WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
d(s(X)) -> s(s(d(X)))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
p(X,0()) -> X
p(0(),X) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
q(0()) -> 0()
q(s(X)) -> s(p(q(X),d(X)))
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
d(s(X)) -> s(s(d(X)))
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
p(s(X),s(Y)) -> s(s(p(X,Y)))
q(s(X)) -> s(p(q(X),d(X)))
All above mentioned rules can be savely removed.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ 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(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(a) = [1] x1 + [14]
p(cs) = [1] x1 + [6]
p(d) = [0]
p(f) = [1] x1 + [1] x2 + [0]
p(nf) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(ns) = [1] x1 + [0]
p(nt) = [1] x1 + [0]
p(p) = [4] x1 + [6] x2 + [0]
p(q) = [0]
p(r) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(t) = [1] x1 + [1]
Following rules are strictly oriented:
a(X) = [1] X + [14]
> [1] X + [0]
= X
f(0(),X) = [1] X + [4]
> [0]
= nil()
p(X,0()) = [4] X + [24]
> [1] X + [0]
= X
p(0(),X) = [6] X + [16]
> [1] X + [0]
= X
t(X) = [1] X + [1]
> [1] X + [0]
= nt(X)
Following rules are (at-least) weakly oriented:
a(nf(X1,X2)) = [1] X1 + [1] X2 + [14]
>= [1] X1 + [1] X2 + [28]
= f(a(X1),a(X2))
a(ns(X)) = [1] X + [14]
>= [1] X + [14]
= s(a(X))
a(nt(X)) = [1] X + [14]
>= [1] X + [15]
= t(a(X))
d(0()) = [0]
>= [4]
= 0()
f(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= nf(X1,X2)
q(0()) = [0]
>= [4]
= 0()
s(X) = [1] X + [0]
>= [1] X + [0]
= ns(X)
t(N) = [1] N + [1]
>= [6]
= cs(r(q(N)),nt(ns(N)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
- Weak TRS:
a(X) -> X
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ 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(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(a) = [8] x1 + [3]
p(cs) = [1] x1 + [0]
p(d) = [0]
p(f) = [1] x1 + [1] x2 + [2]
p(nf) = [1] x1 + [1] x2 + [0]
p(nil) = [1]
p(ns) = [1] x1 + [0]
p(nt) = [1] x1 + [0]
p(p) = [4] x1 + [1] x2 + [7]
p(q) = [0]
p(r) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(t) = [1] x1 + [0]
Following rules are strictly oriented:
f(X1,X2) = [1] X1 + [1] X2 + [2]
> [1] X1 + [1] X2 + [0]
= nf(X1,X2)
Following rules are (at-least) weakly oriented:
a(X) = [8] X + [3]
>= [1] X + [0]
= X
a(nf(X1,X2)) = [8] X1 + [8] X2 + [3]
>= [8] X1 + [8] X2 + [8]
= f(a(X1),a(X2))
a(ns(X)) = [8] X + [3]
>= [8] X + [3]
= s(a(X))
a(nt(X)) = [8] X + [3]
>= [8] X + [3]
= t(a(X))
d(0()) = [0]
>= [4]
= 0()
f(0(),X) = [1] X + [6]
>= [1]
= nil()
p(X,0()) = [4] X + [11]
>= [1] X + [0]
= X
p(0(),X) = [1] X + [23]
>= [1] X + [0]
= X
q(0()) = [0]
>= [4]
= 0()
s(X) = [1] X + [0]
>= [1] X + [0]
= ns(X)
t(N) = [1] N + [0]
>= [0]
= cs(r(q(N)),nt(ns(N)))
t(X) = [1] X + [0]
>= [1] X + [0]
= nt(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
- Weak TRS:
a(X) -> X
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ 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(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(a) = [2] x1 + [0]
p(cs) = [1] x1 + [9]
p(d) = [0]
p(f) = [1] x1 + [1] x2 + [10]
p(nf) = [1] x1 + [1] x2 + [4]
p(nil) = [1]
p(ns) = [1] x1 + [0]
p(nt) = [1] x1 + [8]
p(p) = [1] x1 + [8] x2 + [8]
p(q) = [1] x1 + [4]
p(r) = [1] x1 + [8]
p(s) = [1] x1 + [0]
p(t) = [1] x1 + [9]
Following rules are strictly oriented:
a(nt(X)) = [2] X + [16]
> [2] X + [9]
= t(a(X))
q(0()) = [6]
> [2]
= 0()
Following rules are (at-least) weakly oriented:
a(X) = [2] X + [0]
>= [1] X + [0]
= X
a(nf(X1,X2)) = [2] X1 + [2] X2 + [8]
>= [2] X1 + [2] X2 + [10]
= f(a(X1),a(X2))
a(ns(X)) = [2] X + [0]
>= [2] X + [0]
= s(a(X))
d(0()) = [0]
>= [2]
= 0()
f(X1,X2) = [1] X1 + [1] X2 + [10]
>= [1] X1 + [1] X2 + [4]
= nf(X1,X2)
f(0(),X) = [1] X + [12]
>= [1]
= nil()
p(X,0()) = [1] X + [24]
>= [1] X + [0]
= X
p(0(),X) = [8] X + [10]
>= [1] X + [0]
