Certification Problem
Input (TPDB SRS_Standard/Zantema_06/beans2)
The rewrite relation of the following TRS is considered.
|
b(a(a(x1))) |
→ |
a(b(c(x1))) |
(1) |
|
c(a(x1)) |
→ |
a(c(x1)) |
(2) |
|
c(b(x1)) |
→ |
b(a(x1)) |
(3) |
|
L(a(a(x1))) |
→ |
L(a(b(c(x1)))) |
(4) |
|
c(R(x1)) |
→ |
b(a(R(x1))) |
(5) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Split
We split R in the relative problem D/R-D and R-D, where the rules D
|
L(a(a(x1))) |
→ |
L(a(b(c(x1)))) |
(4) |
|
c(R(x1)) |
→ |
b(a(R(x1))) |
(5) |
are deleted.
1.1 Closure Under Flat Contexts
Using the flat contexts
{L(☐), R(☐), c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
L(L(a(a(x1)))) |
→ |
L(L(a(b(c(x1))))) |
(6) |
|
L(c(R(x1))) |
→ |
L(b(a(R(x1)))) |
(7) |
|
R(L(a(a(x1)))) |
→ |
R(L(a(b(c(x1))))) |
(8) |
|
R(c(R(x1))) |
→ |
R(b(a(R(x1)))) |
(9) |
|
c(L(a(a(x1)))) |
→ |
c(L(a(b(c(x1))))) |
(10) |
|
c(c(R(x1))) |
→ |
c(b(a(R(x1)))) |
(11) |
|
b(L(a(a(x1)))) |
→ |
b(L(a(b(c(x1))))) |
(12) |
|
b(c(R(x1))) |
→ |
b(b(a(R(x1)))) |
(13) |
|
a(L(a(a(x1)))) |
→ |
a(L(a(b(c(x1))))) |
(14) |
|
a(c(R(x1))) |
→ |
a(b(a(R(x1)))) |
(15) |
|
L(b(a(a(x1)))) |
→ |
L(a(b(c(x1)))) |
(16) |
|
L(c(a(x1))) |
→ |
L(a(c(x1))) |
(17) |
|
L(c(b(x1))) |
→ |
L(b(a(x1))) |
(18) |
|
R(b(a(a(x1)))) |
→ |
R(a(b(c(x1)))) |
(19) |
|
R(c(a(x1))) |
→ |
R(a(c(x1))) |
(20) |
|
R(c(b(x1))) |
→ |
R(b(a(x1))) |
(21) |
|
c(b(a(a(x1)))) |
→ |
c(a(b(c(x1)))) |
(22) |
|
c(c(a(x1))) |
→ |
c(a(c(x1))) |
(23) |
|
c(c(b(x1))) |
→ |
c(b(a(x1))) |
(24) |
|
b(b(a(a(x1)))) |
→ |
b(a(b(c(x1)))) |
(25) |
|
b(c(a(x1))) |
→ |
b(a(c(x1))) |
(26) |
|
b(c(b(x1))) |
→ |
b(b(a(x1))) |
(27) |
|
a(b(a(a(x1)))) |
→ |
a(a(b(c(x1)))) |
(28) |
|
a(c(a(x1))) |
→ |
a(a(c(x1))) |
(29) |
|
a(c(b(x1))) |
→ |
a(b(a(x1))) |
(30) |
1.1.1 Closure Under Flat Contexts
Using the flat contexts
{L(☐), R(☐), c(☐), b(☐), a(☐)}
We obtain the transformed TRS
There are 125 ruless (increase limit for explicit display).
1.1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,24}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 25):
| [L(x1)] |
= |
5x1 + 0 |
| [R(x1)] |
= |
5x1 + 1 |
| [c(x1)] |
= |
5x1 + 2 |
| [b(x1)] |
= |
5x1 + 3 |
| [a(x1)] |
= |
5x1 + 4 |
We obtain the labeled TRS
There are 3125 ruless (increase limit for explicit display).
