We consider the following Problem:
Strict Trs:
{ min(0(), y) -> 0()
, min(x, 0()) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, twice(0()) -> 0()
, twice(s(x)) -> s(s(twice(x)))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, p(s(x)) -> x
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ min(0(), y) -> 0()
, min(x, 0()) -> 0()
, min(s(x), s(y)) -> s(min(x, y))
, max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, twice(0()) -> 0()
, twice(s(x)) -> s(s(twice(x)))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, p(s(x)) -> x
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(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:
{ min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(twice) = {1}, Uargs(-) = {1, 2}, Uargs(p) = {1},
Uargs(f) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
min(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] [1]
max(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
twice(x1) = [1 0] x1 + [1]
[0 0] [1]
-(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
p(x1) = [1 0] x1 + [1]
[0 0] [1]
f(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, max(0(), y) -> y
, max(x, 0()) -> x
, max(s(x), s(y)) -> s(max(x, y))
, twice(s(x)) -> s(s(twice(x)))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, p(s(x)) -> x
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))}
Weak Trs:
{ min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {max(x, 0()) -> x}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(twice) = {1}, Uargs(-) = {1, 2}, Uargs(p) = {1},
Uargs(f) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [0]
0() = [1]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
max(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
twice(x1) = [1 0] x1 + [0]
[0 0] [0]
-(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(x1) = [1 0] x1 + [0]
[0 0] [0]
f(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, twice(s(x)) -> s(s(twice(x)))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, p(s(x)) -> x
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))}
Weak Trs:
{ max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(twice) = {1}, Uargs(-) = {1, 2}, Uargs(p) = {1},
Uargs(f) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[1 0] [1 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [2]
max(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 1] [0 1] [1]
twice(x1) = [1 0] x1 + [0]
[0 1] [0]
-(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [2]
p(x1) = [1 0] x1 + [0]
[0 0] [3]
f(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, twice(s(x)) -> s(s(twice(x)))
, p(s(x)) -> x
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))}
Weak Trs:
{ max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {p(s(x)) -> x}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(twice) = {1}, Uargs(-) = {1, 2}, Uargs(p) = {1},
Uargs(f) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [1 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
max(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
twice(x1) = [1 0] x1 + [0]
[0 0] [1]
-(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [1]
p(x1) = [1 0] x1 + [2]
[0 1] [0]
f(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, twice(s(x)) -> s(s(twice(x)))
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))}
Weak Trs:
{ p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(twice) = {1}, Uargs(-) = {1, 2}, Uargs(p) = {1},
Uargs(f) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
twice(x1) = [1 2] x1 + [0]
[0 0] [0]
-(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
p(x1) = [1 0] x1 + [1]
[0 1] [0]
f(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {min(s(x), s(y)) -> s(min(x, y))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
Uargs(twice) = {1}, Uargs(-) = {1, 2}, Uargs(p) = {1},
Uargs(f) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
min(x1, x2) = [0 0] x1 + [0 1] x2 + [0]
[0 0] [0 1] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [1]
max(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
twice(x1) = [1 0] x1 + [0]
[0 0] [0]
-(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [1]
p(x1) = [1 0] x1 + [0]
[0 1] [0]
f(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Weak DPs:
{ min^#(s(x), s(y)) -> min^#(x, y)
, f^#(s(x), s(y)) ->
f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p^#(s(x)) -> c_4()
, max^#(0(), y) -> c_5()
, max^#(s(x), s(y)) -> max^#(x, y)
, -^#(x, 0()) -> c_7()
, -^#(s(x), s(y)) -> -^#(x, y)
, max^#(x, 0()) -> c_9()
, min^#(0(), y) -> c_10()
, min^#(x, 0()) -> c_11()
, twice^#(0()) -> c_12()}
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs:
{ min^#(s(x), s(y)) -> min^#(x, y)
, f^#(s(x), s(y)) ->
f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p^#(s(x)) -> c_4()
, max^#(0(), y) -> c_5()
, max^#(s(x), s(y)) -> max^#(x, y)
, -^#(x, 0()) -> c_7()
, -^#(s(x), s(y)) -> -^#(x, y)
, max^#(x, 0()) -> c_9()
