Certification Problem
Input (TPDB TRS_Equational/AProVE_AC_04/AC20)
The rewrite relation of the following equational TRS is considered.
|
plus(x,0) |
→ |
x |
(1) |
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
times(x,0) |
→ |
0 |
(3) |
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(4) |
|
minus(x,0) |
→ |
x |
(5) |
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(6) |
|
div(0,s(y)) |
→ |
0 |
(7) |
|
div(s(x),s(y)) |
→ |
s(div(minus(x,y),s(y))) |
(8) |
Associative symbols: plus, times
Commutative symbols: plus, times
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 AC Dependency Pair Transformation
The following set of (strict) dependency pairs is constructed for the TRS.
|
plus#(x,s(y)) |
→ |
plus#(x,y) |
(19) |
|
times#(x,s(y)) |
→ |
plus#(times(x,y),x) |
(20) |
|
times#(x,s(y)) |
→ |
times#(x,y) |
(21) |
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(22) |
|
div#(s(x),s(y)) |
→ |
div#(minus(x,y),s(y)) |
(23) |
|
div#(s(x),s(y)) |
→ |
minus#(x,y) |
(24) |
|
plus#(plus(x,0),ext) |
→ |
plus#(x,ext) |
(25) |
|
plus#(plus(x,s(y)),ext) |
→ |
plus#(s(plus(x,y)),ext) |
(26) |
|
plus#(plus(x,s(y)),ext) |
→ |
plus#(x,y) |
(27) |
|
times#(times(x,0),ext) |
→ |
times#(0,ext) |
(28) |
|
times#(times(x,s(y)),ext) |
→ |
times#(plus(times(x,y),x),ext) |
(29) |
|
times#(times(x,s(y)),ext) |
→ |
plus#(times(x,y),x) |
(30) |
|
times#(times(x,s(y)),ext) |
→ |
times#(x,y) |
(31) |
The extended rules of the TRS
|
plus(plus(x,0),ext) |
→ |
plus(x,ext) |
(32) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
times(times(x,0),ext) |
→ |
times(0,ext) |
(34) |
|
times(times(x,s(y)),ext) |
→ |
times(plus(times(x,y),x),ext) |
(35) |
give rise to even more dependency pairs (by sharping the root symbols of each rule).
Finiteness for all DPs in combination with the equational DPs is proven as follows.
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
times#(times(x,0),ext) |
→ |
times#(0,ext) |
(28) |
|
times#(x,s(y)) |
→ |
times#(x,y) |
(21) |
|
times#(times(x,s(y)),ext) |
→ |
times#(plus(times(x,y),x),ext) |
(29) |
|
times#(times(x,s(y)),ext) |
→ |
times#(x,y) |
(31) |
|
times#(times(x,y),z) |
→ |
times#(y,z) |
(18) |
|
times#(x,y) |
→ |
times#(y,x) |
(14) |
|
times#(times(x,y),z) |
→ |
times#(x,times(y,z)) |
(16) |
1.1.1 AC Reduction Pair Processor with Usable Rules
Using the non-linear polynomial interpretation over the naturals
| [times#(x1, x2)] |
= |
1 · x2 + 1 · x1 + 1 · x1 · x2
|
| [times(x1, x2)] |
= |
1 · x2 + 1 · x1 + 1 · x1 · x2
|
| [s(x1)] |
= |
1 + 1 · x1
|
| [plus(x1, x2)] |
= |
1 · x2 + 1 · x1
|
| [0] |
= |
0 |
together with the usable
rules
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
plus(x,0) |
→ |
x |
(1) |
|
plus(plus(x,0),ext) |
→ |
plus(x,ext) |
(32) |
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(4) |
|
times(times(x,s(y)),ext) |
→ |
times(plus(times(x,y),x),ext) |
(35) |
|
times(times(x,0),ext) |
→ |
times(0,ext) |
(34) |
|
times(x,0) |
→ |
0 |
(3) |
|
plus(plus(x,y),z) |
→ |
plus(x,plus(y,z)) |
(11) |
|
plus(x,y) |
→ |
plus(y,x) |
(9) |
|
times(x,y) |
→ |
times(y,x) |
(10) |
|
times(times(x,y),z) |
→ |
times(x,times(y,z)) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
times#(times(x,s(y)),ext) |
→ |
times#(plus(times(x,y),x),ext) |
(29) |
|
times#(times(x,s(y)),ext) |
→ |
times#(x,y) |
(31) |
|
times#(x,s(y)) |
→ |
times#(x,y) |
(21) |
could be deleted.
