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 matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{B(☐), A(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
B(A(b(x1))) |
→ |
B(b(a(B(A(x1))))) |
(5) |
|
B(B(a(x1))) |
→ |
B(a(b(A(B(x1))))) |
(6) |
|
B(A(a(x1))) |
→ |
B(x1) |
(7) |
|
B(B(b(x1))) |
→ |
B(x1) |
(8) |
|
A(A(b(x1))) |
→ |
A(b(a(B(A(x1))))) |
(9) |
|
A(B(a(x1))) |
→ |
A(a(b(A(B(x1))))) |
(10) |
|
A(A(a(x1))) |
→ |
A(x1) |
(11) |
|
A(B(b(x1))) |
→ |
A(x1) |
(12) |
|
b(A(b(x1))) |
→ |
b(b(a(B(A(x1))))) |
(13) |
|
b(B(a(x1))) |
→ |
b(a(b(A(B(x1))))) |
(14) |
|
b(A(a(x1))) |
→ |
b(x1) |
(15) |
|
b(B(b(x1))) |
→ |
b(x1) |
(16) |
|
a(A(b(x1))) |
→ |
a(b(a(B(A(x1))))) |
(17) |
|
a(B(a(x1))) |
→ |
a(a(b(A(B(x1))))) |
(18) |
|
a(A(a(x1))) |
→ |
a(x1) |
(19) |
|
a(B(b(x1))) |
→ |
a(x1) |
(20) |
1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,3}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 4):
| [B(x1)] |
= |
4x1 + 0 |
| [A(x1)] |
= |
4x1 + 1 |
| [b(x1)] |
= |
4x1 + 2 |
| [a(x1)] |
= |
4x1 + 3 |
We obtain the labeled TRS
|
A1(A2(b1(x1))) |
→ |
A2(b3(a0(B1(A1(x1))))) |
(21) |
|
A1(A2(b2(x1))) |
→ |
A2(b3(a0(B1(A2(x1))))) |
(22) |
|
A1(A2(b3(x1))) |
→ |
A2(b3(a0(B1(A3(x1))))) |
(23) |
|
A1(A2(b0(x1))) |
→ |
A2(b3(a0(B1(A0(x1))))) |
(24) |
|
b1(A2(b1(x1))) |
→ |
b2(b3(a0(B1(A1(x1))))) |
(25) |
|
b1(A2(b2(x1))) |
→ |
b2(b3(a0(B1(A2(x1))))) |
(26) |
|
b1(A2(b3(x1))) |
→ |
b2(b3(a0(B1(A3(x1))))) |
(27) |
|
b1(A2(b0(x1))) |
→ |
b2(b3(a0(B1(A0(x1))))) |
(28) |
|
a1(A2(b1(x1))) |
→ |
a2(b3(a0(B1(A1(x1))))) |
(29) |
|
a1(A2(b2(x1))) |
→ |
a2(b3(a0(B1(A2(x1))))) |
(30) |
|
a1(A2(b3(x1))) |
→ |
a2(b3(a0(B1(A3(x1))))) |
(31) |
|
a1(A2(b0(x1))) |
→ |
a2(b3(a0(B1(A0(x1))))) |
(32) |
|
B1(A2(b1(x1))) |
→ |
B2(b3(a0(B1(A1(x1))))) |
(33) |
|
B1(A2(b2(x1))) |
→ |
B2(b3(a0(B1(A2(x1))))) |
(34) |
|
B1(A2(b3(x1))) |
→ |
B2(b3(a0(B1(A3(x1))))) |
(35) |
|
B1(A2(b0(x1))) |
→ |
B2(b3(a0(B1(A0(x1))))) |
(36) |
|
A0(B3(a1(x1))) |
→ |
A3(a2(b1(A0(B1(x1))))) |
(37) |
|
A0(B3(a2(x1))) |
→ |
A3(a2(b1(A0(B2(x1))))) |
(38) |
|
A0(B3(a3(x1))) |
→ |
A3(a2(b1(A0(B3(x1))))) |
(39) |
|
A0(B3(a0(x1))) |
→ |
A3(a2(b1(A0(B0(x1))))) |
(40) |
|
b0(B3(a1(x1))) |
→ |
b3(a2(b1(A0(B1(x1))))) |
(41) |
|
b0(B3(a2(x1))) |
→ |
b3(a2(b1(A0(B2(x1))))) |
(42) |
|
b0(B3(a3(x1))) |
→ |
b3(a2(b1(A0(B3(x1))))) |
(43) |
|
b0(B3(a0(x1))) |
→ |
b3(a2(b1(A0(B0(x1))))) |
(44) |
|
a0(B3(a1(x1))) |
→ |
a3(a2(b1(A0(B1(x1))))) |
(45) |
|
a0(B3(a2(x1))) |
→ |
a3(a2(b1(A0(B2(x1))))) |
(46) |
|
a0(B3(a3(x1))) |
→ |
a3(a2(b1(A0(B3(x1))))) |
(47) |
|
a0(B3(a0(x1))) |
→ |
a3(a2(b1(A0(B0(x1))))) |
(48) |
|
B0(B3(a1(x1))) |
→ |
B3(a2(b1(A0(B1(x1))))) |
(49) |
|
B0(B3(a2(x1))) |
→ |
B3(a2(b1(A0(B2(x1))))) |
(50) |
|
B0(B3(a3(x1))) |
→ |
B3(a2(b1(A0(B3(x1))))) |
(51) |
|
B0(B3(a0(x1))) |
→ |
B3(a2(b1(A0(B0(x1))))) |
(52) |
|
A1(A3(a1(x1))) |
→ |
A1(x1) |
(53) |
|
A1(A3(a2(x1))) |
→ |
A2(x1) |
(54) |
|
A1(A3(a3(x1))) |
→ |
A3(x1) |
(55) |
|
A1(A3(a0(x1))) |
→ |
A0(x1) |
(56) |
|
b1(A3(a1(x1))) |
→ |
b1(x1) |
(57) |
|
b1(A3(a2(x1))) |
→ |
b2(x1) |
(58) |
|
b1(A3(a3(x1))) |
→ |
b3(x1) |
(59) |
|
b1(A3(a0(x1))) |
→ |
b0(x1) |
(60) |
|
a1(A3(a1(x1))) |
→ |
a1(x1) |
(61) |
|
a1(A3(a2(x1))) |
→ |
a2(x1) |
(62) |
|
a1(A3(a3(x1))) |
→ |
a3(x1) |
(63) |
|
a1(A3(a0(x1))) |
→ |
a0(x1) |
(64) |
|
B1(A3(a1(x1))) |
→ |
B1(x1) |
(65) |
|
B1(A3(a2(x1))) |
→ |
B2(x1) |
(66) |
|
B1(A3(a3(x1))) |
→ |
B3(x1) |
(67) |
|
B1(A3(a0(x1))) |
→ |
B0(x1) |
(68) |
|
A0(B2(b1(x1))) |
→ |
A1(x1) |
(69) |
|
A0(B2(b2(x1))) |
→ |
A2(x1) |
(70) |
|
A0(B2(b3(x1))) |
→ |
A3(x1) |
(71) |
|
A0(B2(b0(x1))) |
→ |
A0(x1) |
(72) |
|
b0(B2(b1(x1))) |
→ |
b1(x1) |
(73) |
|
b0(B2(b2(x1))) |
→ |
b2(x1) |
(74) |
|
b0(B2(b3(x1))) |
→ |
b3(x1) |
(75) |
|
b0(B2(b0(x1))) |
→ |
b0(x1) |
(76) |
|
a0(B2(b1(x1))) |
→ |
a1(x1) |
(77) |
|
a0(B2(b2(x1))) |
→ |
a2(x1) |
(78) |
|
a0(B2(b3(x1))) |
→ |
a3(x1) |
(79) |
|
a0(B2(b0(x1))) |
→ |
a0(x1) |
(80) |
|
B0(B2(b1(x1))) |
→ |
B1(x1) |
(81) |
|
B0(B2(b2(x1))) |
→ |
B2(x1) |
(82) |
|
B0(B2(b3(x1))) |
→ |
B3(x1) |
