*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
append(l1,l2) -> ifappend(l1,l2,l1)
hd(cons(x,l)) -> x
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
Weak DP Rules:
Weak TRS Rules:
Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
Obligation:
Innermost
basic terms: {append,hd,ifappend,is_empty,tl}/{cons,false,nil,true}
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(cons) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(append) = [1] x1 + [2] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [9]
p(false) = [0]
p(hd) = [2] x1 + [0]
p(ifappend) = [2] x2 + [1] x3 + [0]
p(is_empty) = [0]
p(nil) = [0]
p(tl) = [2] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
hd(cons(x,l)) = [2] l + [2] x + [18]
> [1] x + [0]
= x
tl(cons(x,l)) = [2] l + [2] x + [18]
> [1] l + [0]
= l
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l1 + [2] l2 + [0]
>= [1] l1 + [2] l2 + [0]
= ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) = [1] l + [2] l2 + [1] x + [9]
>= [1] l + [2] l2 + [1] x + [9]
= cons(x,append(l,l2))
ifappend(l1,l2,nil()) = [2] l2 + [0]
>= [1] l2 + [0]
= l2
is_empty(cons(x,l)) = [0]
>= [0]
= false()
is_empty(nil()) = [0]
>= [0]
= true()
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:
append(l1,l2) -> ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
Weak DP Rules:
Weak TRS Rules:
hd(cons(x,l)) -> x
tl(cons(x,l)) -> l
Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
Obligation:
Innermost
basic terms: {append,hd,ifappend,is_empty,tl}/{cons,false,nil,true}
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(cons) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(append) = [1] x1 + [2] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(hd) = [1] x1 + [0]
p(ifappend) = [2] x2 + [1] x3 + [0]
p(is_empty) = [1]
p(nil) = [0]
p(tl) = [2] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
is_empty(cons(x,l)) = [1]
> [0]
= false()
is_empty(nil()) = [1]
> [0]
= true()
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l1 + [2] l2 + [0]
>= [1] l1 + [2] l2 + [0]
= ifappend(l1,l2,l1)
hd(cons(x,l)) = [1] l + [1] x + [0]
>= [1] x + [0]
= x
ifappend(l1,l2,cons(x,l)) = [1] l + [2] l2 + [1] x + [0]
>= [1] l + [2] l2 + [1] x + [0]
= cons(x,append(l,l2))
ifappend(l1,l2,nil()) = [2] l2 + [0]
>= [1] l2 + [0]
= l2
tl(cons(x,l)) = [2] l + [2] x + [0]
>= [1] l + [0]
= l
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:
append(l1,l2) -> ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
Weak DP Rules:
Weak TRS Rules:
hd(cons(x,l)) -> x
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
Obligation:
Innermost
basic terms: {append,hd,ifappend,is_empty,tl}/{cons,false,nil,true}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2}
Following symbols are considered usable:
{append,hd,ifappend,is_empty,tl}
TcT has computed the following interpretation:
p(append) = [4] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(false) = [1]
p(hd) = [1] x1 + [9]
p(ifappend) = [1] x2 + [4] x3 + [0]
p(is_empty) = [10]
p(nil) = [2]
p(tl) = [1] x1 + [0]
p(true) = [2]
Following rules are strictly oriented:
ifappend(l1,l2,nil()) = [1] l2 + [8]
> [1] l2 + [0]
= l2
Following rules are (at-least) weakly oriented:
append(l1,l2) = [4] l1 + [1] l2 + [0]
>= [4] l1 + [1] l2 + [0]
= ifappend(l1,l2,l1)
hd(cons(x,l)) = [1] l + [1] x + [9]
>= [1] x + [0]
= x
ifappend(l1,l2,cons(x,l)) = [4] l + [1] l2 + [4] x + [0]
>= [4] l + [1] l2 + [1] x + [0]
= cons(x,append(l,l2))
is_empty(cons(x,l)) = [10]
>= [1]
= false()
is_empty(nil()) = [10]
>= [2]
= true()
tl(cons(x,l)) = [1] l + [1] x + [0]
>= [1] l + [0]
= l
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
append(l1,l2) -> ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
Weak DP Rules:
Weak TRS Rules:
hd(cons(x,l)) -> x
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
Obligation:
Innermost
basic terms: {append,hd,ifappend,is_empty,tl}/{cons,false,nil,true}
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(cons) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(append) = [8] x1 + [1] x2 + [10]
p(cons) = [1] x1 + [1] x2 + [2]
p(false) = [2]
p(hd) = [1] x1 + [10]
p(ifappend) = [1] x2 + [8] x3 + [8]
p(is_empty) = [4]
p(nil) = [2]
p(tl) = [2] x1 + [0]
p(true) = [4]
Following rules are strictly oriented:
append(l1,l2) = [8] l1 + [1] l2 + [10]
> [8] l1 + [1] l2 + [8]
= ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) = [8] l + [1] l2 + [8] x + [24]
> [8] l + [1] l2 + [1] x + [12]
= cons(x,append(l,l2))
Following rules are (at-least) weakly oriented:
hd(cons(x,l)) = [1] l + [1] x + [12]
>= [1] x + [0]
= x
ifappend(l1,l2,nil()) = [1] l2 + [24]
>= [1] l2 + [0]
= l2
is_empty(cons(x,l)) = [4]
>= [2]
= false()
is_empty(nil()) = [4]
>= [4]
= true()
tl(cons(x,l)) = [2] l + [2] x + [4]
>= [1] l + [0]
= l
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(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
append(l1,l2) -> ifappend(l1,l2,l1)
hd(cons(x,l)) -> x
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
Obligation:
Innermost
basic terms: {append,hd,ifappend,is_empty,tl}/{cons,false,nil,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).