* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__plus(x,y)} =
activate(n__plus(x,y)) ->^+ plus(activate(x),activate(y))
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ 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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [0]
p(U21) = [1] x2 + [1] x3 + [0]
p(activate) = [1] x1 + [9]
p(and) = [1] x1 + [1] x2 + [7]
p(isNat) = [1] x1 + [0]
p(n__0) = [9]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [3]
p(s) = [1] x1 + [0]
p(tt) = [0]
Following rules are strictly oriented:
activate(X) = [1] X + [9]
> [1] X + [0]
= X
activate(n__0()) = [18]
> [0]
= 0()
activate(n__isNat(X)) = [1] X + [9]
> [1] X + [0]
= isNat(X)
isNat(n__0()) = [9]
> [0]
= tt()
plus(X1,X2) = [1] X1 + [1] X2 + [3]
> [1] X1 + [1] X2 + [1]
= n__plus(X1,X2)
Following rules are (at-least) weakly oriented:
0() = [0]
>= [9]
= n__0()
U11(tt(),N) = [1] N + [0]
>= [1] N + [9]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [21]
= s(plus(activate(N),activate(M)))
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [10]
>= [1] X1 + [1] X2 + [21]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [9]
>= [1] X + [9]
= s(activate(X))
and(tt(),X) = [1] X + [7]
>= [1] X + [9]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [1]
>= [1] V1 + [1] V2 + [25]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [9]
= isNat(activate(V1))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
s(X) -> n__s(X)
- Weak TRS:
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
isNat(n__0()) -> tt()
plus(X1,X2) -> n__plus(X1,X2)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ 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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [1] x1 + [9] x2 + [9]
p(U21) = [9] x2 + [3] x3 + [2]
p(activate) = [1] x1 + [2]
p(and) = [1] x1 + [1] x2 + [2]
p(isNat) = [1] x1 + [2]
p(n__0) = [9]
p(n__isNat) = [1] x1 + [14]
p(n__plus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [10]
p(plus) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [2]
p(tt) = [8]
Following rules are strictly oriented:
U11(tt(),N) = [9] N + [17]
> [1] N + [2]
= activate(N)
activate(n__s(X)) = [1] X + [12]
> [1] X + [4]
= s(activate(X))
and(tt(),X) = [1] X + [10]
> [1] X + [2]
= activate(X)
isNat(n__s(V1)) = [1] V1 + [12]
> [1] V1 + [4]
= isNat(activate(V1))
Following rules are (at-least) weakly oriented:
0() = [1]
>= [9]
= n__0()
U21(tt(),M,N) = [9] M + [3] N + [2]
>= [1] M + [1] N + [7]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__0()) = [11]
>= [1]
= 0()
activate(n__isNat(X)) = [1] X + [16]
>= [1] X + [2]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [5]
= plus(activate(X1),activate(X2))
isNat(X) = [1] X + [2]
>= [1] X + [14]
= n__isNat(X)
isNat(n__0()) = [11]
>= [8]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [22]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [10]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(X) -> n__isNat(X)
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
s(X) -> n__s(X)
- Weak TRS:
U11(tt(),N) -> activate(N)
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ 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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [8] x2 + [0]
p(U21) = [1] x2 + [8] x3 + [9]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [4]
p(isNat) = [1] x1 + [0]
p(n__0) = [3]
p(n__isNat) = [1] x1 + [4]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [3]
p(s) = [1] x1 + [0]
p(tt) = [1]
Following rules are strictly oriented:
U21(tt(),M,N) = [1] M + [8] N + [9]
> [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [3]
= n__0()
U11(tt(),N) = [8] N + [0]
>= [1] N + [0]
= activate(N)