1.1.1.1 AC Reduction Pair Processor with Usable Rules
Using the non-linear polynomial interpretation over the naturals
| [times#(x1, x2)] |
= |
2 · x2 + 2 · x1 + 1 · x1 · x2
|
| [times(x1, x2)] |
= |
2 + 2 · x2 + 2 · x1 + 1 · x1 · x2
|
| [0] |
= |
0 |
| [s(x1)] |
= |
2 + 1 · x1
|
| [plus(x1, x2)] |
= |
1 · x2 + 1 · x1
|
together with the usable
rules
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(4) |
|
times(times(x,s(y)),ext) |
→ |
times(plus(times(x,y),x),ext) |
(35) |
|
times(times(x,0),ext) |
→ |
times(0,ext) |
(34) |
|
times(x,0) |
→ |
0 |
(3) |
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
plus(x,0) |
→ |
x |
(1) |
|
plus(plus(x,0),ext) |
→ |
plus(x,ext) |
(32) |
|
times(x,y) |
→ |
times(y,x) |
(10) |
|
times(times(x,y),z) |
→ |
times(x,times(y,z)) |
(12) |
|
plus(plus(x,y),z) |
→ |
plus(x,plus(y,z)) |
(11) |
|
plus(x,y) |
→ |
plus(y,x) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
times#(times(x,y),z) |
→ |
times#(y,z) |
(18) |
|
times#(times(x,0),ext) |
→ |
times#(0,ext) |
(28) |
could be deleted.
1.1.1.1.1 AC Dependency Pair Problem is trivial
There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
-
The
2nd
component contains the
pair
|
plus#(plus(x,0),ext) |
→ |
plus#(x,ext) |
(25) |
|
plus#(x,s(y)) |
→ |
plus#(x,y) |
(19) |
|
plus#(plus(x,s(y)),ext) |
→ |
plus#(s(plus(x,y)),ext) |
(26) |
|
plus#(plus(x,s(y)),ext) |
→ |
plus#(x,y) |
(27) |
|
plus#(plus(x,y),z) |
→ |
plus#(y,z) |
(17) |
|
plus#(x,y) |
→ |
plus#(y,x) |
(13) |
|
plus#(plus(x,y),z) |
→ |
plus#(x,plus(y,z)) |
(15) |
1.1.2 AC Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [plus#(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [s(x1)] |
= |
1 · x1
|
| [0] |
= |
0 |
together with the usable
rules
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
plus(x,0) |
→ |
x |
(1) |
|
plus(plus(x,0),ext) |
→ |
plus(x,ext) |
(32) |
|
plus(plus(x,y),z) |
→ |
plus(x,plus(y,z)) |
(11) |
|
plus(x,y) |
→ |
plus(y,x) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
plus#(plus(x,0),ext) |
→ |
plus#(x,ext) |
(25) |
and
the
rules
|
plus(x,0) |
→ |
x |
(1) |
|
times(x,0) |
→ |
0 |
(3) |
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(4) |
|
minus(x,0) |
→ |
x |
(5) |
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(6) |
|
div(0,s(y)) |
→ |
0 |
(7) |
|
div(s(x),s(y)) |
→ |
s(div(minus(x,y),s(y))) |
(8) |
|
plus(plus(x,0),ext) |
→ |
plus(x,ext) |
(32) |
|
times(times(x,0),ext) |
→ |
times(0,ext) |
(34) |
|
times(times(x,s(y)),ext) |
→ |
times(plus(times(x,y),x),ext) |
(35) |
could be deleted.
