Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/PEANO_nosorts_C)
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) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(12) |
proper(tt) |
→ |
ok(tt) |
(13) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(14) |
proper(0) |
→ |
ok(0) |
(15) |
proper(s(X)) |
→ |
s(proper(X)) |
(16) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
top(mark(X)) |
→ |
top(proper(X)) |
(20) |
top(ok(X)) |
→ |
top(active(X)) |
(21) |
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.
active#(plus(X1,X2)) |
→ |
active#(X2) |
(22) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(23) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(24) |
active#(s(X)) |
→ |
s#(active(X)) |
(25) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(26) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(27) |
active#(plus(N,s(M))) |
→ |
s#(plus(N,M)) |
(28) |
proper#(s(X)) |
→ |
proper#(X) |
(29) |
top#(mark(X)) |
→ |
proper#(X) |
(30) |
active#(s(X)) |
→ |
active#(X) |
(31) |
active#(plus(N,s(M))) |
→ |
plus#(N,M) |
(32) |
top#(ok(X)) |
→ |
top#(active(X)) |
(33) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(34) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(35) |
proper#(and(X1,X2)) |
→ |
and#(proper(X1),proper(X2)) |
(36) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(37) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(38) |
active#(and(X1,X2)) |
→ |
and#(active(X1),X2) |
(39) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(40) |
s#(ok(X)) |
→ |
s#(X) |
(41) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(42) |
top#(ok(X)) |
→ |
active#(X) |
(43) |
active#(plus(X1,X2)) |
→ |
plus#(active(X1),X2) |
(44) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(45) |
active#(plus(X1,X2)) |
→ |
plus#(X1,active(X2)) |
(46) |
proper#(plus(X1,X2)) |
→ |
plus#(proper(X1),proper(X2)) |
(47) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(48) |
s#(mark(X)) |
→ |
s#(X) |
(49) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(50) |
1.1 Dependency Graph Processor
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(27) |
top#(ok(X)) |
→ |
top#(active(X)) |
(33) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(top#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(s#) |
= |
1 |
π(active) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(s) |
= |
4 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(and) |
= |
5 |
|
status(and) |
= |
[1, 2] |
|
list-extension(and) |
= |
Lex |
prec(plus#) |
= |
0 |
|
status(plus#) |
= |
[] |
|
list-extension(plus#) |
= |
Lex |
prec(0) |
= |
2 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(mark) |
= |
4 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(plus) |
= |
5 |
|
status(plus) |
= |
[1, 2] |
|
list-extension(plus) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
prec(tt) |
= |
3 |
|
status(tt) |
= |
[] |
|
list-extension(tt) |
= |
Lex |
prec(and#) |
= |
0 |
|
status(and#) |
= |
[1, 2] |
|
list-extension(and#) |
= |
Lex |
and the following
Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[plus#(x1, x2)] |
=
|
max(0) |
[0] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
0 |
[and#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
proper(0) |
→ |
ok(0) |
(15) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
proper(s(X)) |
→ |
s(proper(X)) |
(16) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(14) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(12) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
proper(tt) |
→ |
ok(tt) |
(13) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(27) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
proper#(s(X)) |
→ |
proper#(X) |
(29) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(26) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(24) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(35) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(34) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
12457 |
[proper#(x1)] |
=
|
x1 + 0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
x1 + 36229 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
proper(0) |
→ |
ok(0) |
(15) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
proper(tt) |
→ |
ok(tt) |
(13) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(s(X)) |
→ |
proper#(X) |
(29) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(26) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(24) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(35) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(34) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
active#(s(X)) |
→ |
active#(X) |
(31) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(40) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(37) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(22) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
47422 |
[ok(x1)] |
=
|
x1 + 47421 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 6879 |
[active(x1)] |
=
|
x1 + 59821 |
[active#(x1)] |
=
|
x1 + 0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
proper(0) |
→ |
ok(0) |
(15) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
proper(tt) |
→ |
ok(tt) |
(13) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(s(X)) |
→ |
active#(X) |
(31) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(40) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(37) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(22) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(49) |
s#(ok(X)) |
→ |
s#(X) |
(41) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 37894 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
proper(0) |
→ |
ok(0) |
(15) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
proper(tt) |
→ |
ok(tt) |
(13) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(mark(X)) |
→ |
s#(X) |
(49) |
s#(ok(X)) |
→ |
s#(X) |
(41) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(48) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(45) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(23) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 23217 |
[plus#(x1, x2)] |
=
|
x2 + 0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 0 |
[active(x1)] |
=
|
x1 + 41065 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
proper(0) |
→ |
ok(0) |
(15) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
proper(tt) |
→ |
ok(tt) |
(13) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(48) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(23) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
6th
component contains the
pair
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(50) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(42) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 0 |
[active(x1)] |
=
|
x1 + 56247 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(18) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(4) |
proper(0) |
→ |
ok(0) |
(15) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(17) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(5) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(9) |
proper(tt) |
→ |
ok(tt) |
(13) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(50) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(42) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.