 · x1
|
| [bu(x1)] |
= |
1 · x1
|
| [bs(x1)] |
= |
1 · x1
|
| [bt(x1)] |
= |
1 · x1
|
| [bt#(x1)] |
= |
1 + 1 · x1
|
| [ut#(x1)] |
= |
1 · x1
|
the
pairsThere are 108 ruless (increase limit for explicit display).
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
tt#(tu(uu(x1))) |
→ |
ut#(tu(x1)) |
(279) |
|
ut#(tu(uu(x1))) |
→ |
ut#(tu(x1)) |
(283) |
|
ut#(tu(ut(x1))) |
→ |
ut#(tt(x1)) |
(284) |
|
ut#(tu(ut(x1))) |
→ |
tt#(x1) |
(285) |
|
tt#(tu(ut(x1))) |
→ |
ut#(tt(x1)) |
(280) |
|
tt#(tu(ut(x1))) |
→ |
tt#(x1) |
(281) |
1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [tt#(x1)] |
= |
1 · x1
|
| [tu(x1)] |
= |
1 + 1 · x1
|
| [uu(x1)] |
= |
1 + 1 · x1
|
| [ut#(x1)] |
= |
1 · x1
|
| [ut(x1)] |
= |
1 + 1 · x1
|
| [tt(x1)] |
= |
1 + 1 · x1
|
| [tb(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
0 |
| [bs(x1)] |
= |
1 + 1 · x1
|
| [sb(x1)] |
= |
0 |
| [bu(x1)] |
= |
1 + 1 · x1
|
| [su(x1)] |
= |
1 + 1 · x1
|
| [ss(x1)] |
= |
1 + 1 · x1
|
| [bt(x1)] |
= |
1 + 1 · x1
|
| [st(x1)] |
= |
1 + 1 · x1
|
| [ub(x1)] |
= |
1 · x1
|
| [ts(x1)] |
= |
0 |
together with the usable
rules
|
tt(tb(bb(x1))) |
→ |
tb(bs(sb(x1))) |
(81) |
|
tt(tb(bu(x1))) |
→ |
tb(bs(su(x1))) |
(82) |
|
tt(tb(bs(x1))) |
→ |
tb(bs(ss(x1))) |
(83) |
|
tt(tb(bt(x1))) |
→ |
tb(bs(st(x1))) |
(84) |
|
tt(tb(bs(sb(x1)))) |
→ |
tu(ut(tb(bb(x1)))) |
(97) |
|
tt(tb(bs(su(x1)))) |
→ |
tu(ut(tb(bu(x1)))) |
(98) |
|
tt(tb(bs(ss(x1)))) |
→ |
tu(ut(tb(bs(x1)))) |
(99) |
|
tt(tb(bs(st(x1)))) |
→ |
tu(ut(tb(bt(x1)))) |
(100) |
|
tt(tu(ub(x1))) |
→ |
tu(ut(tb(x1))) |
(113) |
|
tt(tu(uu(x1))) |
→ |
tu(ut(tu(x1))) |
(114) |
|
tt(tu(ut(x1))) |
→ |
tu(ut(tt(x1))) |
(116) |
|
bs(sb(bb(x1))) |
→ |
bb(bs(ss(ss(sb(x1))))) |
(45) |
|
bs(sb(bu(x1))) |
→ |
bb(bs(ss(ss(su(x1))))) |
(46) |
|
bs(sb(bs(x1))) |
→ |
bb(bs(ss(ss(ss(x1))))) |
(47) |
|
bs(sb(bt(x1))) |
→ |
bb(bs(ss(ss(st(x1))))) |
(48) |
|
bs(sb(bs(sb(x1)))) |
→ |
bb(bt(tb(x1))) |
(61) |
|
bs(sb(bs(su(x1)))) |
→ |
bb(bt(tu(x1))) |
(62) |
|