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [3]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [4]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [3]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [0]
= s(activate(X))
and(tt(),X) = [1] X + [5]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [4]
= n__isNat(X)
isNat(n__0()) = [3]
>= [1]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [8]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [1]
>= [1] V1 + [0]
= isNat(activate(V1))
plus(X1,X2) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [1]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(X) -> n__isNat(X)
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
s(X) -> n__s(X)
- Weak TRS:
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ 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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [4] x2 + [2]
p(U21) = [1] x1 + [1] x2 + [3] x3 + [1]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [3]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [0]
p(tt) = [0]
Following rules are strictly oriented:
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [1]
> [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [3]
= n__0()
U11(tt(),N) = [4] N + [2]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [3] N + [1]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [3]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
and(tt(),X) = [1] X + [0]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [3]
>= [0]
= tt()
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= isNat(activate(V1))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(X) -> n__isNat(X)
s(X) -> n__s(X)
- Weak TRS:
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ 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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [3]
p(U11) = [5] x2 + [4]
p(U21) = [5] x1 + [1] x2 + [4] x3 + [0]
p(activate) = [1] x1 + [2]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [1]
p(n__0) = [2]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [7]
p(n__s) = [1] x1 + [4]
p(plus) = [1] x1 + [1] x2 + [7]
p(s) = [1] x1 + [0]
p(tt) = [3]
Following rules are strictly oriented:
0() = [3]
> [2]
= n__0()
isNat(X) = [1] X + [1]
> [1] X + [0]
= n__isNat(X)
Following rules are (at-least) weakly oriented:
U11(tt(),N) = [5] N + [4]
>= [1] N + [2]
= activate(N)
U21(tt(),M,N) = [1] M + [4] N + [15]
>= [1] M + [1] N + [11]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__0()) = [4]
>= [3]
= 0()
activate(n__isNat(X)) = [1] X + [2]
>= [1] X + [1]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [9]
>= [1] X1 + [1] X2 + [11]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [6]
>= [1] X + [2]
= s(activate(X))
and(tt(),X) = [1] X + [3]
>= [1] X + [2]
= activate(X)
isNat(n__0()) = [3]
>= [3]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8]
>= [1] V1 + [1] V2 + [5]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [5]
>= [1] V1 + [3]
= isNat(activate(V1))
plus(X1,X2) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [4]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,U11,U21,activate,and,isNat,plus,s}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(U11) = [0 0] x_1 + [2 1] x_2 + [5]
[1 0] [0 1] [0]
p(U21) = [0 6] x_1 + [4 3] x_2 + [1 7] x_3 + [0]
[1 0] [0 4] [4 4] [4]
p(activate) = [1 1] x_1 + [0]
[0 1] [0]
p(and) = [1 4] x_1 + [1 1] x_2 + [2]
[0 0] [0 1] [0]
p(isNat) = [1 4] x_1 + [0]
[0 0] [2]
p(n__0) = [0]
[0]
p(n__isNat) = [1 4] x_1 + [0]
[0 0] [2]
p(n__plus) = [1 4] x_1 + [1 1] x_2 + [4]
[0 1] [0 1] [2]
p(n__s) = [1 1] x_1 + [0]
[0 1] [0]
p(plus) = [1 4] x_1 + [1 1] x_2 + [5]
[0 1] [0 1] [2]
p(s) = [1 1] x_1 + [0]
[0 1] [0]
p(tt) = [0]
[2]
Following rules are strictly oriented:
