* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(N,0()) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(N,0()) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),M,N)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(N,0()) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(N,0()) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),M,N)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
x(N,0()) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),M,N)
All above mentioned rules can be savely removed.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x_1 + [8] x_2 + [8]
p(U12) = [1] x_1 + [0]
p(U21) = [1] x_1 + [0]
p(U31) = [1] x_1 + [8] x_2 + [8]
p(U32) = [1] x_1 + [1]
p(U41) = [1] x_2 + [1]
p(U51) = [1] x_1 + [5] x_2 + [10] x_3 + [12]
p(U52) = [1] x_1 + [4] x_2 + [2] x_3 + [1]
p(U61) = [8]
p(U71) = [1] x_1 + [1] x_2 + [11] x_3 + [15]
p(U72) = [1] x_1 + [1] x_2 + [3] x_3 + [8]
p(activate) = [1] x_1 + [0]
p(isNat) = [8] x_1 + [7]
p(n__0) = [0]
p(n__plus) = [1] x_1 + [1] x_2 + [1]
p(n__s) = [1] x_1 + [0]
p(n__x) = [1] x_1 + [1] x_2 + [1]
p(plus) = [1] x_1 + [1] x_2 + [1]
p(s) = [1] x_1 + [0]
p(tt) = [0]
p(x) = [1] x_1 + [1] x_2 + [1]
Following rules are strictly oriented:
U11(tt(),V2) = [8] V2 + [8]
> [8] V2 + [7]
= U12(isNat(activate(V2)))
U32(tt()) = [1]
> [0]
= tt()
U41(tt(),N) = [1] N + [1]
> [1] N + [0]
= activate(N)
U51(tt(),M,N) = [5] M + [10] N + [12]
> [4] M + [10] N + [8]
= U52(isNat(activate(N)),activate(M),activate(N))
U61(tt()) = [8]
> [0]
= 0()
U72(tt(),M,N) = [1] M + [3] N + [8]
> [1] M + [2] N + [2]
= plus(x(activate(N),activate(M)),activate(N))
isNat(n__0()) = [7]
> [0]
= tt()
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U12(tt()) = [0]
>= [0]
= tt()
U21(tt()) = [0]
>= [0]
= tt()
U31(tt(),V2) = [8] V2 + [8]
>= [8] V2 + [8]
= U32(isNat(activate(V2)))
U52(tt(),M,N) = [4] M + [2] N + [1]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
U71(tt(),M,N) = [1] M + [11] N + [15]
>= [1] M + [11] N + [15]
= U72(isNat(activate(N)),activate(M),activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= x(X1,X2)
isNat(n__plus(V1,V2)) = [8] V1 + [8] V2 + [15]
>= [8] V1 + [8] V2 + [15]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) = [8] V1 + [7]
>= [8] V1 + [7]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [8] V1 + [8] V2 + [15]
>= [8] V1 + [8] V2 + [15]
= U31(isNat(activate(V1)),activate(V2))
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)
x(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__x(X1,X2)
* Step 4: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Weak TRS:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U61(tt()) -> 0()
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
isNat(n__0()) -> tt()
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x_1 + [4] x_2 + [4]
p(U12) = [1] x_1 + [0]
p(U21) = [1] x_1 + [10]
p(U31) = [1] x_1 + [4] x_2 + [0]
p(U32) = [1] x_1 + [0]
p(U41) = [1] x_1 + [2] x_2 + [13]
p(U51) = [8] x_1 + [2] x_2 + [12] x_3 + [9]
p(U52) = [1] x_1 + [2] x_2 + [8] x_3 + [7]
p(U61) = [1] x_1 + [6]
p(U71) = [1] x_2 + [13] x_3 + [3]
p(U72) = [2] x_1 + [1] x_2 + [4] x_3 + [3]
p(activate) = [1] x_1 + [0]
p(isNat) = [4] x_1 + [0]
p(n__0) = [0]
p(n__plus) = [1] x_1 + [1] x_2 + [1]
p(n__s) = [1] x_1 + [6]