bs(sb(bs(ss(x1)))) |
→ |
bb(bt(ts(x1))) |
(63) |
|
bs(sb(bs(st(x1)))) |
→ |
bb(bt(tt(x1))) |
(64) |
|
su(ub(x1)) |
→ |
ss(sb(x1)) |
(37) |
|
su(uu(x1)) |
→ |
ss(su(x1)) |
(38) |
|
su(ut(x1)) |
→ |
ss(st(x1)) |
(40) |
|
ss(sb(bb(x1))) |
→ |
sb(bs(ss(ss(sb(x1))))) |
(41) |
|
ss(sb(bu(x1))) |
→ |
sb(bs(ss(ss(su(x1))))) |
(42) |
|
ss(sb(bs(x1))) |
→ |
sb(bs(ss(ss(ss(x1))))) |
(43) |
|
ss(sb(bt(x1))) |
→ |
sb(bs(ss(ss(st(x1))))) |
(44) |
|
ss(sb(bs(sb(x1)))) |
→ |
sb(bt(tb(x1))) |
(57) |
|
ss(sb(bs(su(x1)))) |
→ |
sb(bt(tu(x1))) |
(58) |
|
ss(sb(bs(ss(x1)))) |
→ |
sb(bt(ts(x1))) |
(59) |
|
ss(sb(bs(st(x1)))) |
→ |
sb(bt(tt(x1))) |
(60) |
|
ut(tb(bb(x1))) |
→ |
ub(bs(sb(x1))) |
(85) |
|
ut(tb(bu(x1))) |
→ |
ub(bs(su(x1))) |
(86) |
|
ut(tb(bs(x1))) |
→ |
ub(bs(ss(x1))) |
(87) |
|
ut(tb(bt(x1))) |
→ |
ub(bs(st(x1))) |
(88) |
|
ut(tb(bs(sb(x1)))) |
→ |
uu(ut(tb(bb(x1)))) |
(101) |
|
ut(tb(bs(su(x1)))) |
→ |
uu(ut(tb(bu(x1)))) |
(102) |
|
ut(tb(bs(ss(x1)))) |
→ |
uu(ut(tb(bs(x1)))) |
(103) |
|
ut(tb(bs(st(x1)))) |
→ |
uu(ut(tb(bt(x1)))) |
(104) |
|
st(tb(bb(x1))) |
→ |
sb(bs(sb(x1))) |
(73) |
|
st(tb(bu(x1))) |
→ |
sb(bs(su(x1))) |
(74) |
|
st(tb(bs(x1))) |
→ |
sb(bs(ss(x1))) |
(75) |
|
st(tb(bt(x1))) |
→ |
sb(bs(st(x1))) |
(76) |
|
st(tb(bs(sb(x1)))) |
→ |
su(ut(tb(bb(x1)))) |
(89) |
|
st(tb(bs(su(x1)))) |
→ |
su(ut(tb(bu(x1)))) |
(90) |
|
st(tb(bs(ss(x1)))) |
→ |
su(ut(tb(bs(x1)))) |
(91) |
|
st(tb(bs(st(x1)))) |
→ |
su(ut(tb(bt(x1)))) |
(92) |
|
st(tu(ub(x1))) |
→ |
su(ut(tb(x1))) |
(105) |
|
st(tu(uu(x1))) |
→ |
su(ut(tu(x1))) |
(106) |
|
st(tu(ut(x1))) |
→ |
su(ut(tt(x1))) |
(108) |
|
bu(ub(x1)) |
→ |
bs(sb(x1)) |
(29) |
|
bu(uu(x1)) |
→ |
bs(su(x1)) |
(30) |
|
bu(ut(x1)) |
→ |
bs(st(x1)) |
(32) |
|
bt(tb(bb(x1))) |
→ |
bb(bs(sb(x1))) |
(77) |
|
bt(tb(bu(x1))) |
→ |
bb(bs(su(x1))) |
(78) |
|
bt(tb(bs(x1))) |
→ |
bb(bs(ss(x1))) |
(79) |
|
bt(tb(bt(x1))) |
→ |
bb(bs(st(x1))) |
(80) |
|
bt(tb(bs(sb(x1)))) |
→ |
bu(ut(tb(bb(x1)))) |
(93) |
|
bt(tb(bs(su(x1)))) |
