Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/PEANO_nosorts_iGM)
The rewrite relation of the following TRS is considered.
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
mark(and(X1,X2)) |
→ |
active(and(mark(X1),X2)) |
(4) |
mark(tt) |
→ |
active(tt) |
(5) |
mark(plus(X1,X2)) |
→ |
active(plus(mark(X1),mark(X2))) |
(6) |
mark(0) |
→ |
active(0) |
(7) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(8) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(9) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(10) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(11) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(12) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(13) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(14) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(15) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(16) |
s(mark(X)) |
→ |
s(X) |
(17) |
s(active(X)) |
→ |
s(X) |
(18) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(19) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(20) |
mark#(tt) |
→ |
active#(tt) |
(21) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(22) |
mark#(0) |
→ |
active#(0) |
(23) |
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(24) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(25) |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(26) |
s#(mark(X)) |
→ |
s#(X) |
(27) |
mark#(and(X1,X2)) |
→ |
and#(mark(X1),X2) |
(28) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(29) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(30) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(31) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(32) |
mark#(s(X)) |
→ |
mark#(X) |
(33) |
s#(active(X)) |
→ |
s#(X) |
(34) |
active#(plus(N,s(M))) |
→ |
s#(plus(N,M)) |
(35) |
active#(and(tt,X)) |
→ |
mark#(X) |
(36) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(37) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(38) |
active#(plus(N,s(M))) |
→ |
mark#(s(plus(N,M))) |
(39) |
mark#(plus(X1,X2)) |
→ |
plus#(mark(X1),mark(X2)) |
(40) |
active#(plus(N,0)) |
→ |
mark#(N) |
(41) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(42) |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(43) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(44) |
active#(plus(N,s(M))) |
→ |
plus#(N,M) |
(45) |
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(44) |
active#(plus(N,0)) |
→ |
mark#(N) |
(41) |
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(24) |
active#(plus(N,s(M))) |
→ |
mark#(s(plus(N,M))) |
(39) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(37) |
active#(and(tt,X)) |
→ |
mark#(X) |
(36) |
mark#(s(X)) |
→ |
mark#(X) |
(33) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(32) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(19) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(29) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
8366 |
[and(x1, x2)] |
=
|
17222 |
[plus#(x1, x2)] |
=
|
0 |
[mark#(x1)] |
=
|
17222 |
[0] |
=
|
2438 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 17223 |
[plus(x1, x2)] |
=
|
17222 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[tt] |
=
|
44083 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
s(active(X)) |
→ |
s(X) |
(18) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(15) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(16) |
s(mark(X)) |
→ |
s(X) |
(17) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(10) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(14) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(12) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(11) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(9) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(29) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(27) |
s#(active(X)) |
→ |
s#(X) |
(34) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
31598 |
[and(x1, x2)] |
=
|
x1 + 18755 |
[plus#(x1, x2)] |
=
|
0 |
[mark#(x1)] |
=
|
x1 + 32279 |
[0] |
=
|
60645 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 0 |
[plus(x1, x2)] |
=
|
x2 + 19091 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
48808 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
s(active(X)) |
→ |
s(X) |
(18) |
s(mark(X)) |
→ |
s(X) |
(17) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
s#(active(X)) |
→ |
s#(X) |
(34) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(43) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(25) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(20) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(31) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + 1 |
[plus#(x1, x2)] |
=
|
x1 + 0 |
[mark#(x1)] |
=
|
x1 + 32279 |
[0] |
=
|
53793 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
1 |
[active(x1)] |
=
|
x1 + 2 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
25785 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
s(active(X)) |
→ |
s(X) |
(18) |
s(mark(X)) |
→ |
s(X) |
(17) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(25) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(20) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
4th
component contains the
pair
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(26) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(42) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(22) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(30) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[mark#(x1)] |
=
|
x1 + 32279 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
1 |
[active(x1)] |
=
|
x1 + 2 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
25785 |
[and#(x1, x2)] |
=
|
x2 + 0 |
together with the usable
rules
s(active(X)) |
→ |
s(X) |
(18) |
s(mark(X)) |
→ |
s(X) |
(17) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(26) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(22) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.