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