We consider the following Problem:
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)
, g(x, c(y)) -> c(g(x, y))
, g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)
, g(x, c(y)) -> c(g(x, y))
, g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1, 2}, Uargs(g) = {2},
Uargs(c) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
true() = [3]
[2]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [1]
[0 0] [2]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[1 1] [1 0] [0 1] [1]
g(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 1] [1 1] [1]
c(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(0()) -> true()
, f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))
, g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))}
Weak Trs:
{ f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(0()) -> true()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1, 2}, Uargs(g) = {2},
Uargs(c) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [1]
[1 0] [1]
0() = [0]
[0]
true() = [0]
[0]
1() = [3]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [0]
[0 0] [1 1] [1 1] [0]
g(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
c(x1) = [1 0] x1 + [1]
[0 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))
, g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))}
Weak Trs:
{ f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1, 2}, Uargs(g) = {2},
Uargs(c) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
true() = [0]
[0]
1() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
if(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
g(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
c(x1) = [1 0] x1 + [0]
[0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs:
{ f^#(s(x)) -> f^#(x)
, g^#(x, c(y)) -> g^#(x, y)}
Weak DPs:
{ g^#(x, c(y)) -> g^#(x, if(f(x), c(g(s(x), y)), c(y)))
, f^#(0()) -> c_4()
, f^#(1()) -> c_5()
, if^#(true(), s(x), s(y)) -> c_6()
, if^#(false(), s(x), s(y)) -> c_7()}
We consider the following Problem:
Strict DPs:
{ f^#(s(x)) -> f^#(x)
, g^#(x, c(y)) -> g^#(x, y)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs:
{ g^#(x, c(y)) -> g^#(x, if(f(x), c(g(s(x), y)), c(y)))
, f^#(0()) -> c_4()
, f^#(1()) -> c_5()
, if^#(true(), s(x), s(y)) -> c_6()
, if^#(false(), s(x), s(y)) -> c_7()}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs:
{ f^#(s(x)) -> f^#(x)
, g^#(x, c(y)) -> g^#(x, y)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs:
{ g^#(x, c(y)) -> g^#(x, if(f(x), c(g(s(x), y)), c(y)))
, f^#(0()) -> c_4()
, f^#(1()) -> c_5()
, if^#(true(), s(x), s(y)) -> c_6()
, if^#(false(), s(x), s(y)) -> c_7()}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->5:{1} [ YES(?,O(n^1)) ]
|
|->6:{4} [ YES(O(1),O(1)) ]
|
`->7:{5} [ YES(O(1),O(1)) ]
->3:{2} [ YES(?,O(n^1)) ]
|
`->4:{3} [ YES(O(1),O(1)) ]
->2:{6} [ YES(O(1),O(1)) ]
->1:{7} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{ 1: f^#(s(x)) -> f^#(x)
, 2: g^#(x, c(y)) -> g^#(x, y)}
WeakDPs DPs:
{ 3: g^#(x, c(y)) -> g^#(x, if(f(x), c(g(s(x), y)), c(y)))
, 4: f^#(0()) -> c_4()
, 5: f^#(1()) -> c_5()
, 6: if^#(true(), s(x), s(y)) -> c_6()
, 7: if^#(false(), s(x), s(y)) -> c_7()}
* Path 5:{1}: YES(?,O(n^1))
-------------------------
We consider the following Problem:
Strict DPs: {f^#(s(x)) -> f^#(x)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {f^#(s(x)) -> f^#(x)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {f^#(s(x)) -> f^#(x)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {f^#(s(x)) -> f^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ s_0(2) -> 2
, f^#_0(2) -> 1
, f^#_1(2) -> 1}
* Path 5:{1}->6:{4}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {f^#(s(x)) -> f^#(x)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {f^#(s(x)) -> f^#(x)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {f^#(s(x)) -> f^#(x)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {f^#(s(x)) -> f^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{1}->7:{5}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {f^#(s(x)) -> f^#(x)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {f^#(s(x)) -> f^#(x)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {f^#(s(x)) -> f^#(x)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {f^#(s(x)) -> f^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 3:{2}: YES(?,O(n^1))
-------------------------
We consider the following Problem:
Strict DPs: {g^#(x, c(y)) -> g^#(x, y)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {g^#(x, c(y)) -> g^#(x, y)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {g^#(x, c(y)) -> g^#(x, y)}
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {g^#(x, c(y)) -> g^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, g^#_0(2, 2) -> 1
, g^#_1(2, 2) -> 1}
* Path 3:{2}->4:{3}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {g^#(x, c(y)) -> g^#(x, y)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {g^#(x, c(y)) -> g^#(x, y)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak DPs: {g^#(x, c(y)) -> g^#(x, y)}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {g^#(x, c(y)) -> g^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 2:{6}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{7}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(s(x)) -> f(x)
, g(x, c(y)) -> c(g(x, y))}
Weak Trs:
{ g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
, f(0()) -> true()
, f(1()) -> false()
, if(true(), s(x), s(y)) -> s(x)
, if(false(), s(x), s(y)) -> s(y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))