Certification Problem
Input (TPDB TRS_Standard/CiME_04/ternary-hard)
The rewrite relation of the following TRS is considered.
|
0(#) |
→ |
# |
(1) |
|
+(#,x) |
→ |
x |
(2) |
|
+(x,#) |
→ |
x |
(3) |
|
+(0(x),0(y)) |
→ |
0(+(x,y)) |
(4) |
|
+(0(x),1(y)) |
→ |
1(+(x,y)) |
(5) |
|
+(1(x),0(y)) |
→ |
1(+(x,y)) |
(6) |
|
+(0(x),j(y)) |
→ |
j(+(x,y)) |
(7) |
|
+(j(x),0(y)) |
→ |
j(+(x,y)) |
(8) |
|
+(1(x),1(y)) |
→ |
j(+(+(x,y),1(#))) |
(9) |
|
+(j(x),j(y)) |
→ |
1(+(+(x,y),j(#))) |
(10) |
|
+(1(x),j(y)) |
→ |
0(+(x,y)) |
(11) |
|
+(j(x),1(y)) |
→ |
0(+(x,y)) |
(12) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(13) |
|
opp(#) |
→ |
# |
(14) |
|
opp(0(x)) |
→ |
0(opp(x)) |
(15) |
|
opp(1(x)) |
→ |
j(opp(x)) |
(16) |
|
opp(j(x)) |
→ |
1(opp(x)) |
(17) |
|
-(x,y) |
→ |
+(x,opp(y)) |
(18) |
|
*(#,x) |
→ |
# |
(19) |
|
*(0(x),y) |
→ |
0(*(x,y)) |
(20) |
|
*(1(x),y) |
→ |
+(0(*(x,y)),y) |
(21) |
|
*(j(x),y) |
→ |
-(0(*(x,y)),y) |
(22) |
|
*(*(x,y),z) |
→ |
*(x,*(y,z)) |
(23) |
|
*(+(x,y),z) |
→ |
+(*(x,z),*(y,z)) |
(24) |
|
*(x,+(y,z)) |
→ |
+(*(x,y),*(x,z)) |
(25) |
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.
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(26) |
|
*#(1(x),y) |
→ |
0#(*(x,y)) |
(27) |
|
+#(0(x),0(y)) |
→ |
0#(+(x,y)) |
(28) |
|
-#(x,y) |
→ |
+#(x,opp(y)) |
(29) |
|
-#(x,y) |
→ |
opp#(y) |
(30) |
|
+#(1(x),j(y)) |
→ |
0#(+(x,y)) |
(31) |
|
+#(j(x),1(y)) |
→ |
0#(+(x,y)) |
(32) |
|
*#(+(x,y),z) |
→ |
*#(x,z) |
(33) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(34) |
|
*#(j(x),y) |
→ |
-#(0(*(x,y)),y) |
(35) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(36) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(37) |
|
opp#(1(x)) |
→ |
opp#(x) |
(38) |
|
+#(j(x),1(y)) |
→ |
+#(x,y) |
(39) |
|
*#(1(x),y) |
→ |
*#(x,y) |
(40) |
|
*#(j(x),y) |
→ |
0#(*(x,y)) |
(41) |
|
+#(0(x),j(y)) |
→ |
+#(x,y) |
(42) |
|
+#(1(x),j(y)) |
→ |
+#(x,y) |
(43) |
|
*#(+(x,y),z) |
→ |
*#(y,z) |
(44) |
|
opp#(0(x)) |
→ |
0#(opp(x)) |
(45) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(46) |
|
*#(1(x),y) |
→ |
+#(0(*(x,y)),y) |
(47) |
|
+#(j(x),j(y)) |
→ |
+#(x,y) |
(48) |
|
opp#(0(x)) |
→ |
opp#(x) |
(49) |
|
opp#(j(x)) |
→ |
opp#(x) |
(50) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(51) |
|
+#(j(x),0(y)) |
→ |
+#(x,y) |
(52) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(53) |
|
*#(j(x),y) |
→ |
*#(x,y) |
(54) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(55) |
|
+#(j(x),j(y)) |
→ |
+#(+(x,y),j(#)) |
(56) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(57) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(58) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(59) |
|
*#(+(x,y),z) |
→ |
+#(*(x,z),*(y,z)) |
(60) |
|
*#(x,+(y,z)) |
→ |
+#(*(x,y),*(x,z)) |
(61) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(62) |
|
*#(0(x),y) |
→ |
0#(*(x,y)) |
(63) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
*#(1(x),y) |
→ |
*#(x,y) |
(40) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(37) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(57) |
|
*#(j(x),y) |
→ |
*#(x,y) |
(54) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(34) |
|
*#(+(x,y),z) |
→ |
*#(x,z) |
(33) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(53) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(51) |
|
*#(+(x,y),z) |
→ |
*#(y,z) |
(44) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [0#(x1)] |
=
|
0 |
| [1(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
x1 + 0 |
| [#] |
=
|
1 |
| [opp#(x1)] |
=
|
0 |
| [0(x1)] |
=
|
x1 + 3 |
| [-(x1, x2)] |
=
|
x1 + x2 + 0 |
| [j(x1)] |
=