1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [L0(x1)] |
= |
x1 +
|
| [L10(x1)] |
= |
x1 +
|
| [L15(x1)] |
= |
x1 +
|
| [L20(x1)] |
= |
x1 +
|
| [L1(x1)] |
= |
x1 +
|
| [L11(x1)] |
= |
x1 +
|
| [L16(x1)] |
= |
x1 +
|
| [L21(x1)] |
= |
x1 +
|
| [L2(x1)] |
= |
x1 +
|
| [L7(x1)] |
= |
x1 +
|
| [L12(x1)] |
= |
x1 +
|
| [L17(x1)] |
= |
x1 +
|
| [L22(x1)] |
= |
x1 +
|
| [L3(x1)] |
= |
x1 +
|
| [L13(x1)] |
= |
x1 +
|
| [L18(x1)] |
= |
x1 +
|
| [L23(x1)] |
= |
x1 +
|
| [L4(x1)] |
= |
x1 +
|
| [L14(x1)] |
= |
x1 +
|
| [L19(x1)] |
= |
x1 +
|
| [L24(x1)] |
= |
x1 +
|
| [R0(x1)] |
= |
x1 +
|
| [R5(x1)] |
= |
x1 +
|
| [R10(x1)] |
= |
x1 +
|
| [R15(x1)] |
= |
x1 +
|
| [R20(x1)] |
= |
x1 +
|
| [R1(x1)] |
= |
x1 +
|
| [R6(x1)] |
= |
x1 +
|
| [R11(x1)] |
= |
x1 +
|
| [R16(x1)] |
= |
x1 +
|
| [R21(x1)] |
= |
x1 +
|
| [R2(x1)] |
= |
x1 +
|
| [R7(x1)] |
= |
x1 +
|
| [R12(x1)] |
= |
x1 +
|
| [R17(x1)] |
= |
x1 +
|
| [R22(x1)] |
= |
x1 +
|
| [R3(x1)] |
= |
x1 +
|
| [R8(x1)] |
= |
x1 +
|
| [R13(x1)] |
= |
x1 +
|
| [R18(x1)] |
= |
x1 +
|
| [R23(x1)] |
= |
x1 +
|
| [R4(x1)] |
= |
x1 +
|
| [R9(x1)] |
= |
x1 +
|
| [R14(x1)] |
= |
x1 +
|
| [R19(x1)] |
= |
x1 +
|
| [R24(x1)] |
= |
x1 +
|
| [c0(x1)] |
= |
x1 +
|
| [c5(x1)] |
= |
x1 +
|
| [c10(x1)] |
= |
x1 +
|
| [c15(x1)] |
= |
x1 +
|
| [c20(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c6(x1)] |
= |
x1 +
|
| [c11(x1)] |
= |
x1 +
|
| [c16(x1)] |
= |
x1 +
|
| [c21(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [c7(x1)] |
= |
x1 +
|
| [c12(x1)] |
= |
x1 +
|
| [c17(x1)] |
= |
x1 +
|
| [c22(x1)] |
= |
x1 +
|
| [c3(x1)] |
= |
x1 +
|
| [c8(x1)] |
= |
x1 +
|
| [c13(x1)] |
= |
x1 +
|
| [c18(x1)] |
= |
x1 +
|
| [c23(x1)] |
= |
x1 +
|
| [c4(x1)] |
= |
x1 +
|
| [c9(x1)] |
= |
x1 +
|
| [c14(x1)] |
= |
x1 +
|
| [c19(x1)] |
= |
x1 +
|
| [c24(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b5(x1)] |
= |
x1 +
|
| [b10(x1)] |
= |
x1 +
|
| [b15(x1)] |
= |
x1 +
|
| [b20(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b6(x1)] |
= |
x1 +
|
| [b11(x1)] |
= |
x1 +
|
| [b16(x1)] |
= |
x1 +
|
| [b21(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b7(x1)] |
= |
x1 +
|
| [b12(x1)] |
= |
x1 +
|
| [b17(x1)] |
= |
x1 +
|
| [b22(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b8(x1)] |
= |
x1 +
|
| [b13(x1)] |
= |
x1 +
|
| [b18(x1)] |
= |
x1 +
|
| [b23(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [b9(x1)] |
= |
x1 +
|
| [b14(x1)] |
= |
x1 +
|
| [b19(x1)] |
= |
x1 +
|
| [b24(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a5(x1)] |
= |
x1 +
|
| [a10(x1)] |
= |
x1 +
|
| [a15(x1)] |
= |
x1 +
|
| [a20(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a6(x1)] |
= |
x1 +
|
| [a11(x1)] |
= |
x1 +
|
| [a16(x1)] |
= |
x1 +
|
| [a21(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a7(x1)] |
= |
x1 +
|
| [a12(x1)] |
= |
x1 +
|
| [a17(x1)] |
= |
x1 +
|
| [a22(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [a8(x1)] |
= |
x1 +
|
| [a13(x1)] |
= |
x1 +
|
| [a18(x1)] |
= |
x1 +
|
| [a23(x1)] |
= |
x1 +
|
| [a4(x1)] |
= |
x1 +
|
| [a9(x1)] |
= |
x1 +
|
| [a14(x1)] |
= |
x1 +
|
| [a19(x1)] |
= |
x1 +
|
| [a24(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
There are 2925 ruless (increase limit for explicit display).
