We consider the following Problem:
Strict Trs:
{ minus(x, y) -> cond(equal(min(x, y), y), x, y)
, cond(true(), x, y) -> s(minus(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ minus(x, y) -> cond(equal(min(x, y), y), x, y)
, cond(true(), x, y) -> s(minus(x, s(y)))
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {cond(true(), x, y) -> s(minus(x, s(y)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {1}, Uargs(equal) = {1},
Uargs(min) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 1] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
[0 0] [0 0] [0 0] [1]
equal(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
true() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ minus(x, y) -> cond(equal(min(x, y), y), x, y)
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()
, equal(s(x), 0()) -> false()
, equal(0(), s(y)) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
Weak Trs: {cond(true(), x, y) -> s(minus(x, 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:
{ equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {1}, Uargs(equal) = {1},
Uargs(min) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 1] x1 + [1 2] x2 + [0]
[0 0] [0 0] [0]
cond(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
[0 0] [0 0] [0 0] [1]
equal(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
true() = [3]
[0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ minus(x, y) -> cond(equal(min(x, y), y), x, y)
, min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()
, equal(0(), s(y)) -> false()}
Weak Trs:
{ equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)
, cond(true(), x, y) -> s(minus(x, 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: {minus(x, y) -> cond(equal(min(x, y), y), x, y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {1}, Uargs(equal) = {1},
Uargs(min) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 1] x1 + [1 2] x2 + [2]
[0 0] [0 0] [2]
cond(x1, x2, x3) = [1 3] x1 + [1 1] x2 + [1 1] x3 + [1]
[0 0] [0 0] [0 0] [1]
equal(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
true() = [2]
[2]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()
, equal(0(), s(y)) -> false()}
Weak Trs:
{ minus(x, y) -> cond(equal(min(x, y), y), x, y)
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)
, cond(true(), x, y) -> s(minus(x, 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:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(minus) = {}, Uargs(cond) = {1}, Uargs(equal) = {1},
Uargs(min) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [2]
cond(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
[0 0] [0 0] [0 0] [1]
equal(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
min(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [1 0] [1]
true() = [3]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, minus(x, y) -> cond(equal(min(x, y), y), x, y)
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)
, cond(true(), x, y) -> s(minus(x, s(y)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, minus(x, y) -> cond(equal(min(x, y), y), x, y)
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)
, cond(true(), x, y) -> s(minus(x, s(y)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs:
{ min^#(s(u), s(v)) -> min^#(u, v)
, equal^#(0(), 0()) -> c_2()}
Weak DPs:
{ min^#(0(), v) -> c_3()
, min^#(u, 0()) -> c_4()
, equal^#(0(), s(y)) -> c_5()
, minus^#(x, y) -> cond^#(equal(min(x, y), y), x, y)
, equal^#(s(x), 0()) -> c_7()
, equal^#(s(x), s(y)) -> equal^#(x, y)
, cond^#(true(), x, y) -> minus^#(x, s(y))}
We consider the following Problem:
Strict DPs:
{ min^#(s(u), s(v)) -> min^#(u, v)
, equal^#(0(), 0()) -> c_2()}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs:
{ min^#(0(), v) -> c_3()
, min^#(u, 0()) -> c_4()
, equal^#(0(), s(y)) -> c_5()
, minus^#(x, y) -> cond^#(equal(min(x, y), y), x, y)
, equal^#(s(x), 0()) -> c_7()
, equal^#(s(x), s(y)) -> equal^#(x, y)
, cond^#(true(), x, y) -> minus^#(x, s(y))}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, minus(x, y) -> cond(equal(min(x, y), y), x, y)
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)
, cond(true(), x, y) -> s(minus(x, s(y)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Usable Rules:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
We consider the following Problem:
Strict DPs:
{ min^#(s(u), s(v)) -> min^#(u, v)
, equal^#(0(), 0()) -> c_2()}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs:
{ min^#(0(), v) -> c_3()
, min^#(u, 0()) -> c_4()
, equal^#(0(), s(y)) -> c_5()
, minus^#(x, y) -> cond^#(equal(min(x, y), y), x, y)
, equal^#(s(x), 0()) -> c_7()
, equal^#(s(x), s(y)) -> equal^#(x, y)
, cond^#(true(), x, y) -> minus^#(x, s(y))}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs:
{ min^#(s(u), s(v)) -> min^#(u, v)
, equal^#(0(), 0()) -> c_2()}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs:
{ min^#(0(), v) -> c_3()
, min^#(u, 0()) -> c_4()
, equal^#(0(), s(y)) -> c_5()
, minus^#(x, y) -> cond^#(equal(min(x, y), y), x, y)
, equal^#(s(x), 0()) -> c_7()
, equal^#(s(x), s(y)) -> equal^#(x, y)
, cond^#(true(), x, y) -> minus^#(x, s(y))}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->6:{1} [ YES(?,O(n^1)) ]
|
|->7:{3} [ YES(O(1),O(1)) ]
|
`->8:{4} [ YES(O(1),O(1)) ]
->3:{6,9} [ YES(O(1),O(1)) ]
->1:{8} [ subsumed ]
|
|->5:{2} [ YES(?,O(n^1)) ]
|
|->4:{5} [ YES(O(1),O(1)) ]
|
`->2:{7} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{ 1: min^#(s(u), s(v)) -> min^#(u, v)
, 2: equal^#(0(), 0()) -> c_2()}
WeakDPs DPs:
{ 3: min^#(0(), v) -> c_3()
, 4: min^#(u, 0()) -> c_4()
, 5: equal^#(0(), s(y)) -> c_5()
, 6: minus^#(x, y) -> cond^#(equal(min(x, y), y), x, y)
, 7: equal^#(s(x), 0()) -> c_7()
, 8: equal^#(s(x), s(y)) -> equal^#(x, y)
, 9: cond^#(true(), x, y) -> minus^#(x, s(y))}
* Path 6:{1}: YES(?,O(n^1))
-------------------------
We consider the following Problem:
Strict DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
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
, min^#_0(2, 2) -> 1
, min^#_1(2, 2) -> 1}
* Path 6:{1}->7:{3}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 6:{1}->8:{4}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {min^#(s(u), s(v)) -> min^#(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 3:{6,9}: YES(O(1),O(1))
----------------------------
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, 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:{8}: subsumed
--------------------
This path is subsumed by the proof of paths 1:{8}->5:{2},
1:{8}->4:{5},
1:{8}->2:{7}.
* Path 1:{8}->5:{2}: YES(?,O(n^1))
--------------------------------
We consider the following Problem:
Strict DPs: {equal^#(0(), 0()) -> c_2()}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {equal^#(0(), 0()) -> c_2()}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {equal^#(0(), 0()) -> c_2()}
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {equal^#(0(), 0()) -> c_2()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(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:
{ s_0(2) -> 2
, 0_0() -> 2
, equal^#_0(2, 2) -> 1
, c_2_1() -> 1}
* Path 1:{8}->4:{5}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{8}->2:{7}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(s(u), s(v)) -> s(min(u, v))
, equal(0(), 0()) -> true()}
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
Weak Trs:
{ min(0(), v) -> 0()
, min(u, 0()) -> 0()
, equal(0(), s(y)) -> false()
, equal(s(x), 0()) -> false()
, equal(s(x), s(y)) -> equal(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {equal^#(s(x), s(y)) -> equal^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))