p(n__x) = [1] x_1 + [1] x_2 + [0]
p(plus) = [1] x_1 + [1] x_2 + [1]
p(s) = [1] x_1 + [6]
p(tt) = [0]
p(x) = [1] x_1 + [1] x_2 + [0]
Following rules are strictly oriented:
U21(tt()) = [10]
> [0]
= tt()
isNat(n__s(V1)) = [4] V1 + [24]
> [4] V1 + [10]
= U21(isNat(activate(V1)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [4] V2 + [4]
>= [4] V2 + [0]
= U12(isNat(activate(V2)))
U12(tt()) = [0]
>= [0]
= tt()
U31(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= U32(isNat(activate(V2)))
U32(tt()) = [0]
>= [0]
= tt()
U41(tt(),N) = [2] N + [13]
>= [1] N + [0]
= activate(N)
U51(tt(),M,N) = [2] M + [12] N + [9]
>= [2] M + [12] N + [7]
= U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) = [2] M + [8] N + [7]
>= [1] M + [1] N + [7]
= s(plus(activate(N),activate(M)))
U61(tt()) = [6]
>= [0]
= 0()
U71(tt(),M,N) = [1] M + [13] N + [3]
>= [1] M + [12] N + [3]
= U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) = [1] M + [4] N + [3]
>= [1] M + [2] N + [1]
= plus(x(activate(N),activate(M)),activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [6]
>= [1] X + [6]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= x(X1,X2)
isNat(n__0()) = [0]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [4] V1 + [4] V2 + [4]
>= [4] V1 + [4] V2 + [4]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__plus(X1,X2)
s(X) = [1] X + [6]
>= [1] X + [6]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
* Step 5: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U12(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Weak TRS:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U21(tt()) -> tt()
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U61(tt()) -> 0()
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
isNat(n__0()) -> tt()
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [8]
p(U11) = [1] x_1 + [2] x_2 + [0]
p(U12) = [1] x_1 + [2]
p(U21) = [1] x_1 + [0]
p(U31) = [1] x_1 + [2] x_2 + [8]
p(U32) = [1] x_1 + [10]
p(U41) = [8] x_1 + [1] x_2 + [0]
p(U51) = [1] x_2 + [6] x_3 + [0]
p(U52) = [2] x_1 + [1] x_2 + [2] x_3 + [0]
p(U61) = [8]
p(U71) = [2] x_1 + [4] x_2 + [10] x_3 + [3]
p(U72) = [1] x_1 + [4] x_2 + [8] x_3 + [6]
p(activate) = [1] x_1 + [0]
p(isNat) = [2] x_1 + [0]
p(n__0) = [8]
p(n__plus) = [1] x_1 + [1] x_2 + [1]
p(n__s) = [1] x_1 + [0]
p(n__x) = [1] x_1 + [1] x_2 + [7]
p(plus) = [1] x_1 + [1] x_2 + [1]
p(s) = [1] x_1 + [0]
p(tt) = [2]
p(x) = [1] x_1 + [1] x_2 + [7]
Following rules are strictly oriented:
U12(tt()) = [4]
> [2]
= tt()
U52(tt(),M,N) = [1] M + [2] N + [4]
> [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
U71(tt(),M,N) = [4] M + [10] N + [7]
> [4] M + [10] N + [6]
= U72(isNat(activate(N)),activate(M),activate(N))
isNat(n__plus(V1,V2)) = [2] V1 + [2] V2 + [2]
> [2] V1 + [2] V2 + [0]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__x(V1,V2)) = [2] V1 + [2] V2 + [14]
> [2] V1 + [2] V2 + [8]
= U31(isNat(activate(V1)),activate(V2))
Following rules are (at-least) weakly oriented:
0() = [8]
>= [8]
= n__0()
U11(tt(),V2) = [2] V2 + [2]
>= [2] V2 + [2]
= U12(isNat(activate(V2)))
U21(tt()) = [2]
>= [2]
= tt()
U31(tt(),V2) = [2] V2 + [10]
>= [2] V2 + [10]
= U32(isNat(activate(V2)))
U32(tt()) = [12]
>= [2]
= tt()
U41(tt(),N) = [1] N + [16]
>= [1] N + [0]
= activate(N)
U51(tt(),M,N) = [1] M + [6] N + [0]
>= [1] M + [6] N + [0]
= U52(isNat(activate(N)),activate(M),activate(N))
U61(tt()) = [8]
>= [8]
= 0()
U72(tt(),M,N) = [4] M + [8] N + [8]
>= [1] M + [2] N + [8]
= plus(x(activate(N),activate(M)),activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [8]
>= [8]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= x(X1,X2)
isNat(n__0()) = [16]
>= [2]
= tt()
isNat(n__s(V1)) = [2] V1 + [0]
>= [2] V1 + [0]
= U21(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)
x(X1,X2) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= n__x(X1,X2)
* Step 6: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U31(tt(),V2) -> U32(isNat(activate(V2)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Weak TRS:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x_1 + [1] x_2 + [0]
p(U12) = [1] x_1 + [0]
p(U21) = [1] x_1 + [0]
p(U31) = [1] x_1 + [1] x_2 + [2]
p(U32) = [1] x_1 + [0]
p(U41) = [1] x_2 + [0]
p(U51) = [1] x_1 + [2] x_2 + [8] x_3 + [4]
p(U52) = [1] x_1 + [2] x_2 + [1] x_3 + [4]
p(U61) = [8]
p(U71) = [1] x_1 + [2] x_2 + [4] x_3 + [11]
p(U72) = [2] x_1 + [2] x_2 + [2] x_3 + [3]
p(activate) = [1] x_1 + [0]
p(isNat) = [1] x_1 + [8]
p(n__0) = [0]
p(n__plus) = [1] x_1 + [1] x_2 + [0]
p(n__s) = [1] x_1 + [0]
p(n__x) = [1] x_1 + [1] x_2 + [2]
p(plus) = [1] x_1 + [1] x_2 + [0]
p(s) = [1] x_1 + [0]
p(tt) = [8]
p(x) = [1] x_1 + [1] x_2 + [2]
Following rules are strictly oriented:
U31(tt(),V2) = [1] V2 + [10]
> [1] V2 + [8]
= U32(isNat(activate(V2)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [1] V2 + [8]
>= [1] V2 + [8]
= U12(isNat(activate(V2)))
U12(tt()) = [8]
>= [8]
= tt()
U21(tt()) = [8]
>= [8]
= tt()
U32(tt()) = [8]
>= [8]
= tt()
U41(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U51(tt(),M,N) = [2] M + [8] N + [12]
>= [2] M + [2] N + [12]
= U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) = [2] M + [1] N + [12]
>= [1] M + [1] N + [0]
= s(plus(activate(N),activate(M)))
U61(tt()) = [8]
>= [0]
= 0()
U71(tt(),M,N) = [2] M + [4] N + [19]
>= [2] M + [4] N + [19]
= U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) = [2] M + [2] N + [19]
>= [1] M + [2] N + [2]
= plus(x(activate(N),activate(M)),activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= x(X1,X2)
isNat(n__0()) = [8]
>= [8]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8]
>= [1] V1 + [1] V2 + [8]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) = [1] V1 + [8]
>= [1] V1 + [8]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [10]
>= [1] V1 + [1] V2 + [10]
= U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__x(X1,X2)
* Step 7: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Weak TRS:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [1] x_1 + [4] x_2 + [0]
p(U12) = [1] x_1 + [0]
p(U21) = [1] x_1 + [0]
p(U31) = [1] x_1 + [4] x_2 + [4]
p(U32) = [1] x_1 + [4]
p(U41) = [2] x_1 + [1] x_2 + [12]
p(U51) = [4] x_1 + [1] x_2 + [5] x_3 + [8]
p(U52) = [1] x_1 + [1] x_2 + [1] x_3 + [1]
p(U61) = [2] x_1 + [2]
p(U71) = [1] x_1 + [8] x_2 + [10] x_3 + [15]
p(U72) = [2] x_1 + [1] x_2 + [2] x_3 + [3]
p(activate) = [1] x_1 + [1]
p(isNat) = [4] x_1 + [0]
p(n__0) = [1]
p(n__plus) = [1] x_1 + [1] x_2 + [2]
p(n__s) = [1] x_1 + [1]
p(n__x) = [1] x_1 + [1] x_2 + [5]
p(plus) = [1] x_1 + [1] x_2 + [2]
p(s) = [1] x_1 + [1]