= X
s(X) = [1] X + [0]
>= [1] X + [0]
= ns(X)
t(N) = [1] N + [9]
>= [1] N + [21]
= cs(r(q(N)),nt(ns(N)))
t(X) = [1] X + [9]
>= [1] X + [8]
= nt(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
d(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
- Weak TRS:
a(X) -> X
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ 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(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [8]
p(a) = [1] x1 + [3]
p(cs) = [1] x1 + [8]
p(d) = [2] x1 + [2]
p(f) = [1] x1 + [1] x2 + [0]
p(nf) = [1] x1 + [1] x2 + [0]
p(nil) = [4]
p(ns) = [1] x1 + [0]
p(nt) = [1] x1 + [0]
p(p) = [1] x1 + [2] x2 + [5]
p(q) = [8]
p(r) = [1] x1 + [3]
p(s) = [1] x1 + [0]
p(t) = [1] x1 + [0]
Following rules are strictly oriented:
d(0()) = [18]
> [8]
= 0()
Following rules are (at-least) weakly oriented:
a(X) = [1] X + [3]
>= [1] X + [0]
= X
a(nf(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [6]
= f(a(X1),a(X2))
a(ns(X)) = [1] X + [3]
>= [1] X + [3]
= s(a(X))
a(nt(X)) = [1] X + [3]
>= [1] X + [3]
= t(a(X))
f(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= nf(X1,X2)
f(0(),X) = [1] X + [8]
>= [4]
= nil()
p(X,0()) = [1] X + [21]
>= [1] X + [0]
= X
p(0(),X) = [2] X + [13]
>= [1] X + [0]
= X
q(0()) = [8]
>= [8]
= 0()
s(X) = [1] X + [0]
>= [1] X + [0]
= ns(X)
t(N) = [1] N + [0]
>= [19]
= cs(r(q(N)),nt(ns(N)))
t(X) = [1] X + [0]
>= [1] X + [0]
= nt(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
- Weak TRS:
a(X) -> X
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ 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(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a) = [10] x1 + [8]
p(cs) = [1] x1 + [0]
p(d) = [0]
p(f) = [1] x1 + [1] x2 + [11]
p(nf) = [1] x1 + [1] x2 + [2]
p(nil) = [0]
p(ns) = [1] x1 + [1]
p(nt) = [1] x1 + [1]
p(p) = [2] x1 + [2] x2 + [0]
p(q) = [8]
p(r) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(t) = [1] x1 + [10]
Following rules are strictly oriented:
a(nf(X1,X2)) = [10] X1 + [10] X2 + [28]
> [10] X1 + [10] X2 + [27]
= f(a(X1),a(X2))
a(ns(X)) = [10] X + [18]
> [10] X + [9]
= s(a(X))
t(N) = [1] N + [10]
> [8]
= cs(r(q(N)),nt(ns(N)))
Following rules are (at-least) weakly oriented:
a(X) = [10] X + [8]
>= [1] X + [0]
= X
a(nt(X)) = [10] X + [18]
>= [10] X + [18]
= t(a(X))
d(0()) = [0]
>= [0]
= 0()
f(X1,X2) = [1] X1 + [1] X2 + [11]
>= [1] X1 + [1] X2 + [2]
= nf(X1,X2)
f(0(),X) = [1] X + [11]
>= [0]
= nil()
p(X,0()) = [2] X + [0]
>= [1] X + [0]
= X
p(0(),X) = [2] X + [0]
>= [1] X + [0]
= X
q(0()) = [8]
>= [0]
= 0()
s(X) = [1] X + [1]
>= [1] X + [1]
= ns(X)
t(X) = [1] X + [10]
>= [1] X + [1]
= nt(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
s(X) -> ns(X)
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ 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(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a) = [11] x1 + [0]
p(cs) = [1] x1 + [0]
p(d) = [0]
p(f) = [1] x1 + [1] x2 + [0]
p(nf) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(ns) = [1] x1 + [1]
p(nt) = [1] x1 + [0]
p(p) = [2] x1 + [2] x2 + [0]
p(q) = [1] x1 + [0]
p(r) = [1] x1 + [0]
p(s) = [1] x1 + [11]
p(t) = [1] x1 + [0]
Following rules are strictly oriented:
s(X) = [1] X + [11]
> [1] X + [1]
= ns(X)
Following rules are (at-least) weakly oriented:
a(X) = [11] X + [0]
>= [1] X + [0]
= X
a(nf(X1,X2)) = [11] X1 + [11] X2 + [0]
>= [11] X1 + [11] X2 + [0]
= f(a(X1),a(X2))
a(ns(X)) = [11] X + [11]
>= [11] X + [11]
= s(a(X))
a(nt(X)) = [11] X + [0]
>= [11] X + [0]
= t(a(X))
d(0()) = [0]
>= [0]
= 0()
f(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= nf(X1,X2)
f(0(),X) = [1] X + [0]
>= [0]
= nil()
p(X,0()) = [2] X + [0]
>= [1] X + [0]
= X
p(0(),X) = [2] X + [0]
>= [1] X + [0]
= X
q(0()) = [0]
>= [0]
= 0()
t(N) = [1] N + [0]
>= [1] N + [0]
= cs(r(q(N)),nt(ns(N)))
t(X) = [1] X + [0]
>= [1] X + [0]
= nt(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))