→ |
bu(ut(tb(bu(x1)))) |
(94) |
|
bt(tb(bs(ss(x1)))) |
→ |
bu(ut(tb(bs(x1)))) |
(95) |
|
bt(tb(bs(st(x1)))) |
→ |
bu(ut(tb(bt(x1)))) |
(96) |
|
bt(tu(ub(x1))) |
→ |
bu(ut(tb(x1))) |
(109) |
|
bt(tu(uu(x1))) |
→ |
bu(ut(tu(x1))) |
(110) |
|
bt(tu(ut(x1))) |
→ |
bu(ut(tt(x1))) |
(112) |
|
ut(tu(ub(x1))) |
→ |
uu(ut(tb(x1))) |
(117) |
|
ut(tu(uu(x1))) |
→ |
uu(ut(tu(x1))) |
(118) |
|
ut(tu(ut(x1))) |
→ |
uu(ut(tt(x1))) |
(120) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
tt#(tu(uu(x1))) |
→ |
ut#(tu(x1)) |
(279) |
|
ut#(tu(uu(x1))) |
→ |
ut#(tu(x1)) |
(283) |
|
ut#(tu(ut(x1))) |
→ |
ut#(tt(x1)) |
(284) |
|
ut#(tu(ut(x1))) |
→ |
tt#(x1) |
(285) |
|
tt#(tu(ut(x1))) |
→ |
ut#(tt(x1)) |
(280) |
|
tt#(tu(ut(x1))) |
→ |
tt#(x1) |
(281) |
could be deleted.
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
ut#(tb(bs(ss(x1)))) |
→ |
ut#(tb(bs(x1))) |
(260) |
|
ut#(tb(bs(su(x1)))) |
→ |
ut#(tb(bu(x1))) |
(258) |
|
ut#(tb(bs(st(x1)))) |
→ |
ut#(tb(bt(x1))) |
(262) |
1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [ut#(x1)] |
= |
1 · x1
|
| [tb(x1)] |
= |
1 + 1 · x1
|
| [bs(x1)] |
= |
1 · x1
|
| [ss(x1)] |
= |
1 + 1 · x1
|
| [su(x1)] |
= |
1 + 1 · x1
|
| [bu(x1)] |
= |
1 · x1
|
| [st(x1)] |
= |
1 + 1 · x1
|
| [bt(x1)] |
= |
1 · x1
|
| [sb(x1)] |
= |
1 |
| [bb(x1)] |
= |
0 |
| [tu(x1)] |
= |
1 + 1 · x1
|
| [ts(x1)] |
= |
0 |
| [tt(x1)] |
= |
1 + 1 · x1
|
| [ub(x1)] |
= |
1 + 1 · x1
|
| [uu(x1)] |
= |
1 + 1 · x1
|
| [ut(x1)] |
= |
1 + 1 · x1
|
together with the usable
rules
|
bs(sb(bb(x1))) |
→ |
bb(bs(ss(ss(sb(x1))))) |
(45) |
|
bs(sb(bu(x1))) |
→ |
bb(bs(ss(ss(su(x1))))) |
(46) |
|
bs(sb(bs(x1))) |
→ |
bb(bs(ss(ss(ss(x1))))) |
(47) |
|
bs(sb(bt(x1))) |
→ |
bb(bs(ss(ss(st(x1))))) |
(48) |
|
bs(sb(bs(sb(x1)))) |
→ |
bb(bt(tb(x1))) |
(61) |
|
bs(sb(bs(su(x1)))) |
→ |
bb(bt(tu(x1))) |
(62) |
|
bs(sb(bs(ss(x1)))) |
→ |
bb(bt(ts(x1))) |
(63) |
|
bs(sb(bs(st(x1)))) |
→ |
bb(bt(tt(x1))) |
(64) |
|
bu(ub(x1)) |
→ |
bs(sb(x1)) |
(29) |
|
