*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
Weak DP Rules:
Weak TRS Rules:
Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
Obligation:
Innermost
basic terms: {admit,cond}/{.,=,carry,nil,sum,true,w}
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(cond) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(.) = [1] x2 + [0]
p(=) = [1] x2 + [1]
p(admit) = [0]
p(carry) = [1] x1 + [1] x2 + [1] x3 + [0]
p(cond) = [8] x1 + [1] x2 + [7]
p(nil) = [0]
p(sum) = [1] x1 + [1] x2 + [1] x3 + [0]
p(true) = [0]
p(w) = [0]
Following rules are strictly oriented:
cond(true(),y) = [1] y + [7]
> [1] y + [0]
= y
Following rules are (at-least) weakly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [0]
>= [15]
= cond(=(sum(x,u,v),w())
,.(u
,.(v
,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) = [0]
>= [0]
= nil()
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:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
Weak DP Rules:
Weak TRS Rules:
cond(true(),y) -> y
Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
Obligation:
Innermost
basic terms: {admit,cond}/{.,=,carry,nil,sum,true,w}
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(cond) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(.) = [1] x2 + [0]
p(=) = [1] x2 + [7]
p(admit) = [1]
p(carry) = [1] x1 + [1] x2 + [1] x3 + [0]
p(cond) = [3] x1 + [1] x2 + [0]
p(nil) = [0]
p(sum) = [1] x1 + [1] x2 + [1] x3 + [0]
p(true) = [0]
p(w) = [0]
Following rules are strictly oriented:
admit(x,nil()) = [1]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [1]
>= [22]
= cond(=(sum(x,u,v),w())
,.(u
,.(v
,.(w(),admit(carry(x,u,v),z)))))
cond(true(),y) = [1] y + [0]
>= [1] y + [0]
= y
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:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
Weak DP Rules:
Weak TRS Rules:
admit(x,nil()) -> nil()
cond(true(),y) -> y
Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
Obligation:
Innermost
basic terms: {admit,cond}/{.,=,carry,nil,sum,true,w}
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(cond) = {2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(.) = [1] x1 + [1] x2 + [1]
p(=) = [1] x2 + [0]
p(admit) = [8] x2 + [0]
p(carry) = [1] x2 + [1] x3 + [1]
p(cond) = [8] x1 + [1] x2 + [1]
p(nil) = [2]
p(sum) = [1] x1 + [1] x2 + [0]
p(true) = [0]
p(w) = [0]
Following rules are strictly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [8] u + [8] v + [8] z + [24]
> [1] u + [1] v + [8] z + [4]
= cond(=(sum(x,u,v),w())
,.(u
,.(v
,.(w(),admit(carry(x,u,v),z)))))
Following rules are (at-least) weakly oriented:
admit(x,nil()) = [16]
>= [2]
= nil()
cond(true(),y) = [1] y + [1]
>= [1] y + [0]
= y
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
Obligation:
Innermost
basic terms: {admit,cond}/{.,=,carry,nil,sum,true,w}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).