*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+Full(0(),y) -> y
+Full(S(x),y) -> +Full(x,S(y))
f(x) -> *(x,x)
goal(xs) -> map(xs)
map(Cons(x,xs)) -> Cons(f(x),map(xs))
map(Nil()) -> Nil()
Weak DP Rules:
Weak TRS Rules:
*(x,0()) -> 0()
*(x,S(0())) -> x
*(x,S(S(y))) -> +(x,*(x,S(y)))
*(0(),y) -> 0()
Signature:
{*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
Obligation:
Innermost
basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {2},
uargs(Cons) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(*) = [12] x1 + [3]
p(+) = [1] x2 + [0]
p(+Full) = [3] x2 + [0]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [0]
p(Nil) = [0]
p(S) = [8]
p(f) = [12] x1 + [1]
p(goal) = [13] x1 + [1]
p(map) = [13] x1 + [12]
Following rules are strictly oriented:
map(Nil()) = [12]
> [0]
= Nil()
Following rules are (at-least) weakly oriented:
*(x,0()) = [12] x + [3]
>= [0]
= 0()
*(x,S(0())) = [12] x + [3]
>= [1] x + [0]
= x
*(x,S(S(y))) = [12] x + [3]
>= [12] x + [3]
= +(x,*(x,S(y)))
*(0(),y) = [3]
>= [0]
= 0()
+Full(0(),y) = [3] y + [0]
>= [1] y + [0]
= y
+Full(S(x),y) = [3] y + [0]
>= [24]
= +Full(x,S(y))
f(x) = [12] x + [1]
>= [12] x + [3]
= *(x,x)
goal(xs) = [13] xs + [1]
>= [13] xs + [12]
= map(xs)
map(Cons(x,xs)) = [13] x + [13] xs + [12]
>= [12] x + [13] xs + [13]
= Cons(f(x),map(xs))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+Full(0(),y) -> y
+Full(S(x),y) -> +Full(x,S(y))
f(x) -> *(x,x)
goal(xs) -> map(xs)
map(Cons(x,xs)) -> Cons(f(x),map(xs))
Weak DP Rules:
Weak TRS Rules:
*(x,0()) -> 0()
*(x,S(0())) -> x
*(x,S(S(y))) -> +(x,*(x,S(y)))
*(0(),y) -> 0()
map(Nil()) -> Nil()
Signature:
{*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
Obligation:
Innermost
basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {2},
uargs(Cons) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(*) = [2] x1 + [12]
p(+) = [1] x2 + [0]
p(+Full) = [8] x2 + [1]
p(0) = [4]
p(Cons) = [1] x1 + [1] x2 + [3]
p(Nil) = [4]
p(S) = [0]
p(f) = [2] x1 + [4]
p(goal) = [7] x1 + [0]
p(map) = [7] x1 + [2]
Following rules are strictly oriented:
+Full(0(),y) = [8] y + [1]
> [1] y + [0]
= y
map(Cons(x,xs)) = [7] x + [7] xs + [23]
> [2] x + [7] xs + [9]
= Cons(f(x),map(xs))
Following rules are (at-least) weakly oriented:
*(x,0()) = [2] x + [12]
>= [4]
= 0()
*(x,S(0())) = [2] x + [12]
>= [1] x + [0]
= x
*(x,S(S(y))) = [2] x + [12]
>= [2] x + [12]
= +(x,*(x,S(y)))
*(0(),y) = [20]
>= [4]
= 0()
+Full(S(x),y) = [8] y + [1]
>= [1]
= +Full(x,S(y))
f(x) = [2] x + [4]
>= [2] x + [12]
= *(x,x)
goal(xs) = [7] xs + [0]
>= [7] xs + [2]
= map(xs)
map(Nil()) = [30]
>= [4]
= Nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+Full(S(x),y) -> +Full(x,S(y))
f(x) -> *(x,x)
goal(xs) -> map(xs)
Weak DP Rules:
Weak TRS Rules:
*(x,0()) -> 0()
*(x,S(0())) -> x
*(x,S(S(y))) -> +(x,*(x,S(y)))
*(0(),y) -> 0()
+Full(0(),y) -> y
map(Cons(x,xs)) -> Cons(f(x),map(xs))
map(Nil()) -> Nil()
Signature:
{*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
Obligation:
Innermost
basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {2},
uargs(Cons) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(*) = [1] x1 + [0]
p(+) = [1] x2 + [0]
p(+Full) = [8] x1 + [1] x2 + [5]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [0]
p(Nil) = [1]
p(S) = [1] x1 + [0]
p(f) = [2] x1 + [0]
p(goal) = [4] x1 + [10]
p(map) = [4] x1 + [1]
Following rules are strictly oriented:
goal(xs) = [4] xs + [10]
> [4] xs + [1]
= map(xs)
Following rules are (at-least) weakly oriented:
*(x,0()) = [1] x + [0]
>= [0]
= 0()
*(x,S(0())) = [1] x + [0]
>= [1] x + [0]
= x