p(tt) = [4]
p(x) = [1] x_1 + [1] x_2 + [6]
Following rules are strictly oriented:
activate(X) = [1] X + [1]
> [1] X + [0]
= X
activate(n__0()) = [2]
> [1]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3]
> [1] X1 + [1] X2 + [2]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [2]
> [1] X + [1]
= s(X)
x(X1,X2) = [1] X1 + [1] X2 + [6]
> [1] X1 + [1] X2 + [5]
= n__x(X1,X2)
Following rules are (at-least) weakly oriented:
0() = [1]
>= [1]
= n__0()
U11(tt(),V2) = [4] V2 + [4]
>= [4] V2 + [4]
= U12(isNat(activate(V2)))
U12(tt()) = [4]
>= [4]
= tt()
U21(tt()) = [4]
>= [4]
= tt()
U31(tt(),V2) = [4] V2 + [8]
>= [4] V2 + [8]
= U32(isNat(activate(V2)))
U32(tt()) = [8]
>= [4]
= tt()
U41(tt(),N) = [1] N + [20]
>= [1] N + [1]
= activate(N)
U51(tt(),M,N) = [1] M + [5] N + [24]
>= [1] M + [5] N + [7]
= U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) = [1] M + [1] N + [5]
>= [1] M + [1] N + [5]
= s(plus(activate(N),activate(M)))
U61(tt()) = [10]
>= [1]
= 0()
U71(tt(),M,N) = [8] M + [10] N + [19]
>= [1] M + [10] N + [14]
= U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) = [1] M + [2] N + [11]
>= [1] M + [2] N + [11]
= plus(x(activate(N),activate(M)),activate(N))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [6]
= x(X1,X2)
isNat(n__0()) = [4]
>= [4]
= tt()
isNat(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [8]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [4]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [4] V1 + [4] V2 + [20]
>= [4] V1 + [4] V2 + [12]
= U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
* Step 8: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(n__x(X1,X2)) -> x(X1,X2)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Weak TRS:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [1] x_1 + [2] x_2 + [10]
p(U12) = [1] x_1 + [7]
p(U21) = [1] x_1 + [6]
p(U31) = [1] x_1 + [2] x_2 + [4]
p(U32) = [1] x_1 + [1]
p(U41) = [2] x_1 + [1] x_2 + [1]
p(U51) = [3] x_1 + [8] x_2 + [6] x_3 + [11]
p(U52) = [1] x_1 + [8] x_2 + [2] x_3 + [8]
p(U61) = [4] x_1 + [10]
p(U71) = [5] x_1 + [8] x_2 + [10] x_3 + [4]
p(U72) = [3] x_1 + [1] x_2 + [4] x_3 + [0]
p(activate) = [1] x_1 + [1]
p(isNat) = [2] x_1 + [6]
p(n__0) = [0]
p(n__plus) = [1] x_1 + [1] x_2 + [7]
p(n__s) = [1] x_1 + [4]
p(n__x) = [1] x_1 + [1] x_2 + [4]
p(plus) = [1] x_1 + [1] x_2 + [7]
p(s) = [1] x_1 + [4]
p(tt) = [5]
p(x) = [1] x_1 + [1] x_2 + [5]
Following rules are strictly oriented:
0() = [1]
> [0]
= n__0()
Following rules are (at-least) weakly oriented:
U11(tt(),V2) = [2] V2 + [15]
>= [2] V2 + [15]
= U12(isNat(activate(V2)))
U12(tt()) = [12]
>= [5]
= tt()
U21(tt()) = [11]
>= [5]
= tt()
U31(tt(),V2) = [2] V2 + [9]
>= [2] V2 + [9]
= U32(isNat(activate(V2)))
U32(tt()) = [6]
>= [5]
= tt()
U41(tt(),N) = [1] N + [11]
>= [1] N + [1]
= activate(N)
U51(tt(),M,N) = [8] M + [6] N + [26]
>= [8] M + [4] N + [26]
= U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) = [8] M + [2] N + [13]
>= [1] M + [1] N + [13]
= s(plus(activate(N),activate(M)))
U61(tt()) = [30]
>= [1]
= 0()
U71(tt(),M,N) = [8] M + [10] N + [29]
>= [1] M + [10] N + [29]
= U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) = [1] M + [4] N + [15]
>= [1] M + [2] N + [15]