|
x1 + 1 |
| [opp(x1)] |
=
|
x1 + 21277 |
| [-#(x1, x2)] |
=
|
0 |
| [+(x1, x2)] |
=
|
x1 + x2 + 7720 |
| [+#(x1, x2)] |
=
|
0 |
| [*(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
|
opp(0(x)) |
→ |
0(opp(x)) |
(15) |
|
0(#) |
→ |
# |
(1) |
|
opp(1(x)) |
→ |
j(opp(x)) |
(16) |
|
opp(j(x)) |
→ |
1(opp(x)) |
(17) |
|
opp(#) |
→ |
# |
(14) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
*#(1(x),y) |
→ |
*#(x,y) |
(40) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(37) |
|
*#(j(x),y) |
→ |
*#(x,y) |
(54) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(34) |
|
*#(+(x,y),z) |
→ |
*#(x,z) |
(33) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(53) |
|
*#(+(x,y),z) |
→ |
*#(y,z) |
(44) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
|
opp#(1(x)) |
→ |
opp#(x) |
(38) |
|
opp#(j(x)) |
→ |
opp#(x) |
(50) |
|
opp#(0(x)) |
→ |
opp#(x) |
(49) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [0#(x1)] |
=
|
0 |
| [1(x1)] |
=
|
x1 + 3 |
| [*#(x1, x2)] |
=
|
0 |
| [#] |
=
|
1 |
| [opp#(x1)] |
=
|
x1 + 0 |
| [0(x1)] |
=
|
x1 + 2 |
| [-(x1, x2)] |
=
|
x1 + x2 + 0 |
| [j(x1)] |
=
|
x1 + 1 |
| [opp(x1)] |
=
|
x1 + 1 |
| [-#(x1, x2)] |
=
|
0 |
| [+(x1, x2)] |
=
|
x2 + 5 |
| [+#(x1, x2)] |
=
|
0 |
| [*(x1, x2)] |
=
|
x1 + x2 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
|
opp#(1(x)) |
→ |
opp#(x) |
(38) |
|
opp#(j(x)) |
→ |
opp#(x) |
(50) |
|
opp#(0(x)) |
→ |
opp#(x) |
(49) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(62) |
|
+#(j(x),1(y)) |
→ |
+#(x,y) |
(39) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(59) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(36) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(58) |
|
+#(j(x),j(y)) |
→ |
+#(+(x,y),j(#)) |
(56) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(55) |
|
+#(j(x),0(y)) |
→ |
+#(x,y) |
(52) |
|
+#(j(x),j(y)) |
→ |
+#(x,y) |
(48) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(46) |
|
+#(1(x),j(y)) |
→ |
+#(x,y) |
(43) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(26) |
|
+#(0(x),j(y)) |
→ |
+#(x,y) |
(42) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [0#(x1)] |
=
|
0 |
| [1(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
0 |
| [#] |
=
|
0 |
| [opp#(x1)] |
=
|
0 |
| [0(x1)] |
=
|
x1 + 2 |
| [-(x1, x2)] |
=
|
x1 + x2 + 38475 |
| [j(x1)] |
=
|
x1 + 1 |
| [opp(x1)] |
=
|
x1 + 38476 |
| [-#(x1, x2)] |
=
|
0 |
| [+(x1, x2)] |
=
|
x1 + x2 + 0 |
| [+#(x1, x2)] |
=
|
x1 + x2 + 0 |
| [*(x1, x2)] |
=
|
28224 |
together with the usable
rules
|
+(0(x),0(y)) |
→ |
0(+(x,y)) |
(4) |
|
+(j(x),0(y)) |
→ |
j(+(x,y)) |
(8) |
|
0(#) |
→ |
# |
(1) |
|
+(x,#) |
→ |
x |
(3) |
|
+(0(x),1(y)) |
→ |
1(+(x,y)) |
(5) |
|
+(j(x),j(y)) |
→ |
1(+(+(x,y),j(#))) |
(10) |
|
+(0(x),j(y)) |
→ |
j(+(x,y)) |
(7) |
|
+(j(x),1(y)) |
→ |
0(+(x,y)) |
(12) |
|
+(1(x),j(y)) |
→ |
0(+(x,y)) |
(11) |
|
+(1(x),1(y)) |
→ |
j(+(+(x,y),1(#))) |
(9) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(13) |
|
+(1(x),0(y)) |
→ |
1(+(x,y)) |
(6) |
|
+(#,x) |
→ |
x |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(62) |
|
+#(j(x),1(y)) |
→ |
+#(x,y) |
(39) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(59) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(36) |
|
+#(j(x),j(y)) |
→ |
+#(+(x,y),j(#)) |
(56) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(55) |
|
+#(j(x),0(y)) |
→ |
+#(x,y) |
(52) |
|
+#(j(x),j(y)) |
→ |
+#(x,y) |
(48) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(46) |
|
+#(1(x),j(y)) |
→ |
+#(x,y) |
(43) |
|
+#(0(x),j(y)) |
→ |
+#(x,y) |
(42) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 1
component.