activate(n__plus(X1,X2)) = [1 5] X1 + [1 2] X2 + [6]
[0 1] [0 1] [2]
> [1 5] X1 + [1 2] X2 + [5]
[0 1] [0 1] [2]
= plus(activate(X1),activate(X2))
Following rules are (at-least) weakly oriented:
0() = [0]
[0]
>= [0]
[0]
= n__0()
U11(tt(),N) = [2 1] N + [5]
[0 1] [0]
>= [1 1] N + [0]
[0 1] [0]
= activate(N)
U21(tt(),M,N) = [4 3] M + [1 7] N + [12]
[0 4] [4 4] [4]
>= [1 3] M + [1 6] N + [7]
[0 1] [0 1] [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1 1] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [0]
[0]
>= [0]
[0]
= 0()
activate(n__isNat(X)) = [1 4] X + [2]
[0 0] [2]
>= [1 4] X + [0]
[0 0] [2]
= isNat(X)
activate(n__s(X)) = [1 2] X + [0]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= s(activate(X))
and(tt(),X) = [1 1] X + [10]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= activate(X)
isNat(X) = [1 4] X + [0]
[0 0] [2]
>= [1 4] X + [0]
[0 0] [2]
= n__isNat(X)
isNat(n__0()) = [0]
[2]
>= [0]
[2]
= tt()
isNat(n__plus(V1,V2)) = [1 8] V1 + [1 5] V2 + [12]
[0 0] [0 0] [2]
>= [1 5] V1 + [1 5] V2 + [12]
[0 0] [0 0] [2]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) = [1 5] V1 + [0]
[0 0] [2]
>= [1 5] V1 + [0]
[0 0] [2]
= isNat(activate(V1))
plus(X1,X2) = [1 4] X1 + [1 1] X2 + [5]
[0 1] [0 1] [2]
>= [1 4] X1 + [1 1] X2 + [4]
[0 1] [0 1] [2]
= n__plus(X1,X2)
s(X) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= n__s(X)
** Step 1.b:8: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,U11,U21,activate,and,isNat,plus,s}
TcT has computed the following interpretation:
p(0) = [6]
[1]
p(U11) = [2 1] x_1 + [1 2] x_2 + [1]
[0 4] [4 1] [3]
p(U21) = [4 0] x_1 + [4 6] x_2 + [4 4] x_3 + [6]
[7 0] [0 4] [0 4] [0]
p(activate) = [1 1] x_1 + [0]
[0 1] [0]
p(and) = [1 4] x_1 + [1 4] x_2 + [3]
[0 1] [0 1] [0]
p(isNat) = [1 0] x_1 + [0]
[0 0] [0]
p(n__0) = [5]
[1]
p(n__isNat) = [1 0] x_1 + [0]
[0 0] [0]
p(n__plus) = [1 1] x_1 + [1 1] x_2 + [3]
[0 1] [0 1] [0]
p(n__s) = [1 2] x_1 + [0]
[0 1] [1]
p(plus) = [1 1] x_1 + [1 1] x_2 + [3]
[0 1] [0 1] [0]
p(s) = [1 2] x_1 + [1]
[0 1] [1]
p(tt) = [2]
[0]
Following rules are strictly oriented:
s(X) = [1 2] X + [1]
[0 1] [1]
> [1 2] X + [0]
[0 1] [1]
= n__s(X)
Following rules are (at-least) weakly oriented:
0() = [6]
[1]
>= [5]
[1]
= n__0()
U11(tt(),N) = [1 2] N + [5]
[4 1] [3]
>= [1 1] N + [0]
[0 1] [0]
= activate(N)
U21(tt(),M,N) = [4 6] M + [4 4] N + [14]
[0 4] [0 4] [14]
>= [1 4] M + [1 4] N + [4]
[0 1] [0 1] [1]
= s(plus(activate(N),activate(M)))
activate(X) = [1 1] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [6]
[1]
>= [6]
[1]
= 0()
activate(n__isNat(X)) = [1 0] X + [0]
[0 0] [0]
>= [1 0] X + [0]
[0 0] [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1 2] X1 + [1 2] X2 + [3]
[0 1] [0 1] [0]
>= [1 2] X1 + [1 2] X2 + [3]
[0 1] [0 1] [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1 3] X + [1]
[0 1] [1]
>= [1 3] X + [1]
[0 1] [1]
= s(activate(X))
and(tt(),X) = [1 4] X + [5]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= activate(X)
isNat(X) = [1 0] X + [0]
[0 0] [0]
>= [1 0] X + [0]
[0 0] [0]
= n__isNat(X)
isNat(n__0()) = [5]
[0]
>= [2]
[0]
= tt()
isNat(n__plus(V1,V2)) = [1 1] V1 + [1 1] V2 + [3]
[0 0] [0 0] [0]
>= [1 1] V1 + [1 1] V2 + [3]
[0 0] [0 0] [0]
= and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) = [1 2] V1 + [0]
[0 0] [0]
>= [1 1] V1 + [0]
[0 0] [0]
= isNat(activate(V1))
plus(X1,X2) = [1 1] X1 + [1 1] X2 + [3]
[0 1] [0 1] [0]
>= [1 1] X1 + [1 1] X2 + [3]
[0 1] [0 1] [0]
= n__plus(X1,X2)
** Step 1.b:9: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/2,U21/3,activate/1,and/2,isNat/1,plus/2,s/1} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,activate,and,isNat,plus
,s} and constructors {n__0,n__isNat,n__plus,n__s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))