*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
Weak DP Rules:
Weak TRS Rules:
Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
Obligation:
Full
basic terms: {U11,U12,activate,plus}/{0,s,tt}
Applied Processor:
ToInnermost
Proof:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
Weak DP Rules:
Weak TRS Rules:
Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
Obligation:
Innermost
basic terms: {U11,U12,activate,plus}/{0,s,tt}
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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [1] x3 + [0]
p(U12) = [1] x2 + [1] x3 + [3]
p(activate) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [3]
p(s) = [1] x1 + [0]
p(tt) = [0]
Following rules are strictly oriented:
plus(N,0()) = [1] N + [3]
> [1] N + [0]
= N
plus(N,s(M)) = [1] M + [1] N + [3]
> [1] M + [1] N + [0]
= U11(tt(),M,N)
Following rules are (at-least) weakly oriented:
U11(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [3]
= U12(tt()
,activate(M)
,activate(N))
U12(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
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:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
Weak DP Rules:
Weak TRS Rules:
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
Obligation:
Innermost
basic terms: {U11,U12,activate,plus}/{0,s,tt}
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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [1] x3 + [0]
p(U12) = [7] x1 + [1] x2 + [1] x3 + [0]
p(activate) = [1] x1 + [3]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [0]
p(tt) = [3]
Following rules are strictly oriented:
U12(tt(),M,N) = [1] M + [1] N + [21]
> [1] M + [1] N + [6]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [3]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
U11(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [27]
= U12(tt()
,activate(M)
,activate(N))
plus(N,0()) = [1] N + [0]
>= [1] N + [0]
= N
plus(N,s(M)) = [1] M + [1] N + [0]
>= [1] M + [1] N + [0]
= U11(tt(),M,N)
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:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
Weak DP Rules:
Weak TRS Rules:
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
Obligation:
Innermost
basic terms: {U11,U12,activate,plus}/{0,s,tt}
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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{U11,U12,activate,plus}
TcT has computed the following interpretation:
p(0) = [2]
p(U11) = [4] x2 + [4] x3 + [12]
p(U12) = [4] x2 + [4] x3 + [3]
p(activate) = [1] x1 + [0]
p(plus) = [4] x1 + [4] x2 + [0]
p(s) = [1] x1 + [3]
p(tt) = [0]
Following rules are strictly oriented:
U11(tt(),M,N) = [4] M + [4] N + [12]
> [4] M + [4] N + [3]
= U12(tt()
,activate(M)
,activate(N))
Following rules are (at-least) weakly oriented:
U12(tt(),M,N) = [4] M + [4] N + [3]
>= [4] M + [4] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
plus(N,0()) = [4] N + [8]
>= [1] N + [0]
= N
plus(N,s(M)) = [4] M + [4] N + [12]
>= [4] M + [4] N + [12]
= U11(tt(),M,N)
*** 1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
Obligation:
Innermost
basic terms: {U11,U12,activate,plus}/{0,s,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).