bu(uu(x1)) |
→ |
bs(su(x1)) |
(30) |
|
bu(ut(x1)) |
→ |
bs(st(x1)) |
(32) |
|
bt(tb(bb(x1))) |
→ |
bb(bs(sb(x1))) |
(77) |
|
bt(tb(bu(x1))) |
→ |
bb(bs(su(x1))) |
(78) |
|
bt(tb(bs(x1))) |
→ |
bb(bs(ss(x1))) |
(79) |
|
bt(tb(bt(x1))) |
→ |
bb(bs(st(x1))) |
(80) |
|
bt(tb(bs(sb(x1)))) |
→ |
bu(ut(tb(bb(x1)))) |
(93) |
|
bt(tb(bs(su(x1)))) |
→ |
bu(ut(tb(bu(x1)))) |
(94) |
|
bt(tb(bs(ss(x1)))) |
→ |
bu(ut(tb(bs(x1)))) |
(95) |
|
bt(tb(bs(st(x1)))) |
→ |
bu(ut(tb(bt(x1)))) |
(96) |
|
bt(tu(ub(x1))) |
→ |
bu(ut(tb(x1))) |
(109) |
|
bt(tu(uu(x1))) |
→ |
bu(ut(tu(x1))) |
(110) |
|
bt(tu(ut(x1))) |
→ |
bu(ut(tt(x1))) |
(112) |
|
ss(sb(bb(x1))) |
→ |
sb(bs(ss(ss(sb(x1))))) |
(41) |
|
ss(sb(bu(x1))) |
→ |
sb(bs(ss(ss(su(x1))))) |
(42) |
|
ss(sb(bs(x1))) |
→ |
sb(bs(ss(ss(ss(x1))))) |
(43) |
|
ss(sb(bt(x1))) |
→ |
sb(bs(ss(ss(st(x1))))) |
(44) |
|
ss(sb(bs(sb(x1)))) |
→ |
sb(bt(tb(x1))) |
(57) |
|
ss(sb(bs(su(x1)))) |
→ |
sb(bt(tu(x1))) |
(58) |
|
ss(sb(bs(ss(x1)))) |
→ |
sb(bt(ts(x1))) |
(59) |
|
ss(sb(bs(st(x1)))) |
→ |
sb(bt(tt(x1))) |
(60) |
|
su(ub(x1)) |
→ |
ss(sb(x1)) |
(37) |
|
su(uu(x1)) |
→ |
ss(su(x1)) |
(38) |
|
su(ut(x1)) |
→ |
ss(st(x1)) |
(40) |
|
st(tb(bb(x1))) |
→ |
sb(bs(sb(x1))) |
(73) |
|
st(tb(bu(x1))) |
→ |
sb(bs(su(x1))) |
(74) |
|
st(tb(bs(x1))) |
→ |
sb(bs(ss(x1))) |
(75) |
|
st(tb(bt(x1))) |
→ |
sb(bs(st(x1))) |
(76) |
|
st(tb(bs(sb(x1)))) |
→ |
su(ut(tb(bb(x1)))) |
(89) |
|
st(tb(bs(su(x1)))) |
→ |
su(ut(tb(bu(x1)))) |
(90) |
|
st(tb(bs(ss(x1)))) |
→ |
su(ut(tb(bs(x1)))) |
(91) |
|
st(tb(bs(st(x1)))) |
→ |
su(ut(tb(bt(x1)))) |
(92) |
|
st(tu(ub(x1))) |
→ |
su(ut(tb(x1))) |
(105) |
|
st(tu(uu(x1))) |
→ |
su(ut(tu(x1))) |
(106) |
|
st(tu(ut(x1))) |
→ |
su(ut(tt(x1))) |
(108) |
|
ut(tb(bb(x1))) |
→ |
ub(bs(sb(x1))) |
(85) |
|
ut(tb(bu(x1))) |
→ |
ub(bs(su(x1))) |
(86) |
|
ut(tb(bs(x1))) |
→ |
ub(bs(ss(x1))) |
(87) |
|
ut(tb(bt(x1))) |
→ |
ub(bs(st(x1))) |
(88) |
|
ut(tb(bs(sb(x1)))) |
→ |
uu(ut(tb(bb(x1)))) |
(101) |
|
ut(tb(bs(su(x1)))) |
