We consider the following Problem:
Strict Trs:
{ f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, trunc(s(s(x))) -> s(s(trunc(x)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, trunc(s(s(x))) -> s(s(trunc(x)))
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()
, gt(s(u), s(v)) -> gt(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1, 2}, Uargs(gt) = {}, Uargs(trunc) = {},
Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [1]
[0 0] [0 0] [1 0] [1]
true() = [0]
[0]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
trunc(x1) = [0 0] x1 + [3]
[0 0] [1]
s(x1) = [1 0] x1 + [0]
[0 1] [2]
0() = [0]
[0]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(s(s(x))) -> s(s(trunc(x)))
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs:
{ trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1, 2}, Uargs(gt) = {}, Uargs(trunc) = {},
Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [1 1] x1 + [1 0] x2 + [0 2] x3 + [0]
[0 0] [0 0] [0 0] [1]
true() = [1]
[3]
gt(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 1] [1]
trunc(x1) = [0 0] x1 + [0]
[0 0] [2]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
0() = [0]
[2]
false() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ trunc(s(s(x))) -> s(s(trunc(x)))
, gt(s(u), s(v)) -> gt(u, v)}
Weak Trs:
{ f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {gt(s(u), s(v)) -> gt(u, v)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(f) = {1, 2}, Uargs(gt) = {}, Uargs(trunc) = {},
Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 0] x3 + [3]
[0 0] [0 0] [0 0] [1]
true() = [2]
[0]
gt(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [1]
trunc(x1) = [0 0] x1 + [0]
[0 0] [1]
s(x1) = [1 0] x1 + [2]
[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: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak DPs:
{ gt^#(s(u), s(v)) -> gt^#(u, v)
, f^#(true(), x, y) -> f^#(gt(x, y), trunc(x), s(y))
, trunc^#(0()) -> c_4()
, trunc^#(s(0())) -> c_5()
, gt^#(0(), v) -> c_6()
, gt^#(s(u), 0()) -> c_7()}
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs:
{ gt^#(s(u), s(v)) -> gt^#(u, v)
, f^#(true(), x, y) -> f^#(gt(x, y), trunc(x), s(y))
, trunc^#(0()) -> c_4()
, trunc^#(s(0())) -> c_5()
, gt^#(0(), v) -> c_6()
, gt^#(s(u), 0()) -> c_7()}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Usable Rules:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs:
{ gt^#(s(u), s(v)) -> gt^#(u, v)
, f^#(true(), x, y) -> f^#(gt(x, y), trunc(x), s(y))
, trunc^#(0()) -> c_4()
, trunc^#(s(0())) -> c_5()
, gt^#(0(), v) -> c_6()
, gt^#(s(u), 0()) -> c_7()}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs:
{ gt^#(s(u), s(v)) -> gt^#(u, v)
, f^#(true(), x, y) -> f^#(gt(x, y), trunc(x), s(y))
, trunc^#(0()) -> c_4()
, trunc^#(s(0())) -> c_5()
, gt^#(0(), v) -> c_6()
, gt^#(s(u), 0()) -> c_7()}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
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)) ]
->2:{2} [ subsumed ]
|
|->3:{6} [ YES(O(1),O(1)) ]
|
`->4:{7} [ YES(O(1),O(1)) ]
->1:{3} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: trunc^#(s(s(x))) -> trunc^#(x)}
WeakDPs DPs:
{ 2: gt^#(s(u), s(v)) -> gt^#(u, v)
, 3: f^#(true(), x, y) -> f^#(gt(x, y), trunc(x), s(y))
, 4: trunc^#(0()) -> c_4()
, 5: trunc^#(s(0())) -> c_5()
, 6: gt^#(0(), v) -> c_6()
, 7: gt^#(s(u), 0()) -> c_7()}
* Path 5:{1}: YES(?,O(n^1))
-------------------------
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {trunc^#(s(s(x))) -> trunc^#(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(5) -> 5
, trunc^#_0(5) -> 8
, trunc^#_1(5) -> 8}
* Path 5:{1}->6:{4}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(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: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {trunc^#(s(s(x))) -> trunc^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 2:{2}: subsumed
--------------------
This path is subsumed by the proof of paths 2:{2}->4:{7},
2:{2}->3:{6}.
* Path 2:{2}->3:{6}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 2:{2}->4:{7}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {gt^#(s(u), s(v)) -> gt^#(u, v)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{3}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {trunc(s(s(x))) -> s(s(trunc(x)))}
Weak Trs:
{ gt(s(u), s(v)) -> gt(u, v)
, trunc(0()) -> 0()
, trunc(s(0())) -> 0()
, gt(0(), v) -> false()
, gt(s(u), 0()) -> true()}
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))