MAYBE
* Step 1: WeightGap MAYBE
+ Considered Problem:
- Strict TRS:
diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
leq(0(),Y) -> true()
leq(s(X),0()) -> false()
leq(s(X),s(Y)) -> leq(X,Y)
p(0()) -> 0()
p(s(X)) -> X
- Signature:
{diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(diff) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(diff) = [1] x1 + [0]
p(false) = [0]
p(if) = [1] x1 + [2] x2 + [1] x3 + [0]
p(leq) = [0]
p(p) = [1] x1 + [9]
p(s) = [1] x1 + [12]
p(true) = [0]
Following rules are strictly oriented:
p(0()) = [9]
> [0]
= 0()
p(s(X)) = [1] X + [21]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
diff(X,Y) = [1] X + [0]
>= [1] X + [21]
= if(leq(X,Y),0(),s(diff(p(X),Y)))
if(false(),X,Y) = [2] X + [1] Y + [0]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [2] X + [1] Y + [0]
>= [1] X + [0]
= X
leq(0(),Y) = [0]
>= [0]
= true()
leq(s(X),0()) = [0]
>= [0]
= false()
leq(s(X),s(Y)) = [0]
>= [0]
= leq(X,Y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap MAYBE
+ Considered Problem:
- Strict TRS:
diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
leq(0(),Y) -> true()
leq(s(X),0()) -> false()
leq(s(X),s(Y)) -> leq(X,Y)
- Weak TRS:
p(0()) -> 0()
p(s(X)) -> X
- Signature:
{diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(diff) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(diff) = [1] x1 + [4] x2 + [0]
p(false) = [5]
p(if) = [1] x1 + [4] x2 + [1] x3 + [0]
p(leq) = [3]
p(p) = [1] x1 + [8]
p(s) = [1] x1 + [8]
p(true) = [0]
Following rules are strictly oriented:
if(false(),X,Y) = [4] X + [1] Y + [5]
> [1] Y + [0]
= Y
leq(0(),Y) = [3]
> [0]
= true()
Following rules are (at-least) weakly oriented:
diff(X,Y) = [1] X + [4] Y + [0]
>= [1] X + [4] Y + [19]
= if(leq(X,Y),0(),s(diff(p(X),Y)))
if(true(),X,Y) = [4] X + [1] Y + [0]
>= [1] X + [0]
= X
leq(s(X),0()) = [3]
>= [5]
= false()
leq(s(X),s(Y)) = [3]
>= [3]
= leq(X,Y)
p(0()) = [8]
>= [0]
= 0()
p(s(X)) = [1] X + [16]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap MAYBE
+ Considered Problem:
- Strict TRS:
diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
if(true(),X,Y) -> X
leq(s(X),0()) -> false()
leq(s(X),s(Y)) -> leq(X,Y)
- Weak TRS:
if(false(),X,Y) -> Y
leq(0(),Y) -> true()
p(0()) -> 0()
p(s(X)) -> X
- Signature:
{diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(diff) = {1},
uargs(if) = {1,3},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(diff) = [1] x1 + [1] x2 + [2]
p(false) = [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(leq) = [4]
p(p) = [1] x1 + [8]
p(s) = [1] x1 + [0]
p(true) = [3]
Following rules are strictly oriented:
if(true(),X,Y) = [1] X + [1] Y + [3]
> [1] X + [0]
= X
leq(s(X),0()) = [4]
> [1]
= false()
Following rules are (at-least) weakly oriented:
diff(X,Y) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [16]
= if(leq(X,Y),0(),s(diff(p(X),Y)))
if(false(),X,Y) = [1] X + [1] Y + [1]
>= [1] Y + [0]
= Y
leq(0(),Y) = [4]
>= [3]
= true()
leq(s(X),s(Y)) = [4]
>= [4]
= leq(X,Y)
p(0()) = [10]
>= [2]
= 0()
p(s(X)) = [1] X + [8]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: Failure MAYBE
+ Considered Problem:
- Strict TRS:
diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y)))
leq(s(X),s(Y)) -> leq(X,Y)
- Weak TRS:
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
leq(0(),Y) -> true()
leq(s(X),0()) -> false()
p(0()) -> 0()
p(s(X)) -> X
- Signature:
{diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE