Certification Problem
Input (TPDB TRS_Standard/AProVE_04/Liveness8)
The rewrite relation of the following TRS is considered.
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
top(active(c)) |
→ |
top(mark(c)) |
(2) |
|
top(mark(x)) |
→ |
top(check(x)) |
(3) |
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
top(found(x)) |
→ |
top(active(x)) |
(13) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
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.
|
match#(f(x),f(y)) |
→ |
f#(match(x,y)) |
(16) |
|
proper#(f(x)) |
→ |
f#(proper(x)) |
(17) |
|
check#(x) |
→ |
start#(match(f(X),x)) |
(18) |
|
active#(f(x)) |
→ |
f#(active(x)) |
(19) |
|
match#(X,x) |
→ |
proper#(x) |
(20) |
|
f#(ok(x)) |
→ |
f#(x) |
(21) |
|
check#(x) |
→ |
match#(f(X),x) |
(22) |
|
check#(f(x)) |
→ |
check#(x) |
(23) |
|
proper#(f(x)) |
→ |
proper#(x) |
(24) |
|
active#(f(x)) |
→ |
active#(x) |
(25) |
|
top#(found(x)) |
→ |
active#(x) |
(26) |
|
top#(mark(x)) |
→ |
check#(x) |
(27) |
|
check#(f(x)) |
→ |
f#(check(x)) |
(28) |
|
top#(mark(x)) |
→ |
top#(check(x)) |
(29) |
|
match#(f(x),f(y)) |
→ |
match#(x,y) |
(30) |
|
top#(active(c)) |
→ |
top#(mark(c)) |
(31) |
|
f#(found(x)) |
→ |
f#(x) |
(32) |
|
check#(x) |
→ |
f#(X) |
(33) |
|
top#(found(x)) |
→ |
top#(active(x)) |
(34) |
|
f#(mark(x)) |
→ |
f#(x) |
(35) |
1.1 Dependency Graph Processor
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
|
top#(found(x)) |
→ |
top#(active(x)) |
(34) |
|
top#(active(c)) |
→ |
top#(mark(c)) |
(31) |
|
top#(mark(x)) |
→ |
top#(check(x)) |
(29) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [top(x1)] |
=
|
0 |
| [check#(x1)] |
=
|
0 |
| [top#(x1)] |
=
|
x1 + 0 |
| [c] |
=
|
52869 |
| [f(x1)] |
=
|
52868 |
| [check(x1)] |
=
|
52868 |
| [proper(x1)] |
=
|
52869 |
| [ok(x1)] |
=
|
x1 + 0 |
| [match#(x1, x2)] |
=
|
0 |
| [mark(x1)] |
=
|
52868 |
| [f#(x1)] |
=
|
0 |
| [found(x1)] |
=
|
x1 + 0 |
| [proper#(x1)] |
=
|
0 |
| [match(x1, x2)] |
=
|
x1 + 0 |
| [start#(x1)] |
=
|
0 |
| [active(x1)] |
=
|
x1 + 0 |
| [active#(x1)] |
=
|
0 |
| [start(x1)] |
=
|
x1 + 0 |
| [X] |
=
|
52869 |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
top#(active(c)) |
→ |
top#(mark(c)) |
(31) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
|
active#(f(x)) |
→ |
active#(x) |
(25) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [top(x1)] |
=
|
0 |
| [check#(x1)] |
=
|
0 |
| [top#(x1)] |
=
|
x1 + 0 |
| [c] |
=
|
1 |
| [f(x1)] |
=
|
x1 + 1 |
| [check(x1)] |
=
|
x1 + 35287 |
| [proper(x1)] |
=
|
x1 + 1 |
| [ok(x1)] |
=
|
x1 + 0 |
| [match#(x1, x2)] |
=
|
0 |
| [mark(x1)] |
=
|
x1 + 28224 |
| [f#(x1)] |
=
|
0 |
| [found(x1)] |
=
|
x1 + 7126 |
| [proper#(x1)] |
=
|
0 |
| [match(x1, x2)] |
=
|
x2 + 1 |
| [start#(x1)] |
=
|
0 |
| [active(x1)] |
=
|
x1 + 61131 |
| [active#(x1)] |
=
|
x1 + 0 |
| [start(x1)] |
=
|
x1 + 13459 |
| [X] |
=
|
24766 |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
active#(f(x)) |
→ |
active#(x) |
(25) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
check#(f(x)) |
→ |
check#(x) |
(23) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [top(x1)] |
=
|
0 |
| [check#(x1)] |
=
|
x1 + 0 |
| [top#(x1)] |
=
|
x1 + 0 |
| [c] |
=
|
1 |
| [f(x1)] |
=
|
x1 + 1 |
| [check(x1)] |
=
|
x1 + 35287 |
| [proper(x1)] |
=
|
x1 + 1 |
| [ok(x1)] |
=
|
x1 + 0 |
| [match#(x1, x2)] |
=
|
0 |
| [mark(x1)] |
=
|
x1 + 28224 |
| [f#(x1)] |
=
|
0 |
| [found(x1)] |
=
|
x1 + 7126 |
| [proper#(x1)] |
=
|
0 |
| [match(x1, x2)] |
=
|
x2 + 1 |
| [start#(x1)] |
=
|
0 |
| [active(x1)] |
=
|
x1 + 61131 |
| [active#(x1)] |
=
|
0 |
| [start(x1)] |
=
|
x1 + 13459 |
| [X] |
=
|
38690 |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
check#(f(x)) |
→ |
check#(x) |
(23) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
|
match#(f(x),f(y)) |
→ |
match#(x,y) |
(30) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [top(x1)] |
=
|
0 |
| [check#(x1)] |
=
|
0 |
| [top#(x1)] |
=
|
x1 + 0 |
| [c] |
=
|
1 |
| [f(x1)] |
=
|
x1 + 1 |
| [check(x1)] |
=
|
x1 + 35287 |
| [proper(x1)] |
=
|
x1 + 1 |
| [ok(x1)] |
=
|
x1 + 0 |
| [match#(x1, x2)] |
=
|
x1 + 0 |
| [mark(x1)] |
=
|
x1 + 12330 |
| [f#(x1)] |
=
|
0 |
| [found(x1)] |
=
|
x1 + 5806 |
| [proper#(x1)] |
=
|
0 |
| [match(x1, x2)] |
=
|
x2 + 1 |
| [start#(x1)] |
=
|
0 |
| [active(x1)] |
=
|
x1 + 26720 |
| [active#(x1)] |
=
|
0 |
| [start(x1)] |
=
|
x1 + 13459 |
| [X] |
=
|
18826 |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
match#(f(x),f(y)) |
→ |
match#(x,y) |
(30) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
|
proper#(f(x)) |
→ |
proper#(x) |
(24) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [top(x1)] |
=
|
0 |
| [check#(x1)] |
=
|
0 |
| [top#(x1)] |
=
|
x1 + 0 |
| [c] |
=
|
1 |
| [f(x1)] |
=
|
x1 + 1 |
| [check(x1)] |
=
|
x1 + 2 |
| [proper(x1)] |
=
|
x1 + 1 |
| [ok(x1)] |
=
|
x1 + 0 |
| [match#(x1, x2)] |
=
|
0 |
| [mark(x1)] |
=
|
x1 + 1136 |
| [f#(x1)] |
=
|
0 |
| [found(x1)] |
=
|
x1 + 1 |
| [proper#(x1)] |
=
|
x1 + 0 |
| [match(x1, x2)] |
=
|
x2 + 1 |
| [start#(x1)] |
=
|
0 |
| [active(x1)] |
=
|
x1 + 26720 |
| [active#(x1)] |
=
|
0 |
| [start(x1)] |
=
|
x1 + 1 |
| [X] |
=
|
18826 |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
proper#(f(x)) |
→ |
proper#(x) |
(24) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
|
f#(mark(x)) |
→ |
f#(x) |
(35) |
|
f#(ok(x)) |
→ |
f#(x) |
(21) |
|
f#(found(x)) |
→ |
f#(x) |
(32) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [top(x1)] |
=
|
0 |
| [check#(x1)] |
=
|
0 |
| [top#(x1)] |
=
|
x1 + 0 |
| [c] |
=
|
1 |
| [f(x1)] |
=
|
x1 + 1 |
| [check(x1)] |
=
|
x1 + 2 |
| [proper(x1)] |
=
|
x1 + 1 |
| [ok(x1)] |
=
|
x1 + 0 |
| [match#(x1, x2)] |
=
|
0 |
| [mark(x1)] |
=
|
x1 + 1 |
| [f#(x1)] |
=
|
x1 + 0 |
| [found(x1)] |
=
|
x1 + 1 |
| [proper#(x1)] |
=
|
0 |
| [match(x1, x2)] |
=
|
x2 + 1 |
| [start#(x1)] |
=
|
0 |
| [active(x1)] |
=
|
x1 + 20752 |
| [active#(x1)] |
=
|
0 |
| [start(x1)] |
=
|
x1 + 1 |
| [X] |
=
|
56247 |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
f#(mark(x)) |
→ |
f#(x) |
(35) |
|
f#(found(x)) |
→ |
f#(x) |
(32) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 1
component.