(83) |
|
B0(B2(b0(x1))) |
→ |
B0(x1) |
(84) |
1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [B0(x1)] |
= |
x1 +
|
| [B1(x1)] |
= |
x1 +
|
| [B2(x1)] |
= |
x1 +
|
| [B3(x1)] |
= |
x1 +
|
| [A0(x1)] |
= |
x1 +
|
| [A1(x1)] |
= |
x1 +
|
| [A2(x1)] |
= |
x1 +
|
| [A3(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
A1(A2(b0(x1))) |
→ |
A2(b3(a0(B1(A0(x1))))) |
(24) |
|
b1(A2(b0(x1))) |
→ |
b2(b3(a0(B1(A0(x1))))) |
(28) |
|
a1(A2(b1(x1))) |
→ |
a2(b3(a0(B1(A1(x1))))) |
(29) |
|
a1(A2(b2(x1))) |
→ |
a2(b3(a0(B1(A2(x1))))) |
(30) |
|
a1(A2(b3(x1))) |
→ |
a2(b3(a0(B1(A3(x1))))) |
(31) |
|
a1(A2(b0(x1))) |
→ |
a2(b3(a0(B1(A0(x1))))) |
(32) |
|
B1(A2(b0(x1))) |
→ |
B2(b3(a0(B1(A0(x1))))) |
(36) |
|
A0(B3(a1(x1))) |
→ |
A3(a2(b1(A0(B1(x1))))) |
(37) |
|
b0(B3(a1(x1))) |
→ |
b3(a2(b1(A0(B1(x1))))) |
(41) |
|
b0(B3(a2(x1))) |
→ |
b3(a2(b1(A0(B2(x1))))) |
(42) |
|
b0(B3(a3(x1))) |
→ |
b3(a2(b1(A0(B3(x1))))) |
(43) |
|
b0(B3(a0(x1))) |
→ |
b3(a2(b1(A0(B0(x1))))) |
(44) |
|
a0(B3(a1(x1))) |
→ |
a3(a2(b1(A0(B1(x1))))) |
(45) |
|
B0(B3(a1(x1))) |
→ |
B3(a2(b1(A0(B1(x1))))) |
(49) |
|
A1(A3(a1(x1))) |
→ |
A1(x1) |
(53) |
|
A1(A3(a3(x1))) |
→ |
A3(x1) |
(55) |
|
A1(A3(a0(x1))) |
→ |
A0(x1) |
(56) |
|
b1(A3(a1(x1))) |
→ |
b1(x1) |
(57) |
|
b1(A3(a3(x1))) |
→ |
b3(x1) |
(59) |
|
b1(A3(a0(x1))) |
→ |
b0(x1) |
(60) |
|
a1(A3(a1(x1))) |
→ |
a1(x1) |
(61) |
|
a1(A3(a2(x1))) |
→ |
a2(x1) |
(62) |
|
a1(A3(a3(x1))) |
→ |
a3(x1) |
(63) |
|
a1(A3(a0(x1))) |
→ |
a0(x1) |
(64) |
|
B1(A3(a1(x1))) |
→ |
B1(x1) |
(65) |
|
A0(B2(b0(x1))) |
→ |
A0(x1) |
(72) |
|
b0(B2(b1(x1))) |
→ |
b1(x1) |
(73) |
|
b0(B2(b2(x1))) |
→ |
b2(x1) |
(74) |
|
b0(B2(b3(x1))) |
→ |
b3(x1) |
(75) |
|
b0(B2(b0(x1))) |
→ |
b0(x1) |
(76) |
|
a0(B2(b1(x1))) |
→ |
a1(x1) |
(77) |
|
a0(B2(b2(x1))) |
→ |
a2(x1) |
(78) |
|
a0(B2(b0(x1))) |
→ |
a0(x1) |
(80) |
|
B0(B2(b1(x1))) |
→ |
B1(x1) |
(81) |
|
B0(B2(b2(x1))) |
→ |
B2(x1) |
(82) |
|
B0(B2(b0(x1))) |
→ |
B0(x1) |
(84) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