, min^#(0(), y) -> c_10()
, min^#(x, 0()) -> c_11()
, twice^#(0()) -> c_12()}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, f(s(x), s(y)) ->
f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules: {twice(s(x)) -> s(s(twice(x)))}
Weak Usable Rules:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs:
{ min^#(s(x), s(y)) -> min^#(x, y)
, f^#(s(x), s(y)) ->
f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p^#(s(x)) -> c_4()
, max^#(0(), y) -> c_5()
, max^#(s(x), s(y)) -> max^#(x, y)
, -^#(x, 0()) -> c_7()
, -^#(s(x), s(y)) -> -^#(x, y)
, max^#(x, 0()) -> c_9()
, min^#(0(), y) -> c_10()
, min^#(x, 0()) -> c_11()
, twice^#(0()) -> c_12()}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs:
{ min^#(s(x), s(y)) -> min^#(x, y)
, f^#(s(x), s(y)) ->
f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, p^#(s(x)) -> c_4()
, max^#(0(), y) -> c_5()
, max^#(s(x), s(y)) -> max^#(x, y)
, -^#(x, 0()) -> c_7()
, -^#(s(x), s(y)) -> -^#(x, y)
, max^#(x, 0()) -> c_9()
, min^#(0(), y) -> c_10()
, min^#(x, 0()) -> c_11()
, twice^#(0()) -> c_12()}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->11:{1} [ YES(?,O(n^1)) ]
|
`->12:{12} [ YES(O(1),O(1)) ]
->8:{2} [ subsumed ]
|
|->9:{10} [ YES(O(1),O(1)) ]
|
`->10:{11} [ YES(O(1),O(1)) ]
->7:{3} [ YES(O(1),O(1)) ]
->6:{4} [ YES(O(1),O(1)) ]
->3:{6} [ subsumed ]
|
|->5:{5} [ YES(O(1),O(1)) ]
|
`->4:{9} [ YES(O(1),O(1)) ]
->1:{8} [ subsumed ]
|
`->2:{7} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: twice^#(s(x)) -> twice^#(x)}
WeakDPs DPs:
{ 2: min^#(s(x), s(y)) -> min^#(x, y)
, 3: f^#(s(x), s(y)) ->
f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))
, 4: p^#(s(x)) -> c_4()
, 5: max^#(0(), y) -> c_5()
, 6: max^#(s(x), s(y)) -> max^#(x, y)
, 7: -^#(x, 0()) -> c_7()
, 8: -^#(s(x), s(y)) -> -^#(x, y)
, 9: max^#(x, 0()) -> c_9()
, 10: min^#(0(), y) -> c_10()
, 11: min^#(x, 0()) -> c_11()
, 12: twice^#(0()) -> c_12()}
* Path 11:{1}: YES(?,O(n^1))
--------------------------
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(x)}
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {twice^#(s(x)) -> twice^#(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
, twice^#_0(2) -> 1
, twice^#_1(2) -> 1}
* Path 11:{1}->12:{12}: YES(O(1),O(1))
------------------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {twice^#(s(x)) -> twice^#(x)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {twice^#(s(x)) -> twice^#(x)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {twice^#(s(x)) -> twice^#(x)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {twice^#(s(x)) -> twice^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 8:{2}: subsumed
--------------------
This path is subsumed by the proof of paths 8:{2}->10:{11},
8:{2}->9:{10}.
* Path 8:{2}->9:{10}: YES(O(1),O(1))
----------------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {min^#(s(x), s(y)) -> min^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {min^#(s(x), s(y)) -> min^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {min^#(s(x), s(y)) -> min^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
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(x), s(y)) -> min^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 8:{2}->10:{11}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {min^#(s(x), s(y)) -> min^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {min^#(s(x), s(y)) -> min^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {min^#(s(x), s(y)) -> min^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
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(x), s(y)) -> min^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 7:{3}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
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 6:{4}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
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 3:{6}: subsumed
--------------------
This path is subsumed by the proof of paths 3:{6}->5:{5},
3:{6}->4:{9}.
* Path 3:{6}->5:{5}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 3:{6}->4:{9}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {max^#(s(x), s(y)) -> max^#(x, y)}
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}->2:{7}.
* Path 1:{8}->2:{7}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {-^#(s(x), s(y)) -> -^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {-^#(s(x), s(y)) -> -^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Strict Trs: {twice(s(x)) -> s(s(twice(x)))}
Weak DPs: {-^#(s(x), s(y)) -> -^#(x, y)}
Weak Trs:
{ min(s(x), s(y)) -> s(min(x, y))
, p(s(x)) -> x
, max(0(), y) -> y
, max(s(x), s(y)) -> s(max(x, y))
, -(x, 0()) -> x
, -(s(x), s(y)) -> -(x, y)
, max(x, 0()) -> x
, min(0(), y) -> 0()
, min(x, 0()) -> 0()
, twice(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
Weak DPs: {-^#(s(x), s(y)) -> -^#(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))