*(x,S(S(y))) = [1] x + [0]
>= [1] x + [0]
= +(x,*(x,S(y)))
*(0(),y) = [0]
>= [0]
= 0()
+Full(0(),y) = [1] y + [5]
>= [1] y + [0]
= y
+Full(S(x),y) = [8] x + [1] y + [5]
>= [8] x + [1] y + [5]
= +Full(x,S(y))
f(x) = [2] x + [0]
>= [1] x + [0]
= *(x,x)
map(Cons(x,xs)) = [4] x + [4] xs + [1]
>= [2] x + [4] xs + [1]
= Cons(f(x),map(xs))
map(Nil()) = [5]
>= [1]
= Nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+Full(S(x),y) -> +Full(x,S(y))
f(x) -> *(x,x)
Weak DP Rules:
Weak TRS Rules:
*(x,0()) -> 0()
*(x,S(0())) -> x
*(x,S(S(y))) -> +(x,*(x,S(y)))
*(0(),y) -> 0()
+Full(0(),y) -> y
goal(xs) -> map(xs)
map(Cons(x,xs)) -> Cons(f(x),map(xs))
map(Nil()) -> Nil()
Signature:
{*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
Obligation:
Innermost
basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {2},
uargs(Cons) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(*) = [1] x1 + [2] x2 + [0]
p(+) = [1] x2 + [8]
p(+Full) = [4] x1 + [5] x2 + [1]
p(0) = [4]
p(Cons) = [1] x1 + [1] x2 + [8]
p(Nil) = [3]
p(S) = [1] x1 + [4]
p(f) = [3] x1 + [8]
p(goal) = [3] x1 + [4]
p(map) = [3] x1 + [4]
Following rules are strictly oriented:
f(x) = [3] x + [8]
> [3] x + [0]
= *(x,x)
Following rules are (at-least) weakly oriented:
*(x,0()) = [1] x + [8]
>= [4]
= 0()
*(x,S(0())) = [1] x + [16]
>= [1] x + [0]
= x
*(x,S(S(y))) = [1] x + [2] y + [16]
>= [1] x + [2] y + [16]
= +(x,*(x,S(y)))
*(0(),y) = [2] y + [4]
>= [4]
= 0()
+Full(0(),y) = [5] y + [17]
>= [1] y + [0]
= y
+Full(S(x),y) = [4] x + [5] y + [17]
>= [4] x + [5] y + [21]
= +Full(x,S(y))
goal(xs) = [3] xs + [4]
>= [3] xs + [4]
= map(xs)
map(Cons(x,xs)) = [3] x + [3] xs + [28]
>= [3] x + [3] xs + [20]
= Cons(f(x),map(xs))
map(Nil()) = [13]
>= [3]
= Nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
+Full(S(x),y) -> +Full(x,S(y))
Weak DP Rules:
Weak TRS Rules:
*(x,0()) -> 0()
*(x,S(0())) -> x
*(x,S(S(y))) -> +(x,*(x,S(y)))
*(0(),y) -> 0()
+Full(0(),y) -> y
f(x) -> *(x,x)
goal(xs) -> map(xs)
map(Cons(x,xs)) -> Cons(f(x),map(xs))
map(Nil()) -> Nil()
Signature:
{*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
Obligation:
Innermost
basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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(+) = {2},
uargs(Cons) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(*) = [8] x1 + [9]
p(+) = [1] x2 + [0]
p(+Full) = [6] x1 + [4] x2 + [1]
p(0) = [2]
p(Cons) = [1] x1 + [1] x2 + [2]
p(Nil) = [0]
p(S) = [1] x1 + [4]
p(f) = [8] x1 + [9]
p(goal) = [10] x1 + [8]
p(map) = [8] x1 + [6]
Following rules are strictly oriented:
+Full(S(x),y) = [6] x + [4] y + [25]
> [6] x + [4] y + [17]
= +Full(x,S(y))
Following rules are (at-least) weakly oriented:
*(x,0()) = [8] x + [9]
>= [2]
= 0()
*(x,S(0())) = [8] x + [9]
>= [1] x + [0]
= x
*(x,S(S(y))) = [8] x + [9]
>= [8] x + [9]
= +(x,*(x,S(y)))
*(0(),y) = [25]
>= [2]
= 0()
+Full(0(),y) = [4] y + [13]
>= [1] y + [0]
= y
f(x) = [8] x + [9]
>= [8] x + [9]
= *(x,x)
goal(xs) = [10] xs + [8]
>= [8] xs + [6]
= map(xs)
map(Cons(x,xs)) = [8] x + [8] xs + [22]
>= [8] x + [8] xs + [17]
= Cons(f(x),map(xs))
map(Nil()) = [6]
>= [0]
= Nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
*(x,0()) -> 0()
*(x,S(0())) -> x
*(x,S(S(y))) -> +(x,*(x,S(y)))
*(0(),y) -> 0()
+Full(0(),y) -> y
+Full(S(x),y) -> +Full(x,S(y))
f(x) -> *(x,x)
goal(xs) -> map(xs)
map(Cons(x,xs)) -> Cons(f(x),map(xs))
map(Nil()) -> Nil()
Signature:
{*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
Obligation:
Innermost
basic terms: {*,+Full,f,goal,map}/{+,0,Cons,Nil,S}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).