The rewrite relation of the following TRS is considered.
active#(minus(0,Y)) |
→ |
mark#(0) |
(38) |
active#(minus(s(X),s(Y))) |
→ |
mark#(minus(X,Y)) |
(39) |
active#(minus(s(X),s(Y))) |
→ |
minus#(X,Y) |
(40) |
active#(geq(X,0)) |
→ |
mark#(true) |
(41) |
active#(geq(0,s(Y))) |
→ |
mark#(false) |
(42) |
active#(geq(s(X),s(Y))) |
→ |
mark#(geq(X,Y)) |
(43) |
active#(geq(s(X),s(Y))) |
→ |
geq#(X,Y) |
(44) |
active#(div(0,s(Y))) |
→ |
mark#(0) |
(45) |
active#(div(s(X),s(Y))) |
→ |
mark#(if(geq(X,Y),s(div(minus(X,Y),s(Y))),0)) |
(46) |
active#(div(s(X),s(Y))) |
→ |
if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0) |
(47) |
active#(div(s(X),s(Y))) |
→ |
geq#(X,Y) |
(48) |
active#(div(s(X),s(Y))) |
→ |
s#(div(minus(X,Y),s(Y))) |
(49) |
active#(div(s(X),s(Y))) |
→ |
div#(minus(X,Y),s(Y)) |
(50) |
active#(div(s(X),s(Y))) |
→ |
minus#(X,Y) |
(51) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(52) |
active#(if(false,X,Y)) |
→ |
mark#(Y) |
(53) |
mark#(minus(X1,X2)) |
→ |
active#(minus(X1,X2)) |
(54) |
mark#(0) |
→ |
active#(0) |
(55) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(56) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(57) |
mark#(s(X)) |
→ |
mark#(X) |
(58) |
mark#(geq(X1,X2)) |
→ |
active#(geq(X1,X2)) |
(59) |
mark#(true) |
→ |
active#(true) |
(60) |
mark#(false) |
→ |
active#(false) |
(61) |
mark#(div(X1,X2)) |
→ |
active#(div(mark(X1),X2)) |
(62) |
mark#(div(X1,X2)) |
→ |
div#(mark(X1),X2) |
(63) |
mark#(div(X1,X2)) |
→ |
mark#(X1) |
(64) |
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),X2,X3)) |
(65) |
mark#(if(X1,X2,X3)) |
→ |
if#(mark(X1),X2,X3) |
(66) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(67) |
minus#(mark(X1),X2) |
→ |
minus#(X1,X2) |
(68) |
minus#(X1,mark(X2)) |
→ |
minus#(X1,X2) |
(69) |
minus#(active(X1),X2) |
→ |
minus#(X1,X2) |
(70) |
minus#(X1,active(X2)) |
→ |
minus#(X1,X2) |
(71) |
s#(mark(X)) |
→ |
s#(X) |
(72) |
s#(active(X)) |
→ |
s#(X) |
(73) |
geq#(mark(X1),X2) |
→ |
geq#(X1,X2) |
(74) |
geq#(X1,mark(X2)) |
→ |
geq#(X1,X2) |
(75) |
geq#(active(X1),X2) |
→ |
geq#(X1,X2) |
(76) |
geq#(X1,active(X2)) |
→ |
geq#(X1,X2) |
(77) |
div#(mark(X1),X2) |
→ |
div#(X1,X2) |
(78) |
div#(X1,mark(X2)) |
→ |
div#(X1,X2) |
(79) |
div#(active(X1),X2) |
→ |
div#(X1,X2) |
(80) |
div#(X1,active(X2)) |
→ |
div#(X1,X2) |
(81) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(82) |
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(83) |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(84) |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(85) |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(86) |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(87) |
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
active#(minus(s(X),s(Y))) |
→ |
mark#(minus(X,Y)) |
(39) |
mark#(minus(X1,X2)) |
→ |
active#(minus(X1,X2)) |
(54) |
active#(geq(s(X),s(Y))) |
→ |
mark#(geq(X,Y)) |
(43) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(56) |
active#(div(s(X),s(Y))) |
→ |
mark#(if(geq(X,Y),s(div(minus(X,Y),s(Y))),0)) |
(46) |
mark#(s(X)) |
→ |
mark#(X) |
(58) |
mark#(geq(X1,X2)) |
→ |
active#(geq(X1,X2)) |
(59) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(52) |
mark#(div(X1,X2)) |
→ |
active#(div(mark(X1),X2)) |
(62) |
active#(if(false,X,Y)) |
→ |
mark#(Y) |
(53) |
mark#(div(X1,X2)) |
→ |
mark#(X1) |
(64) |
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),X2,X3)) |
(65) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(67) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 · x1
|
[minus(x1, x2)] |
= |
1 |
[s(x1)] |
= |
0 |
[mark#(x1)] |
= |
1 |
[geq(x1, x2)] |
= |
1 |
[mark(x1)] |
= |
0 |
[div(x1, x2)] |
= |
1 |
[if(x1, x2, x3)] |
= |
1 |
[0] |
= |
0 |
[true] |
= |
0 |
[false] |
= |
0 |
[active(x1)] |
= |
0 |
together with the usable
rules
minus(X1,mark(X2)) |
→ |
minus(X1,X2) |
(19) |
minus(mark(X1),X2) |
→ |
minus(X1,X2) |
(18) |
minus(active(X1),X2) |
→ |
minus(X1,X2) |
(20) |
minus(X1,active(X2)) |
→ |
minus(X1,X2) |