B0#(B3(a0(x1))) |
→ |
B0#(x1) |
(85) |
|
B0#(B3(a0(x1))) |
→ |
A0#(B0(x1)) |
(86) |
|
B0#(B3(a0(x1))) |
→ |
b1#(A0(B0(x1))) |
(87) |
|
B0#(B3(a2(x1))) |
→ |
A0#(B2(x1)) |
(88) |
|
B0#(B3(a2(x1))) |
→ |
b1#(A0(B2(x1))) |
(89) |
|
B0#(B3(a3(x1))) |
→ |
A0#(B3(x1)) |
(90) |
|
B0#(B3(a3(x1))) |
→ |
b1#(A0(B3(x1))) |
(91) |
|
B1#(A2(b1(x1))) |
→ |
B1#(A1(x1)) |
(92) |
|
B1#(A2(b1(x1))) |
→ |
A1#(x1) |
(93) |
|
B1#(A2(b1(x1))) |
→ |
a0#(B1(A1(x1))) |
(94) |
|
B1#(A2(b2(x1))) |
→ |
B1#(A2(x1)) |
(95) |
|
B1#(A2(b2(x1))) |
→ |
a0#(B1(A2(x1))) |
(96) |
|
B1#(A2(b3(x1))) |
→ |
B1#(A3(x1)) |
(97) |
|
B1#(A2(b3(x1))) |
→ |
a0#(B1(A3(x1))) |
(98) |
|
B1#(A3(a0(x1))) |
→ |
B0#(x1) |
(99) |
|
A0#(B2(b1(x1))) |
→ |
A1#(x1) |
(100) |
|
A0#(B3(a0(x1))) |
→ |
B0#(x1) |
(101) |
|
A0#(B3(a0(x1))) |
→ |
A0#(B0(x1)) |
(102) |
|
A0#(B3(a0(x1))) |
→ |
b1#(A0(B0(x1))) |
(103) |
|
A0#(B3(a2(x1))) |
→ |
A0#(B2(x1)) |
(104) |
|
A0#(B3(a2(x1))) |
→ |
b1#(A0(B2(x1))) |
(105) |
|
A0#(B3(a3(x1))) |
→ |
A0#(B3(x1)) |
(106) |
|
A0#(B3(a3(x1))) |
→ |
b1#(A0(B3(x1))) |
(107) |
|
A1#(A2(b1(x1))) |
→ |
B1#(A1(x1)) |
(108) |
|
A1#(A2(b1(x1))) |
→ |
A1#(x1) |
(109) |
|
A1#(A2(b1(x1))) |
→ |
a0#(B1(A1(x1))) |
(110) |
|
A1#(A2(b2(x1))) |
→ |
B1#(A2(x1)) |
(111) |
|
A1#(A2(b2(x1))) |
→ |
a0#(B1(A2(x1))) |
(112) |
|
A1#(A2(b3(x1))) |
→ |
B1#(A3(x1)) |
(113) |
|
A1#(A2(b3(x1))) |
→ |
a0#(B1(A3(x1))) |
(114) |
|
b1#(A2(b1(x1))) |
→ |
B1#(A1(x1)) |
(115) |
|
b1#(A2(b1(x1))) |
→ |
A1#(x1) |
(116) |
|
b1#(A2(b1(x1))) |
→ |
a0#(B1(A1(x1))) |
(117) |
|
b1#(A2(b2(x1))) |
→ |
B1#(A2(x1)) |
(118) |
|
b1#(A2(b2(x1))) |
→ |
a0#(B1(A2(x1))) |
(119) |
|
b1#(A2(b3(x1))) |
→ |
B1#(A3(x1)) |
(120) |
|
b1#(A2(b3(x1))) |
→ |
a0#(B1(A3(x1))) |
(121) |
|
a0#(B3(a0(x1))) |
→ |
B0#(x1) |
(122) |
|
a0#(B3(a0(x1))) |
→ |
A0#(B0(x1)) |
(123) |
|
a0#(B3(a0(x1))) |
→ |
b1#(A0(B0(x1))) |
(124) |
|
a0#(B3(a2(x1))) |
→ |
A0#(B2(x1)) |
(125) |
|
a0#(B3(a2(x1))) |
→ |
b1#(A0(B2(x1))) |
(126) |
|
a0#(B3(a3(x1))) |
→ |
A0#(B3(x1)) |
(127) |
|
a0#(B3(a3(x1))) |
→ |