→ |
uu(ut(tb(bu(x1)))) |
(102) |
|
ut(tb(bs(ss(x1)))) |
→ |
uu(ut(tb(bs(x1)))) |
(103) |
|
ut(tb(bs(st(x1)))) |
→ |
uu(ut(tb(bt(x1)))) |
(104) |
|
ut(tu(ub(x1))) |
→ |
uu(ut(tb(x1))) |
(117) |
|
ut(tu(uu(x1))) |
→ |
uu(ut(tu(x1))) |
(118) |
|
ut(tu(ut(x1))) |
→ |
uu(ut(tt(x1))) |
(120) |
|
tt(tb(bb(x1))) |
→ |
tb(bs(sb(x1))) |
(81) |
|
tt(tb(bu(x1))) |
→ |
tb(bs(su(x1))) |
(82) |
|
tt(tb(bs(x1))) |
→ |
tb(bs(ss(x1))) |
(83) |
|
tt(tb(bt(x1))) |
→ |
tb(bs(st(x1))) |
(84) |
|
tt(tb(bs(sb(x1)))) |
→ |
tu(ut(tb(bb(x1)))) |
(97) |
|
tt(tb(bs(su(x1)))) |
→ |
tu(ut(tb(bu(x1)))) |
(98) |
|
tt(tb(bs(ss(x1)))) |
→ |
tu(ut(tb(bs(x1)))) |
(99) |
|
tt(tb(bs(st(x1)))) |
→ |
tu(ut(tb(bt(x1)))) |
(100) |
|
tt(tu(ub(x1))) |
→ |
tu(ut(tb(x1))) |
(113) |
|
tt(tu(uu(x1))) |
→ |
tu(ut(tu(x1))) |
(114) |
|
tt(tu(ut(x1))) |
→ |
tu(ut(tt(x1))) |
(116) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
ut#(tb(bs(ss(x1)))) |
→ |
ut#(tb(bs(x1))) |
(260) |
|
ut#(tb(bs(su(x1)))) |
→ |
ut#(tb(bu(x1))) |
(258) |
|
ut#(tb(bs(st(x1)))) |
→ |
ut#(tb(bt(x1))) |
(262) |
could be deleted.
1.1.1.1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
|
ts#(ss(x1)) |
→ |
ts#(x1) |
(129) |
1.1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [ss(x1)] |
= |
1 · x1
|
| [ts#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
ts#(ss(x1)) |
→ |
ts#(x1) |
(129) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
su#(ut(x1)) |
→ |
st#(x1) |
(136) |
|
st#(tb(bs(sb(x1)))) |
→ |
su#(ut(tb(bb(x1)))) |
(228) |
|
su#(uu(x1)) |
→ |
su#(x1) |
(134) |
|
st#(tb(bs(su(x1)))) |
→ |
su#(ut(tb(bu(x1)))) |
(230) |
|
st#(tb(bs(ss(x1)))) |
→ |
su#(ut(tb(bs(x1)))) |
(233) |
|
st#(tb(bs(st(x1)))) |
→ |
su#(ut(tb(bt(x1)))) |
(236) |
|
st#(tu(ub(x1))) |
→ |
su#(ut(tb(x1))) |
(264) |
|
st#(tu(uu(x1))) |
→ |
su#(ut(tu(x1))) |
(266) |
|
st#(tu(ut(x1))) |
→ |
su#(ut(tt(x1))) |
(268) |
1.1.1.1.1.1.4 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [su#(x1)] |
= |
1 + 1 · x1
|
| [ut(x1)] |
= |
1 + 1 · x1