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(b(a(a(x1)))) |
→ |
c(a(b(c(x1)))) |
(22) |
|
c(c(a(x1))) |
→ |
c(a(c(x1))) |
(23) |
|
c(c(b(x1))) |
→ |
c(b(a(x1))) |
(24) |
|
b(b(a(a(x1)))) |
→ |
b(a(b(c(x1)))) |
(25) |
|
b(c(a(x1))) |
→ |
b(a(c(x1))) |
(26) |
|
b(c(b(x1))) |
→ |
b(b(a(x1))) |
(27) |
|
a(b(a(a(x1)))) |
→ |
a(a(b(c(x1)))) |
(28) |
|
a(c(a(x1))) |
→ |
a(a(c(x1))) |
(29) |
|
a(c(b(x1))) |
→ |
a(b(a(x1))) |
(30) |
1.2.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,1,2}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 3):
| [c(x1)] |
= |
3x1 + 0 |
| [b(x1)] |
= |
3x1 + 1 |
| [a(x1)] |
= |
3x1 + 2 |
We obtain the labeled TRS
|
b1(b2(a2(a1(x1)))) |
→ |
b2(a1(b0(c1(x1)))) |
(3281) |
|
b1(b2(a2(a2(x1)))) |
→ |
b2(a1(b0(c2(x1)))) |
(3282) |
|
b1(b2(a2(a0(x1)))) |
→ |
b2(a1(b0(c0(x1)))) |
(3283) |
|
a1(b2(a2(a1(x1)))) |
→ |
a2(a1(b0(c1(x1)))) |
(3284) |
|
a1(b2(a2(a2(x1)))) |
→ |
a2(a1(b0(c2(x1)))) |
(3285) |
|
a1(b2(a2(a0(x1)))) |
→ |
a2(a1(b0(c0(x1)))) |
(3286) |
|
c1(b2(a2(a1(x1)))) |
→ |
c2(a1(b0(c1(x1)))) |
(3287) |
|
c1(b2(a2(a2(x1)))) |
→ |
c2(a1(b0(c2(x1)))) |
(3288) |
|
c1(b2(a2(a0(x1)))) |
→ |
c2(a1(b0(c0(x1)))) |
(3289) |
|
b0(c2(a1(x1))) |
→ |
b2(a0(c1(x1))) |
(3290) |
|
b0(c2(a2(x1))) |
→ |
b2(a0(c2(x1))) |
(3291) |
|
b0(c2(a0(x1))) |
→ |
b2(a0(c0(x1))) |
(3292) |
|
a0(c2(a1(x1))) |
→ |
a2(a0(c1(x1))) |
(3293) |
|
a0(c2(a2(x1))) |
→ |
a2(a0(c2(x1))) |
(3294) |
|
a0(c2(a0(x1))) |
→ |
a2(a0(c0(x1))) |
(3295) |
|
c0(c2(a1(x1))) |
→ |
c2(a0(c1(x1))) |
(3296) |
|
c0(c2(a2(x1))) |
→ |
c2(a0(c2(x1))) |
(3297) |
|
c0(c2(a0(x1))) |
→ |
c2(a0(c0(x1))) |
(3298) |
|
b0(c1(b1(x1))) |
→ |
b1(b2(a1(x1))) |
(3299) |
|
b0(c1(b2(x1))) |
→ |
b1(b2(a2(x1))) |
(3300) |
|
b0(c1(b0(x1))) |
→ |
b1(b2(a0(x1))) |
(3301) |
|
a0(c1(b1(x1))) |
→ |
a1(b2(a1(x1))) |
(3302) |
|
a0(c1(b2(x1))) |
→ |
a1(b2(a2(x1))) |
(3303) |
|
a0(c1(b0(x1))) |
→ |
a1(b2(a0(x1))) |
(3304) |
|
c0(c1(b1(x1))) |
→ |
c1(b2(a1(x1))) |
(3305) |
|
c0(c1(b2(x1))) |
