We consider the following Problem:
Strict Trs:
{ a__minus(0(), Y) -> 0()
, a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(0(), s(Y)) -> 0()
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, mark(0()) -> 0()
, mark(s(X)) -> s(mark(X))
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ a__minus(0(), Y) -> 0()
, a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(0(), s(Y)) -> 0()
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, mark(0()) -> 0()
, mark(s(X)) -> s(mark(X))
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [1]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [1]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 0] x1 + [1]
[0 0] [1]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [2]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 0] x1 + [0 1] x2 + [1]
[0 0] [0 0] [1]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [0]
[0 0] [1 0] [0 0] [1]
div(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 1] [1]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [2]
[0 0] [1 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(geq(X1, X2)) -> a__geq(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [3]
[0 0] [0 0] [3]
0() = [3]
[3]
s(x1) = [1 0] x1 + [0]
[0 0] [2]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[1 1] [0 1] [1]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [1]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 0] x1 + [3]
[1 0] [1]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(minus(X1, X2)) -> a__minus(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [1]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 0] x1 + [3]
[0 0] [1]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [1]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 0] x1 + [0]
[0 0] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__geq(s(X), s(Y)) -> a__geq(X, Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
0() = [0]
[2]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
a__geq(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [3]
true() = [1]
[3]
false() = [1]
[3]
a__div(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 1] [0 0] [1]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 1] x3 + [0]
[0 0] [0 0] [0 0] [0]
div(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 1] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [3]
mark(x1) = [1 1] x1 + [0]
[0 0] [3]
geq(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [2]
[0 1] [0 0] [0 1] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(true(), X, Y) -> mark(X)
, a__if(false(), X, Y) -> mark(Y)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__if(false(), X, Y) -> mark(Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [2]
true() = [0]
[0]
false() = [1]
[2]
a__div(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 1] [1]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 2] x3 + [0]
[0 1] [0 0] [0 0] [0]
div(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
mark(x1) = [1 1] x1 + [0]
[0 0] [2]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [2]
if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 2] x3 + [3]
[0 1] [0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(true(), X, Y) -> mark(X)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ a__if(false(), X, Y) -> mark(Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__minus(s(X), s(Y)) -> a__minus(X, Y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [1 1] x1 + [0 0] x2 + [3]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [3]
[0 0] [0 0] [1]
true() = [3]
[1]
false() = [0]
[0]
a__div(x1, x2) = [1 0] x1 + [0 1] x2 + [2]
[0 1] [0 1] [3]
a__if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 1] x3 + [0]
[0 0] [0 0] [0 0] [1]
div(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
[0 1] [0 1] [0]
minus(x1, x2) = [1 1] x1 + [0 0] x2 + [3]
[0 0] [0 0] [0]
mark(x1) = [1 1] x1 + [0]
[0 0] [1]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [3]
[0 0] [0 0] [1]
if(x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1 0] x3 + [2]
[0 1] [0 0] [0 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ a__if(true(), X, Y) -> mark(X)
, mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(false(), X, Y) -> mark(Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__if(true(), X, Y) -> mark(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [2]
true() = [0]
[2]
false() = [0]
[2]
a__div(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
[0 1] [0 0] [3]
a__if(x1, x2, x3) = [1 2] x1 + [0 2] x2 + [0 2] x3 + [0]
[0 0] [0 0] [0 0] [2]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 2] x1 + [2]
[0 0] [2]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 1] [0 1] [0 1] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))
, a__if(X1, X2, X3) -> if(X1, X2, X3)}
Weak Trs:
{ a__if(true(), X, Y) -> mark(X)
, a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(false(), X, Y) -> mark(Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {a__if(X1, X2, X3) -> if(X1, X2, X3)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [3]
a__if(x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [1]
[0 1] [0 1] [0 1] [2]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 1] x1 + [0]
[0 1] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 1] [0 1] [0 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(div(X1, X2)) -> a__div(mark(X1), X2)
, mark(s(X)) -> s(mark(X))}
Weak Trs:
{ a__if(X1, X2, X3) -> if(X1, X2, X3)
, a__if(true(), X, Y) -> mark(X)
, a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(false(), X, Y) -> mark(Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(div(X1, X2)) -> a__div(mark(X1), X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [0]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [2]
a__if(x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [0]
[0 1] [0 1] [0 1] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 1] [0 0] [2]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [0 1] x1 + [0]
[0 1] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(div(X1, X2)) -> a__div(mark(X1), X2)
, a__if(X1, X2, X3) -> if(X1, X2, X3)
, a__if(true(), X, Y) -> mark(X)
, a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(false(), X, Y) -> mark(Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {mark(s(X)) -> s(mark(X))}
Weak Trs:
{ mark(div(X1, X2)) -> a__div(mark(X1), X2)
, a__if(X1, X2, X3) -> if(X1, X2, X3)
, a__if(true(), X, Y) -> mark(X)
, a__minus(s(X), s(Y)) -> a__minus(X, Y)
, a__if(false(), X, Y) -> mark(Y)
, a__geq(s(X), s(Y)) -> a__geq(X, Y)
, a__div(s(X), s(Y)) ->
a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0())
, mark(minus(X1, X2)) -> a__minus(X1, X2)
, mark(geq(X1, X2)) -> a__geq(X1, X2)
, mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
, a__minus(0(), Y) -> 0()
, a__geq(X, 0()) -> true()
, a__geq(0(), s(Y)) -> false()
, a__div(0(), s(Y)) -> 0()
, mark(0()) -> 0()
, mark(true()) -> true()
, mark(false()) -> false()
, a__minus(X1, X2) -> minus(X1, X2)
, a__geq(X1, X2) -> geq(X1, X2)
, a__div(X1, X2) -> div(X1, X2)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The following argument positions are usable:
Uargs(a__minus) = {}, Uargs(s) = {1}, Uargs(a__geq) = {},
Uargs(a__div) = {1}, Uargs(a__if) = {1}, Uargs(div) = {},
Uargs(minus) = {}, Uargs(mark) = {}, Uargs(geq) = {},
Uargs(if) = {}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
a__minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 1] x1 + [0]
[0 0] [1]
a__geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
true() = [0]
[0]
false() = [0]
[0]
a__div(x1, x2) = [1 3] x1 + [0 2] x2 + [0]
[0 0] [0 1] [0]
a__if(x1, x2, x3) = [1 0] x1 + [3 2] x2 + [3 2] x3 + [0]
[0 1] [0 1] [0 1] [0]
div(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 0] [0 1] [0]
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [3 2] x1 + [0]
[0 1] [0]
geq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
Hurray, we answered YES(?,O(n^1))