b1#(A0(B3(x1))) |
(128) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [B0(x1)] |
= |
x1 +
|
| [B1(x1)] |
= |
x1 +
|
| [B2(x1)] |
= |
x1 +
|
| [B3(x1)] |
= |
x1 +
|
| [A0(x1)] |
= |
x1 +
|
| [A1(x1)] |
= |
x1 +
|
| [A2(x1)] |
= |
x1 +
|
| [A3(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [B0#(x1)] |
= |
x1 +
|
| [B1#(x1)] |
= |
x1 +
|
| [A0#(x1)] |
= |
x1 +
|
| [A1#(x1)] |
= |
x1 +
|
| [b1#(x1)] |
= |
x1 +
|
| [a0#(x1)] |
= |
x1 +
|
together with the usable
rules
|
A1(A2(b1(x1))) |
→ |
A2(b3(a0(B1(A1(x1))))) |
(21) |
|
A1(A2(b2(x1))) |
→ |
A2(b3(a0(B1(A2(x1))))) |
(22) |
|
A1(A2(b3(x1))) |
→ |
A2(b3(a0(B1(A3(x1))))) |
(23) |
|
b1(A2(b1(x1))) |
→ |
b2(b3(a0(B1(A1(x1))))) |
(25) |
|
b1(A2(b2(x1))) |
→ |
b2(b3(a0(B1(A2(x1))))) |
(26) |
|
b1(A2(b3(x1))) |
→ |
b2(b3(a0(B1(A3(x1))))) |
(27) |
|
B1(A2(b1(x1))) |
→ |
B2(b3(a0(B1(A1(x1))))) |
(33) |
|
B1(A2(b2(x1))) |
→ |
B2(b3(a0(B1(A2(x1))))) |
(34) |
|
B1(A2(b3(x1))) |
→ |
B2(b3(a0(B1(A3(x1))))) |
(35) |
|
A0(B3(a2(x1))) |
→ |
A3(a2(b1(A0(B2(x1))))) |
(38) |
|
A0(B3(a3(x1))) |
→ |
A3(a2(b1(A0(B3(x1))))) |
(39) |
|
A0(B3(a0(x1))) |
→ |
A3(a2(b1(A0(B0(x1))))) |
(40) |
|
a0(B3(a2(x1))) |
→ |
a3(a2(b1(A0(B2(x1))))) |
(46) |
|
a0(B3(a3(x1))) |
→ |
a3(a2(b1(A0(B3(x1))))) |
(47) |
|
a0(B3(a0(x1))) |
→ |
a3(a2(b1(A0(B0(x1))))) |
(48) |
|
B0(B3(a2(x1))) |
→ |
B3(a2(b1(A0(B2(x1))))) |
(50) |
|
B0(B3(a3(x1))) |
→ |
B3(a2(b1(A0(B3(x1))))) |
(51) |
|
B0(B3(a0(x1))) |
→ |
B3(a2(b1(A0(B0(x1))))) |
(52) |
|
A1(A3(a2(x1))) |
→ |
A2(x1) |
(54) |
|
b1(A3(a2(x1))) |
→ |
b2(x1) |
(58) |
|
B1(A3(a2(x1))) |
→ |
B2(x1) |
(66) |
|
B1(A3(a3(x1))) |
→ |
B3(x1) |
(67) |
|
B1(A3(a0(x1))) |
→ |
B0(x1) |
(68) |
|
A0(B2(b1(x1))) |
→ |
A1(x1) |
(69) |
|
A0(B2(b2(x1))) |
→ |
A2(x1) |
(70) |
|
A0(B2(b3(x1))) |
→ |
A3(x1) |
(71) |
|
a0(B2(b3(x1))) |
→ |
a3(x1) |
(79) |
|
B0(B2(b3(x1))) |
→ |
B3(x1) |
(83) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
B0#(B3(a0(x1))) |
→ |
B0#(x1) |
(85) |
|
B0#(B3(a0(x1))) |
→ |
A0#(B0(x1)) |
(86) |
|
B0#(B3(a0(x1))) |