→ |
c1(b2(a2(x1))) |
(3306) |
|
c0(c1(b0(x1))) |
→ |
c1(b2(a0(x1))) |
(3307) |
1.2.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [c0(x1)] |
= |
· x1 +
|
| [c1(x1)] |
= |
· x1 +
|
| [c2(x1)] |
= |
· x1 +
|
| [b0(x1)] |
= |
· x1 +
|
| [b1(x1)] |
= |
· x1 +
|
| [b2(x1)] |
= |
· x1 +
|
| [a0(x1)] |
= |
· x1 +
|
| [a1(x1)] |
= |
· x1 +
|
| [a2(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
b1(b2(a2(a1(x1)))) |
→ |
b2(a1(b0(c1(x1)))) |
(3281) |
|
b1(b2(a2(a2(x1)))) |
→ |
b2(a1(b0(c2(x1)))) |
(3282) |
|
b1(b2(a2(a0(x1)))) |
→ |
b2(a1(b0(c0(x1)))) |
(3283) |
|
a1(b2(a2(a1(x1)))) |
→ |
a2(a1(b0(c1(x1)))) |
(3284) |
|
a1(b2(a2(a2(x1)))) |
→ |
a2(a1(b0(c2(x1)))) |
(3285) |
|
a1(b2(a2(a0(x1)))) |
→ |
a2(a1(b0(c0(x1)))) |
(3286) |
|
c1(b2(a2(a1(x1)))) |
→ |
c2(a1(b0(c1(x1)))) |
(3287) |
|
c1(b2(a2(a2(x1)))) |
→ |
c2(a1(b0(c2(x1)))) |
(3288) |
|
c1(b2(a2(a0(x1)))) |
→ |
c2(a1(b0(c0(x1)))) |
(3289) |
|
b0(c2(a1(x1))) |
→ |
b2(a0(c1(x1))) |
(3290) |
|
b0(c2(a2(x1))) |
→ |
b2(a0(c2(x1))) |
(3291) |
|
b0(c2(a0(x1))) |
→ |
b2(a0(c0(x1))) |
(3292) |
|
a0(c2(a1(x1))) |
→ |
a2(a0(c1(x1))) |
(3293) |
|
a0(c2(a2(x1))) |
→ |
a2(a0(c2(x1))) |
(3294) |
|
a0(c2(a0(x1))) |
→ |
a2(a0(c0(x1))) |
(3295) |
|
c0(c2(a1(x1))) |
→ |
c2(a0(c1(x1))) |
(3296) |
|
c0(c2(a2(x1))) |
→ |
c2(a0(c2(x1))) |
(3297) |
|
c0(c2(a0(x1))) |
→ |
c2(a0(c0(x1))) |
(3298) |
1.2.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c0(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
b0(c1(b1(x1))) |
→ |
b1(b2(a1(x1))) |
(3299) |
|
b0(c1(b2(x1))) |
→ |
b1(b2(a2(x1))) |
(3300) |
|
b0(c1(b0(x1))) |
→ |
b1(b2(a0(x1))) |
(3301) |
|
a0(c1(b1(x1))) |
→ |
a1(b2(a1(x1))) |
(3302) |
|
a0(c1(b2(x1))) |
→ |
a1(b2(a2(x1))) |
(3303) |
|
a0(c1(b0(x1))) |
→ |
a1(b2(a0(x1))) |
(3304) |
|
c0(c1(b1(x1))) |
→ |
c1(b2(a1(x1))) |
(3305) |
|
c0(c1(b2(x1))) |
→ |
c1(b2(a2(x1))) |
(3306) |
|
c0(c1(b0(x1))) |
→ |
c1(b2(a0(x1))) |
(3307) |
1.2.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.