1.1.2.1 AC Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [plus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [times(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [s(x1)] |
= |
1 + 1 · x1
|
together with the usable
rules
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
plus(x,y) |
→ |
plus(y,x) |
(9) |
|
times(x,y) |
→ |
times(y,x) |
(10) |
|
plus(plus(x,y),z) |
→ |
plus(x,plus(y,z)) |
(11) |
|
times(times(x,y),z) |
→ |
times(x,times(y,z)) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
plus#(plus(x,s(y)),ext) |
→ |
plus#(x,y) |
(27) |
|
plus#(x,s(y)) |
→ |
plus#(x,y) |
(19) |
and
no rules
could be deleted.
1.1.2.1.1 AC Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [plus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [plus(x1, x2)] |
= |
2 + 1 · x1 + 1 · x2
|
| [times(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [s(x1)] |
= |
1 · x1
|
together with the usable
rules
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
plus(x,y) |
→ |
plus(y,x) |
(9) |
|
times(x,y) |
→ |
times(y,x) |
(10) |
|
plus(plus(x,y),z) |
→ |
plus(x,plus(y,z)) |
(11) |
|
times(times(x,y),z) |
→ |
times(x,times(y,z)) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
plus#(plus(x,y),z) |
→ |
plus#(y,z) |
(17) |
and
no rules
could be deleted.
1.1.2.1.1.1 AC Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [plus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [plus(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
| [s(x1)] |
= |
0 |
together with the usable
rules
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
plus(plus(x,y),z) |
→ |
plus(x,plus(y,z)) |
(11) |
|
plus(x,y) |
→ |
plus(y,x) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
plus#(plus(x,s(y)),ext) |
→ |
plus#(s(plus(x,y)),ext) |
(26) |
could be deleted.
1.1.2.1.1.1.1 AC Dependency Pair Problem is trivial
There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
-
The
3rd
component contains the
pair
|
div#(s(x),s(y)) |
→ |
div#(minus(x,y),s(y)) |
(23) |
1.1.3 AC Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [div#(x1, x2)] |
= |
2 · x1
|
| [s(x1)] |
= |
2 + 2 · x1
|
| [minus(x1, x2)] |
= |
2 · x1
|
| [0] |
= |
0 |
together with the usable
rules
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(6) |
|
minus(x,0) |
→ |
x |
(5) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
div#(s(x),s(y)) |
→ |
div#(minus(x,y),s(y)) |
(23) |
could be deleted.
1.1.3.1 AC Dependency Pair Problem is trivial
There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
-
The
4th
component contains the
pair
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(22) |
1.1.4 AC Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [minus#(x1, x2)] |
= |
3 · x1 + 3 · x2
|
| [s(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(22) |
and
the
rules
|
plus(x,0) |
→ |
x |
(1) |
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(2) |
|
times(x,0) |
→ |
0 |
(3) |
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(4) |
|
minus(x,0) |
→ |
x |
(5) |
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(6) |
|
div(0,s(y)) |
→ |
0 |
(7) |
|
div(s(x),s(y)) |
→ |
s(div(minus(x,y),s(y))) |
(8) |
|
plus(plus(x,0),ext) |
→ |
plus(x,ext) |
(32) |
|
plus(plus(x,s(y)),ext) |
→ |
plus(s(plus(x,y)),ext) |
(33) |
|
times(times(x,0),ext) |
→ |
times(0,ext) |
(34) |
|
times(times(x,s(y)),ext) |
→ |
times(plus(times(x,y),x),ext) |
(35) |
could be deleted.
1.1.4.1 AC Dependency Pair Problem is trivial
There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.