→ |
b1#(A0(B0(x1))) |
(87) |
|
B0#(B3(a2(x1))) |
→ |
A0#(B2(x1)) |
(88) |
|
B0#(B3(a2(x1))) |
→ |
b1#(A0(B2(x1))) |
(89) |
|
B0#(B3(a3(x1))) |
→ |
A0#(B3(x1)) |
(90) |
|
B0#(B3(a3(x1))) |
→ |
b1#(A0(B3(x1))) |
(91) |
|
B1#(A2(b1(x1))) |
→ |
B1#(A1(x1)) |
(92) |
|
B1#(A2(b1(x1))) |
→ |
A1#(x1) |
(93) |
|
B1#(A2(b1(x1))) |
→ |
a0#(B1(A1(x1))) |
(94) |
|
B1#(A2(b2(x1))) |
→ |
B1#(A2(x1)) |
(95) |
|
B1#(A2(b2(x1))) |
→ |
a0#(B1(A2(x1))) |
(96) |
|
B1#(A2(b3(x1))) |
→ |
B1#(A3(x1)) |
(97) |
|
B1#(A2(b3(x1))) |
→ |
a0#(B1(A3(x1))) |
(98) |
|
B1#(A3(a0(x1))) |
→ |
B0#(x1) |
(99) |
|
A0#(B2(b1(x1))) |
→ |
A1#(x1) |
(100) |
|
A0#(B3(a0(x1))) |
→ |
B0#(x1) |
(101) |
|
A0#(B3(a0(x1))) |
→ |
A0#(B0(x1)) |
(102) |
|
A0#(B3(a0(x1))) |
→ |
b1#(A0(B0(x1))) |
(103) |
|
A0#(B3(a2(x1))) |
→ |
A0#(B2(x1)) |
(104) |
|
A0#(B3(a2(x1))) |
→ |
b1#(A0(B2(x1))) |
(105) |
|
A0#(B3(a3(x1))) |
→ |
A0#(B3(x1)) |
(106) |
|
A0#(B3(a3(x1))) |
→ |
b1#(A0(B3(x1))) |
(107) |
|
A1#(A2(b1(x1))) |
→ |
B1#(A1(x1)) |
(108) |
|
A1#(A2(b1(x1))) |
→ |
A1#(x1) |
(109) |
|
A1#(A2(b1(x1))) |
→ |
a0#(B1(A1(x1))) |
(110) |
|
A1#(A2(b2(x1))) |
→ |
B1#(A2(x1)) |
(111) |
|
A1#(A2(b2(x1))) |
→ |
a0#(B1(A2(x1))) |
(112) |
|
A1#(A2(b3(x1))) |
→ |
B1#(A3(x1)) |
(113) |
|
A1#(A2(b3(x1))) |
→ |
a0#(B1(A3(x1))) |
(114) |
|
b1#(A2(b1(x1))) |
→ |
B1#(A1(x1)) |
(115) |
|
b1#(A2(b1(x1))) |
→ |
A1#(x1) |
(116) |
|
b1#(A2(b1(x1))) |
→ |
a0#(B1(A1(x1))) |
(117) |
|
b1#(A2(b2(x1))) |
→ |
B1#(A2(x1)) |
(118) |
|
b1#(A2(b2(x1))) |
→ |
a0#(B1(A2(x1))) |
(119) |
|
b1#(A2(b3(x1))) |
→ |
B1#(A3(x1)) |
(120) |
|
b1#(A2(b3(x1))) |
→ |
a0#(B1(A3(x1))) |
(121) |
|
a0#(B3(a0(x1))) |
→ |
B0#(x1) |
(122) |
|
a0#(B3(a0(x1))) |
→ |
A0#(B0(x1)) |
(123) |
|
a0#(B3(a0(x1))) |
→ |
b1#(A0(B0(x1))) |
(124) |
|
a0#(B3(a2(x1))) |
→ |
A0#(B2(x1)) |
(125) |
|
a0#(B3(a2(x1))) |
→ |
b1#(A0(B2(x1))) |
(126) |
|
a0#(B3(a3(x1))) |
→ |
A0#(B3(x1)) |
(127) |
|
a0#(B3(a3(x1))) |
→ |
b1#(A0(B3(x1))) |
(128) |
and
no rules
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.