Tool CaT
Execution Time | Unknown |
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Answer | YES(?,O(n^1)) |
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Input | AG01 3.7 |
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stdout:
YES(?,O(n^1))
Problem:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
Proof:
Complexity Transformation Processor:
strict:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
weak:
Arctic Interpretation Processor:
dimension: 1
interpretation:
[log](x0) = 1x0,
[s](x0) = x0,
[half](x0) = x0,
[0] = 6
orientation:
half(0()) = 6 >= 6 = 0()
half(s(s(x))) = x >= x = s(half(x))
log(s(0())) = 7 >= 6 = 0()
log(s(s(x))) = 1x >= 1x = s(log(s(half(x))))
problem:
strict:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(s(x))) -> s(log(s(half(x))))
weak:
log(s(0())) -> 0()
Arctic Interpretation Processor:
dimension: 1
interpretation:
[log](x0) = x0,
[s](x0) = 1x0,
[half](x0) = x0,
[0] = 2
orientation:
half(0()) = 2 >= 2 = 0()
half(s(s(x))) = 2x >= 1x = s(half(x))
log(s(s(x))) = 2x >= 2x = s(log(s(half(x))))
log(s(0())) = 3 >= 2 = 0()
problem:
strict:
half(0()) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
weak:
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
Arctic Interpretation Processor:
dimension: 2
interpretation:
[0 -&]
[log](x0) = [1 -&]x0,
[0 0]
[s](x0) = [2 0]x0,
[0 -&]
[half](x0) = [0 -&]x0,
[0]
[0] = [0]
orientation:
[0] [0]
half(0()) = [0] >= [0] = 0()
[2 0] [1 -&]
log(s(s(x))) = [3 1]x >= [2 -&]x = s(log(s(half(x))))
[2 0] [0 -&]
half(s(s(x))) = [2 0]x >= [2 -&]x = s(half(x))
[0] [0]
log(s(0())) = [1] >= [0] = 0()
problem:
strict:
half(0()) -> 0()
weak:
log(s(s(x))) -> s(log(s(half(x))))
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
Arctic Interpretation Processor:
dimension: 2
interpretation:
[2 0]
[log](x0) = [1 1]x0,
[0 3]
[s](x0) = [0 2]x0,
[1 3 ]
[half](x0) = [-& 0 ]x0,
[0 ]
[0] = [-&]
orientation:
[1 ] [0 ]
half(0()) = [-&] >= [-&] = 0()
[5 7] [5 7]
log(s(s(x))) = [4 6]x >= [4 6]x = s(log(s(half(x))))
[5 7] [1 3]
half(s(s(x))) = [2 4]x >= [1 3]x = s(half(x))
[2] [0 ]
log(s(0())) = [1] >= [-&] = 0()
problem:
strict:
weak:
half(0()) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
QedTool IRC1
Execution Time | Unknown |
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Answer | YES(?,O(n^1)) |
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Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^1))
Tool IRC2
Execution Time | Unknown |
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Answer | YES(?,O(n^1)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
Proof Output:
'matrix-interpretation of dimension 1' proved the best result:
Details:
--------
'matrix-interpretation of dimension 1' succeeded with the following output:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
Proof Output:
The following argument positions are usable:
Uargs(half) = {}, Uargs(s) = {1}, Uargs(log) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
half(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [2]
log(x1) = [3] x1 + [0]Tool RC1
Execution Time | Unknown |
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Answer | YES(?,O(n^1)) |
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Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^1))
Tool RC2
Execution Time | Unknown |
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Answer | YES(?,O(n^1)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
Proof Output:
'matrix-interpretation of dimension 1' proved the best result:
Details:
--------
'matrix-interpretation of dimension 1' succeeded with the following output:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
Proof Output:
The following argument positions are usable:
Uargs(half) = {1}, Uargs(s) = {1}, Uargs(log) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
half(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [3]
log(x1) = [2] x1 + [0]Tool pair1rc
Execution Time | Unknown |
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Answer | YES(?,O(n^2)) |
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Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
half(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [2]
[0]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
log(x1) = [2 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool pair2rc
Execution Time | Unknown |
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Answer | YES(?,O(n^2)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
half(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [2]
[0]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
log(x1) = [2 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool pair3irc
Execution Time | Unknown |
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Answer | YES(?,O(n^2)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: innermost
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(half) = {}, Uargs(s) = {1}, Uargs(log) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
half(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [0]
[1]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
log(x1) = [2 0 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool pair3rc
Execution Time | Unknown |
---|
Answer | YES(?,O(n^2)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
half(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [2]
[0]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
log(x1) = [2 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool rc
Execution Time | Unknown |
---|
Answer | YES(?,O(n^2)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^2))
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
half(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [2]
[0]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
log(x1) = [2 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
Hurray, we answered YES(?,O(n^2))Tool tup3irc
Execution Time | 5.532ms |
---|
Answer | YES(?,O(n^2)) |
---|
Input | AG01 3.7 |
---|
stdout:
YES(?,O(n^2))
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ half(0()) -> 0()
, half(s(s(x))) -> s(half(x))
, log(s(0())) -> 0()
, log(s(s(x))) -> s(log(s(half(x))))}
StartTerms: basic terms
Strategy: innermost
1) 'Fastest' proved the goal fastest:
'Sequentially' proved the goal fastest:
'Fastest' succeeded:
'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' proved the goal fastest:
The following argument positions are usable:
Uargs(half) = {}, Uargs(s) = {1}, Uargs(log) = {1}
We have the following constructor-restricted (at most 2 in the main diagonals) matrix interpretation:
Interpretation Functions:
half(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [0]
[1]
[2]
s(x1) = [1 1 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
log(x1) = [2 0 0] x1 + [0]
[0 0 2] [1]
[0 0 1] [0]
Hurray, we answered YES(?,O(n^2))