= plus(x(activate(N),activate(M)),activate(N))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [1]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [7]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [5]
>= [1] X + [4]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= x(X1,X2)
isNat(n__0()) = [6]
>= [5]
= tt()
isNat(n__plus(V1,V2)) = [2] V1 + [2] V2 + [20]
>= [2] V1 + [2] V2 + [20]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) = [2] V1 + [14]
>= [2] V1 + [14]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [2] V1 + [2] V2 + [14]
>= [2] V1 + [2] V2 + [14]
= U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= n__plus(X1,X2)
s(X) = [1] X + [4]
>= [1] X + [4]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [4]
= n__x(X1,X2)
* Step 9: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(n__x(X1,X2)) -> x(X1,X2)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [4]
p(U11) = [1] x_1 + [1] x_2 + [0]
p(U12) = [1] x_1 + [2]
p(U21) = [1] x_1 + [0]
p(U31) = [1] x_1 + [1] x_2 + [6]
p(U32) = [1] x_1 + [0]
p(U41) = [2] x_1 + [1] x_2 + [9]
p(U51) = [6] x_1 + [1] x_2 + [3] x_3 + [0]
p(U52) = [2] x_1 + [1] x_2 + [1] x_3 + [14]
p(U61) = [1] x_1 + [9]
p(U71) = [8] x_1 + [8] x_2 + [5] x_3 + [0]
p(U72) = [1] x_1 + [4] x_2 + [4] x_3 + [15]
p(activate) = [1] x_1 + [1]
p(isNat) = [1] x_1 + [0]
p(n__0) = [4]
p(n__plus) = [1] x_1 + [1] x_2 + [4]
p(n__s) = [1] x_1 + [1]
p(n__x) = [1] x_1 + [1] x_2 + [8]
p(plus) = [1] x_1 + [1] x_2 + [5]
p(s) = [1] x_1 + [2]
p(tt) = [3]
p(x) = [1] x_1 + [1] x_2 + [8]
Following rules are strictly oriented:
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [9]
> [1] X1 + [1] X2 + [8]
= x(X1,X2)
plus(X1,X2) = [1] X1 + [1] X2 + [5]
> [1] X1 + [1] X2 + [4]
= n__plus(X1,X2)
s(X) = [1] X + [2]
> [1] X + [1]
= n__s(X)
Following rules are (at-least) weakly oriented:
0() = [4]
>= [4]
= n__0()
U11(tt(),V2) = [1] V2 + [3]
>= [1] V2 + [3]
= U12(isNat(activate(V2)))
U12(tt()) = [5]
>= [3]
= tt()
U21(tt()) = [3]
>= [3]
= tt()
U31(tt(),V2) = [1] V2 + [9]
>= [1] V2 + [1]
= U32(isNat(activate(V2)))
U32(tt()) = [3]
>= [3]
= tt()
U41(tt(),N) = [1] N + [15]
>= [1] N + [1]
= activate(N)
U51(tt(),M,N) = [1] M + [3] N + [18]
>= [1] M + [3] N + [18]
= U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) = [1] M + [1] N + [20]
>= [1] M + [1] N + [9]
= s(plus(activate(N),activate(M)))
U61(tt()) = [12]
>= [4]
= 0()
U71(tt(),M,N) = [8] M + [5] N + [24]
>= [4] M + [5] N + [24]
= U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) = [4] M + [4] N + [18]
>= [1] M + [2] N + [16]
= plus(x(activate(N),activate(M)),activate(N))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__0()) = [5]
>= [4]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(X)
isNat(n__0()) = [4]
>= [3]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [2]
= U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) = [1] V1 + [1]
>= [1] V1 + [1]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [8]
>= [1] V1 + [1] V2 + [8]
= U31(isNat(activate(V1)),activate(V2))
x(X1,X2) = [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [8]
= n__x(X1,X2)
* Step 10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1
,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate
,isNat,plus,s,x} and constructors {n__0,n__plus,n__s,n__x,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))