(21) |
geq(X1,mark(X2)) |
→ |
geq(X1,X2) |
(25) |
geq(mark(X1),X2) |
→ |
geq(X1,X2) |
(24) |
geq(active(X1),X2) |
→ |
geq(X1,X2) |
(26) |
geq(X1,active(X2)) |
→ |
geq(X1,X2) |
(27) |
s(active(X)) |
→ |
s(X) |
(23) |
s(mark(X)) |
→ |
s(X) |
(22) |
div(X1,mark(X2)) |
→ |
div(X1,X2) |
(29) |
div(mark(X1),X2) |
→ |
div(X1,X2) |
(28) |
div(active(X1),X2) |
→ |
div(X1,X2) |
(30) |
div(X1,active(X2)) |
→ |
div(X1,X2) |
(31) |
if(X1,mark(X2),X3) |
→ |
if(X1,X2,X3) |
(33) |
if(mark(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(32) |
if(X1,X2,mark(X3)) |
→ |
if(X1,X2,X3) |
(34) |
if(active(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(35) |
if(X1,active(X2),X3) |
→ |
if(X1,X2,X3) |
(36) |
if(X1,X2,active(X3)) |
→ |
if(X1,X2,X3) |
(37) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(56) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 · x1
|
[minus(x1, x2)] |
= |
0 |
[s(x1)] |
= |
1 + 1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[geq(x1, x2)] |
= |
0 |
[div(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[0] |
= |
0 |
[true] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[false] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
the
pair
mark#(s(X)) |
→ |
mark#(X) |
(58) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 · x1
|
[minus(x1, x2)] |
= |
0 |
[s(x1)] |
= |
0 |
[mark#(x1)] |
= |
1 · x1
|
[geq(x1, x2)] |
= |
0 |
[div(x1, x2)] |
= |
1 + 1 · x1
|
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[0] |
= |
0 |
[true] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[false] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
the
pairs
active#(div(s(X),s(Y))) |
→ |
mark#(if(geq(X,Y),s(div(minus(X,Y),s(Y))),0)) |
(46) |
mark#(div(X1,X2)) |
→ |
mark#(X1) |
(64) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 · x1
|
[minus(x1, x2)] |
= |
0 |
[s(x1)] |
= |
0 |
[mark#(x1)] |
= |
1 · x1
|
[geq(x1, x2)] |
= |
1 |
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[true] |
= |
1 |
[div(x1, x2)] |
= |
1 |
[mark(x1)] |
= |
1 · x1
|
[false] |
= |
1 |
[active(x1)] |
= |
1 · x1
|
[0] |
= |
0 |
the
pairs
active#(if(true,X,Y)) |
→ |
mark#(X) |
(52) |
active#(if(false,X,Y)) |
→ |
mark#(Y) |
(53) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 |
[mark#(x1)] |
= |
1 + x1
|
[minus(x1, x2)] |
= |
0 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
1 + 2 · x1
|
[geq(x1, x2)] |
= |
0 |
[div(x1, x2)] |
= |
1 + 2 · x2
|
[if(x1, x2, x3)] |
= |
2 + x1 + 2 · x2 + 2 · x3
|
[s(x1)] |
= |
1 |
[0] |
= |
2 |
[true] |
= |
2 |
[false] |
= |
1 |
together with the usable
rules
minus(X1,mark(X2)) |
→ |
minus(X1,X2) |
(19) |
minus(mark(X1),X2) |
→ |
minus(X1,X2) |
(18) |
minus(active(X1),X2) |
→ |
minus(X1,X2) |
(20) |
minus(X1,active(X2)) |
→ |
minus(X1,X2) |
(21) |
geq(X1,mark(X2)) |
→ |
geq(X1,X2) |
(25) |
geq(mark(X1),X2) |
→ |
geq(X1,X2) |
(24) |
geq(active(X1),X2) |
→ |
geq(X1,X2) |
(26) |
geq(X1,active(X2)) |
→ |
geq(X1,X2) |
(27) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(div(X1,X2)) |
→ |
active#(div(mark(X1),X2)) |
(62) |
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),X2,X3)) |
(65) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(67) |
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[geq(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[minus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
geq(X1,mark(X2)) |
→ |
geq(X1,X2) |
(25) |
geq(mark(X1),X2) |
→ |
geq(X1,X2) |
(24) |
geq(active(X1),X2) |
→ |
geq(X1,X2) |
(26) |
geq(X1,active(X2)) |
→ |
geq(X1,X2) |
(27) |
minus(X1,mark(X2)) |
→ |
minus(X1,X2) |
(19) |
minus(mark(X1),X2) |
→ |
minus(X1,X2) |
(18) |
minus(active(X1),X2) |
→ |
minus(X1,X2) |
(20) |
minus(X1,active(X2)) |
→ |
minus(X1,X2) |
(21) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[geq(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 · x1
|
[minus(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[active#(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
2 + 2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
