Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex16_Luc06_iGM)
The rewrite relation of the following TRS is considered.
active(f(X,X)) |
→ |
mark(f(a,b)) |
(1) |
active(b) |
→ |
mark(a) |
(2) |
mark(f(X1,X2)) |
→ |
active(f(mark(X1),X2)) |
(3) |
mark(a) |
→ |
active(a) |
(4) |
mark(b) |
→ |
active(b) |
(5) |
f(mark(X1),X2) |
→ |
f(X1,X2) |
(6) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(7) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(8) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(9) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
active#(f(X,X)) |
→ |
mark#(f(a,b)) |
(10) |
active#(f(X,X)) |
→ |
f#(a,b) |
(11) |
active#(b) |
→ |
mark#(a) |
(12) |
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(13) |
mark#(f(X1,X2)) |
→ |
f#(mark(X1),X2) |
(14) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(15) |
mark#(a) |
→ |
active#(a) |
(16) |
mark#(b) |
→ |
active#(b) |
(17) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(18) |
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(19) |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(20) |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(21) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(13) |
active#(f(X,X)) |
→ |
mark#(f(a,b)) |
(10) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(15) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
[mark(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[b] |
= |
0 |
[mark#(x1)] |
= |
2 + 2 · x1
|
[active#(x1)] |
= |
2 + 2 · x1
|
the
pair
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(15) |
and
no rules
could be deleted.
1.1.1.1 Instantiation Processor
We instantiate the pair
to the following set of pairs
mark#(f(a,b)) |
→ |
active#(f(mark(a),b)) |
(22) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[f(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[b] |
= |
0 |
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
mark(a) |
→ |
active(a) |
(4) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(7) |
f(mark(X1),X2) |
→ |
f(X1,X2) |
(6) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(8) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
mark#(f(a,b)) |
→ |
active#(f(a,b)) |
(23) |
mark#(f(a,b)) |
→ |
active#(f(active(a),b)) |
(24) |
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(f(a,b)) |
→ |
active#(f(active(a),b)) |
(24) |
active#(f(X,X)) |
→ |
mark#(f(a,b)) |
(10) |
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[b] |
= |
0 |
[active#(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
together with the usable
rules
f(mark(X1),X2) |
→ |
f(X1,X2) |
(6) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(7) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(8) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[b] |
= |
0 |
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(7) |
f(mark(X1),X2) |
→ |
f(X1,X2) |
(6) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(8) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rules
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(7) |
f(mark(X1),X2) |
→ |
f(X1,X2) |
(6) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Semantic Labeling Processor
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,1}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 2):
[active(x1)] |
= |
0 |
[a] |
= |
0 |
[b] |
= |
1 |
[mark#(x1)] |
= |
0 |
[f(x1, x2)] |
= |
0 |
[active#(x1)] |
= |
0 |
We obtain the set of labeled pairs
mark#0(f01(a,b)) |
→ |
active#0(f01(active0(a),b)) |
(25) |
active#0(f00(X,X)) |
→ |
mark#0(f01(a,b)) |
(26) |
active#0(f11(X,X)) |
→ |
mark#0(f01(a,b)) |
(27) |
and the set of labeled rules:
f00(active0(X1),X2) |
→ |
f00(X1,X2) |
(28) |
f01(active0(X1),X2) |
→ |
f01(X1,X2) |
(29) |
f00(active1(X1),X2) |
→ |
f10(X1,X2) |
(30) |
f01(active1(X1),X2) |
→ |
f11(X1,X2) |
(31) |
f00(X1,active0(X2)) |
→ |
f00(X1,X2) |
(32) |
f00(X1,active1(X2)) |
→ |
f01(X1,X2) |
(33) |
f10(X1,active0(X2)) |
→ |
f10(X1,X2) |
(34) |
f10(X1,active1(X2)) |
→ |
f11(X1,X2) |
(35) |
1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f01(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[active0(x1)] |
= |
1 · x1
|
[active1(x1)] |
= |
1 + 1 · x1
|
[f11(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
[mark#0(x1)] |
= |
1 + 1 · x1
|
[a] |
= |
0 |
[b] |
= |
1 |
[active#0(x1)] |
= |
1 + 1 · x1
|
[f00(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
together with the usable
rules
f01(active0(X1),X2) |
→ |
f01(X1,X2) |
(29) |
f01(active1(X1),X2) |
→ |
f11(X1,X2) |
(31) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#0(f00(X,X)) |
→ |
mark#0(f01(a,b)) |
(26) |
and
the
rule
f01(active1(X1),X2) |
→ |
f11(X1,X2) |
(31) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(19) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(18) |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(20) |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(21) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[f#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(19) |
|
1 |
≥ |
1 |
2 |
> |
2 |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(18) |
|
1 |
> |
1 |
2 |
≥ |
2 |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(20) |
|
1 |
> |
1 |
2 |
≥ |
2 |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(21) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.