Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex5_Zan97_C)
The rewrite relation of the following TRS is considered.
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(10) |
proper(if(X1,X2,X3)) |
→ |
if(proper(X1),proper(X2),proper(X3)) |
(11) |
proper(c) |
→ |
ok(c) |
(12) |
proper(true) |
→ |
ok(true) |
(13) |
proper(false) |
→ |
ok(false) |
(14) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
top(mark(X)) |
→ |
top(proper(X)) |
(17) |
top(ok(X)) |
→ |
top(active(X)) |
(18) |
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#(if(X1,X2,X3)) |
→ |
active#(X2) |
(19) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(20) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
proper#(f(X)) |
→ |
f#(proper(X)) |
(22) |
active#(if(X1,X2,X3)) |
→ |
if#(active(X1),X2,X3) |
(23) |
top#(ok(X)) |
→ |
active#(X) |
(24) |
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(25) |
top#(mark(X)) |
→ |
proper#(X) |
(26) |
active#(f(X)) |
→ |
f#(active(X)) |
(27) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(28) |
proper#(f(X)) |
→ |
proper#(X) |
(29) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(30) |
top#(ok(X)) |
→ |
top#(active(X)) |
(31) |
active#(f(X)) |
→ |
f#(true) |
(32) |
f#(ok(X)) |
→ |
f#(X) |
(33) |
active#(f(X)) |
→ |
active#(X) |
(34) |
active#(f(X)) |
→ |
if#(X,c,f(true)) |
(35) |
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(36) |
active#(if(X1,X2,X3)) |
→ |
if#(X1,active(X2),X3) |
(37) |
f#(mark(X)) |
→ |
f#(X) |
(38) |
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(39) |
proper#(if(X1,X2,X3)) |
→ |
if#(proper(X1),proper(X2),proper(X3)) |
(40) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(41) |
1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
top#(ok(X)) |
→ |
top#(active(X)) |
(31) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(top#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(active) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(false) |
= |
4 |
|
status(false) |
= |
[] |
|
list-extension(false) |
= |
Lex |
prec(c) |
= |
3 |
|
status(c) |
= |
[] |
|
list-extension(c) |
= |
Lex |
prec(true) |
= |
2 |
|
status(true) |
= |
[] |
|
list-extension(true) |
= |
Lex |
prec(f) |
= |
5 |
|
status(f) |
= |
[1] |
|
list-extension(f) |
= |
Lex |
prec(if) |
= |
4 |
|
status(if) |
= |
[2, 1] |
|
list-extension(if) |
= |
Lex |
prec(mark) |
= |
1 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(f#) |
= |
0 |
|
status(f#) |
= |
[] |
|
list-extension(f#) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(if#) |
= |
0 |
|
status(if#) |
= |
[2, 1] |
|
list-extension(if#) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
and the following
Max-polynomial interpretation
[top(x1)] |
=
|
1 |
[false] |
=
|
21656 |
[c] |
=
|
0 |
[true] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[if(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 0 |
[mark(x1)] |
=
|
x1 + 0 |
[f#(x1)] |
=
|
1 |
[proper#(x1)] |
=
|
1 |
[if#(x1, x2, x3)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
proper(f(X)) |
→ |
f(proper(X)) |
(10) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
proper(false) |
→ |
ok(false) |
(14) |
proper(c) |
→ |
ok(c) |
(12) |
proper(if(X1,X2,X3)) |
→ |
if(proper(X1),proper(X2),proper(X3)) |
(11) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
proper(true) |
→ |
ok(true) |
(13) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
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#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(20) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(30) |
proper#(f(X)) |
→ |
proper#(X) |
(29) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(28) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[false] |
=
|
32742 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
0 |
[true] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[if(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 0 |
[mark(x1)] |
=
|
0 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 49509 |
[if#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
proper#(f(X)) |
→ |
proper#(X) |
(29) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(25) |
active#(f(X)) |
→ |
active#(X) |
(34) |
active#(if(X1,X2,X3)) |
→ |
active#(X2) |
(19) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[false] |
=
|
1 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
0 |
[true] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[if(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[mark(x1)] |
=
|
0 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 56247 |
[if#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(25) |
active#(f(X)) |
→ |
active#(X) |
(34) |
active#(if(X1,X2,X3)) |
→ |
active#(X2) |
(19) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(41) |
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(39) |
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(36) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[false] |
=
|
240 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
0 |
[true] |
=
|
0 |
[f(x1)] |
=
|
1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[if(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[mark(x1)] |
=
|
2 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[if#(x1, x2, x3)] |
=
|
x3 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(39) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
f#(mark(X)) |
→ |
f#(X) |
(38) |
f#(ok(X)) |
→ |
f#(X) |
(33) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[false] |
=
|
1 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
0 |
[true] |
=
|
0 |
[f(x1)] |
=
|
5969 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 17525 |
[if(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 3578 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3579 |
[if#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(mark(X)) |
→ |
f#(X) |
(38) |
f#(ok(X)) |
→ |
f#(X) |
(33) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.