together with the usable
rules
geq(mark(X1),X2) |
→ |
geq(X1,X2) |
(24) |
geq(X1,mark(X2)) |
→ |
geq(X1,X2) |
(25) |
geq(active(X1),X2) |
→ |
geq(X1,X2) |
(26) |
geq(X1,active(X2)) |
→ |
geq(X1,X2) |
(27) |
minus(mark(X1),X2) |
→ |
minus(X1,X2) |
(18) |
minus(X1,mark(X2)) |
→ |
minus(X1,X2) |
(19) |
minus(active(X1),X2) |
→ |
minus(X1,X2) |
(20) |
minus(X1,active(X2)) |
→ |
minus(X1,X2) |
(21) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(minus(s(X),s(Y))) |
→ |
mark#(minus(X,Y)) |
(39) |
active#(geq(s(X),s(Y))) |
→ |
mark#(geq(X,Y)) |
(43) |
mark#(geq(X1,X2)) |
→ |
active#(geq(X1,X2)) |
(59) |
and
the
rules
geq(mark(X1),X2) |
→ |
geq(X1,X2) |
(24) |
geq(X1,mark(X2)) |
→ |
geq(X1,X2) |
(25) |
geq(active(X1),X2) |
→ |
geq(X1,X2) |
(26) |
geq(X1,active(X2)) |
→ |
geq(X1,X2) |
(27) |
minus(mark(X1),X2) |
→ |
minus(X1,X2) |
(18) |
minus(X1,mark(X2)) |
→ |
minus(X1,X2) |
(19) |
minus(active(X1),X2) |
→ |
minus(X1,X2) |
(20) |
minus(X1,active(X2)) |
→ |
minus(X1,X2) |
(21) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(minus(X1,X2)) |
→ |
active#(minus(X1,X2)) |
(54) |
|
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
minus#(X1,mark(X2)) |
→ |
minus#(X1,X2) |
(69) |
minus#(mark(X1),X2) |
→ |
minus#(X1,X2) |
(68) |
minus#(active(X1),X2) |
→ |
minus#(X1,X2) |
(70) |
minus#(X1,active(X2)) |
→ |
minus#(X1,X2) |
(71) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[minus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
minus#(X1,mark(X2)) |
→ |
minus#(X1,X2) |
(69) |
|
1 |
≥ |
1 |
2 |
> |
2 |
minus#(mark(X1),X2) |
→ |
minus#(X1,X2) |
(68) |
|
1 |
> |
1 |
2 |
≥ |
2 |
minus#(active(X1),X2) |
→ |
minus#(X1,X2) |
(70) |
|
1 |
> |
1 |
2 |
≥ |
2 |
minus#(X1,active(X2)) |
→ |
minus#(X1,X2) |
(71) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(73) |
s#(mark(X)) |
→ |
s#(X) |
(72) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(73) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(72) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
geq#(X1,mark(X2)) |
→ |
geq#(X1,X2) |
(75) |
geq#(mark(X1),X2) |
→ |
geq#(X1,X2) |
(74) |
geq#(active(X1),X2) |
→ |
geq#(X1,X2) |
(76) |
geq#(X1,active(X2)) |
→ |
geq#(X1,X2) |
(77) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[geq#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
geq#(X1,mark(X2)) |
→ |
geq#(X1,X2) |
(75) |
|
1 |
≥ |
1 |
2 |
> |
2 |
geq#(mark(X1),X2) |
→ |
geq#(X1,X2) |
(74) |
|
1 |
> |
1 |
2 |
≥ |
2 |
geq#(active(X1),X2) |
→ |
geq#(X1,X2) |
(76) |
|
1 |
> |
1 |
2 |
≥ |
2 |
geq#(X1,active(X2)) |
→ |
geq#(X1,X2) |
(77) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
div#(X1,mark(X2)) |
→ |
div#(X1,X2) |
(79) |
div#(mark(X1),X2) |
→ |
div#(X1,X2) |
(78) |
div#(active(X1),X2) |
→ |
div#(X1,X2) |
(80) |
div#(X1,active(X2)) |
→ |
div#(X1,X2) |
(81) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[div#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
div#(X1,mark(X2)) |
→ |
div#(X1,X2) |
(79) |
|
1 |
≥ |
1 |
2 |
> |
2 |
div#(mark(X1),X2) |
→ |
div#(X1,X2) |
(78) |
|
1 |
> |
1 |
2 |
≥ |
2 |
div#(active(X1),X2) |
→ |
div#(X1,X2) |
(80) |
|
1 |
> |
1 |
2 |
≥ |
2 |
div#(X1,active(X2)) |
→ |
div#(X1,X2) |
(81) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(83) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(82) |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(84) |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(85) |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(86) |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(87) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[if#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(83) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(82) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(84) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(85) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(86) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(87) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.