|
| [st#(x1)] |
= |
1 + 1 · x1
|
| [tb(x1)] |
= |
1 + 1 · x1
|
| [bs(x1)] |
= |
1 + 1 · x1
|
| [sb(x1)] |
= |
0 |
| [bb(x1)] |
= |
0 |
| [uu(x1)] |
= |
1 + 1 · x1
|
| [su(x1)] |
= |
1 + 1 · x1
|
| [bu(x1)] |
= |
1 + 1 · x1
|
| [ss(x1)] |
= |
1 + 1 · x1
|
| [st(x1)] |
= |
1 + 1 · x1
|
| [bt(x1)] |
= |
1 + 1 · x1
|
| [tu(x1)] |
= |
1 + 1 · x1
|
| [ub(x1)] |
= |
1 + 1 · x1
|
| [tt(x1)] |
= |
1 + 1 · x1
|
| [ts(x1)] |
= |
0 |
together with the usable
rules
|
ut(tb(bb(x1))) |
→ |
ub(bs(sb(x1))) |
(85) |
|
bu(ub(x1)) |
→ |
bs(sb(x1)) |
(29) |
|
bu(uu(x1)) |
→ |
bs(su(x1)) |
(30) |
|
bu(ut(x1)) |
→ |
bs(st(x1)) |
(32) |
|
ut(tb(bu(x1))) |
→ |
ub(bs(su(x1))) |
(86) |
|
ut(tb(bs(x1))) |
→ |
ub(bs(ss(x1))) |
(87) |
|
ut(tb(bt(x1))) |
→ |
ub(bs(st(x1))) |
(88) |
|
ut(tb(bs(sb(x1)))) |
→ |
uu(ut(tb(bb(x1)))) |
(101) |
|
ut(tb(bs(su(x1)))) |
→ |
uu(ut(tb(bu(x1)))) |
(102) |
|
ut(tb(bs(ss(x1)))) |
→ |
uu(ut(tb(bs(x1)))) |
(103) |
|
ut(tb(bs(st(x1)))) |
→ |
uu(ut(tb(bt(x1)))) |
(104) |
|
bs(sb(bb(x1))) |
→ |
bb(bs(ss(ss(sb(x1))))) |
(45) |
|
bs(sb(bu(x1))) |
→ |
bb(bs(ss(ss(su(x1))))) |
(46) |
|
bs(sb(bs(x1))) |
→ |
bb(bs(ss(ss(ss(x1))))) |
(47) |
|
bs(sb(bt(x1))) |
→ |
bb(bs(ss(ss(st(x1))))) |
(48) |
|
bs(sb(bs(sb(x1)))) |
→ |
bb(bt(tb(x1))) |
(61) |
|
bs(sb(bs(su(x1)))) |
→ |
bb(bt(tu(x1))) |
(62) |
|
bs(sb(bs(ss(x1)))) |
→ |
bb(bt(ts(x1))) |
(63) |
|
bs(sb(bs(st(x1)))) |
→ |
bb(bt(tt(x1))) |
(64) |
|
bt(tb(bb(x1))) |
→ |
bb(bs(sb(x1))) |
(77) |
|
bt(tb(bu(x1))) |
→ |
bb(bs(su(x1))) |
(78) |
|
bt(tb(bs(x1))) |
→ |
bb(bs(ss(x1))) |
(79) |
|
bt(tb(bt(x1))) |
→ |
bb(bs(st(x1))) |
(80) |
|
bt(tb(bs(sb(x1)))) |
→ |
bu(ut(tb(bb(x1)))) |
(93) |
|
bt(tb(bs(su(x1)))) |
→ |
bu(ut(tb(bu(x1)))) |
(94) |
|
bt(tb(bs(ss(x1)))) |
→ |
bu(ut(tb(bs(x1)))) |
(95) |
|
bt(tb(bs(st(x1)))) |
→ |
bu(ut(tb(bt(x1)))) |
(96) |
|
bt(tu(ub(x1))) |
→ |
bu(ut(tb(x1))) |
(109) |
|
bt(tu(uu(x1))) |
→ |
bu(ut(tu(x1))) |
(110) |
|
bt(tu(ut(x1))) |
→ |
bu(ut(tt(x1))) |
(112) |
|
ut(tu(ub(x1))) |
→ |
uu(ut(tb(x1))) |
(117) |
|
ut(tu(uu(x1))) |
→ |
uu(ut(tu(x1))) |
(118) |
|
ut(tu(ut(x1))) |
→ |
uu(ut(tt(x1))) |
(120) |
|
tt(tb(bb(x1))) |
→ |
tb(bs(sb(x1))) |
(81) |
|
tt(tb(bu(x1))) |
→ |
tb(bs(su(x1))) |
(82) |
|
tt(tb(bs(x1))) |
→ |
tb(bs(ss(x1))) |
(83) |
|
tt(tb(bt(x1))) |
→ |
tb(bs(st(x1))) |
(84) |
|
tt(tb(bs(sb(x1)))) |
→ |
tu(ut(tb(bb(x1)))) |
(97) |
|
tt(tb(bs(su(x1)))) |
→ |
tu(ut(tb(bu(x1)))) |
(98) |
|
tt(tb(bs(ss(x1)))) |
→ |
tu(ut(tb(bs(x1)))) |
(99) |
|
tt(tb(bs(st(x1)))) |
→ |
tu(ut(tb(bt(x1)))) |
(100) |
|
tt(tu(ub(x1))) |
→ |
tu(ut(tb(x1))) |
(113) |
|
tt(tu(uu(x1))) |
→ |
tu(ut(tu(x1))) |
(114) |
|
tt(tu(ut(x1))) |
→ |
tu(ut(tt(x1))) |
(116) |
|
ss(sb(bb(x1))) |
→ |
sb(bs(ss(ss(sb(x1))))) |
(41) |
|
ss(sb(bu(x1))) |
→ |
sb(bs(ss(ss(su(x1))))) |
(42) |
|
ss(sb(bs(x1))) |
→ |
sb(bs(ss(ss(ss(x1))))) |
(43) |
|
ss(sb(bt(x1))) |
→ |
sb(bs(ss(ss(st(x1))))) |
(44) |
|
ss(sb(bs(sb(x1)))) |
→ |
sb(bt(tb(x1))) |
(57) |
|
ss(sb(bs(su(x1)))) |
→ |
sb(bt(tu(x1))) |
(58) |
|
ss(sb(bs(ss(x1)))) |
→ |
sb(bt(ts(x1))) |
(59) |
|
ss(sb(bs(st(x1)))) |
→ |
sb(bt(tt(x1))) |
(60) |
|
su(ub(x1)) |
→ |
ss(sb(x1)) |
(37) |
|
su(uu(x1)) |
→ |
ss(su(x1)) |
(38) |
|
su(ut(x1)) |
→ |
ss(st(x1)) |
(40) |
|
st(tb(bb(x1))) |
→ |
sb(bs(sb(x1))) |
(73) |
|
st(tb(bu(x1))) |
→ |
sb(bs(su(x1))) |
(74) |
|
st(tb(bs(x1))) |
→ |
sb(bs(ss(x1))) |
(75) |
|
st(tb(bt(x1))) |
→ |
sb(bs(st(x1))) |
(76) |
|
st(tb(bs(sb(x1)))) |
→ |
su(ut(tb(bb(x1)))) |
(89) |
|
st(tb(bs(su(x1)))) |
→ |
su(ut(tb(bu(x1)))) |
(90) |
|
st(tb(bs(ss(x1)))) |
→ |
su(ut(tb(bs(x1)))) |
(91) |
|
st(tb(bs(st(x1)))) |
→ |
su(ut(tb(bt(x1)))) |
(92) |
|
st(tu(ub(x1))) |
→ |
su(ut(tb(x1))) |
(105) |
|
st(tu(uu(x1))) |
→ |
su(ut(tu(x1))) |
(106) |
|
st(tu(ut(x1))) |
→ |
su(ut(tt(x1))) |
(108) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
su#(ut(x1)) |
→ |
st#(x1) |
(136) |
|
su#(uu(x1)) |
→ |
su#(x1) |
(134) |
could be deleted.
1.1.1.1.1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.