*** 1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
plus(N,0()) -> U31(isNat(N),N)
plus(N,s(M)) -> U41(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__s,tt}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U31(isNat(N),N)
plus(N,s(M)) -> U41(isNat(M),M,N)
All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [1] x2 + [0]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [0]
p(U31) = [1] x2 + [3]
p(U41) = [1] x2 + [2] x3 + [0]
p(U42) = [1] x1 + [1] x2 + [1] x3 + [9]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [8]
p(n__0) = [10]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [3]
p(tt) = [0]
Following rules are strictly oriented:
U31(tt(),N) = [1] N + [3]
> [1] N + [0]
= activate(N)
U42(tt(),M,N) = [1] M + [1] N + [9]
> [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
activate(n__0()) = [10]
> [0]
= 0()
isNat(n__0()) = [18]
> [0]
= tt()
s(X) = [1] X + [3]
> [1] X + [0]
= n__s(X)
Following rules are (at-least) weakly oriented:
0() = [0]
>= [10]
= n__0()
U11(tt(),V2) = [1] V2 + [0]
>= [1] V2 + [8]
= U12(isNat(activate(V2)))
U12(tt()) = [0]
>= [0]
= tt()
U21(tt()) = [0]
>= [0]
= tt()
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [17]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [3]
= s(activate(X))
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8]
>= [1] V1 + [1] V2 + [8]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1] V1 + [8]
>= [1] V1 + [8]
= U21(isNat(activate(V1)))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
activate(X) -> X
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
U31(tt(),N) -> activate(N)
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__0()) -> 0()
isNat(n__0()) -> tt()
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [1] x2 + [2]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [0]
p(U31) = [1] x2 + [0]
p(U41) = [1] x2 + [2] x3 + [0]
p(U42) = [1] x1 + [1] x2 + [1] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [4]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [0]
p(tt) = [0]
Following rules are strictly oriented:
U11(tt(),V2) = [1] V2 + [2]
> [1] V2 + [0]
= U12(isNat(activate(V2)))
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4]
> [1] X1 + [1] X2 + [1]
= plus(activate(X1),activate(X2))
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [4]
> [1] V1 + [1] V2 + [2]
= U11(isNat(activate(V1))
,activate(V2))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U12(tt()) = [0]
>= [0]
= tt()
U21(tt()) = [0]
>= [0]
= tt()
U31(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [1]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
U42(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
isNat(n__0()) = [0]
>= [0]
= tt()
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= U21(isNat(activate(V1)))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [4]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U12(tt()) -> tt()
U21(tt()) -> tt()
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
activate(X) -> X
activate(n__s(X)) -> s(activate(X))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U31(tt(),N) -> activate(N)
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [1] x2 + [2]
p(U12) = [1] x1 + [2]
p(U21) = [1] x1 + [0]
p(U31) = [1] x2 + [0]
p(U41) = [1] x2 + [2] x3 + [0]
p(U42) = [1] x1 + [1] x2 + [1] x3 + [5]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [4]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [2]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [7]
p(tt) = [4]
Following rules are strictly oriented:
U12(tt()) = [6]
> [4]
= tt()
isNat(n__s(V1)) = [1] V1 + [6]
> [1] V1 + [4]
= U21(isNat(activate(V1)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [1] V2 + [6]
>= [1] V2 + [6]
= U12(isNat(activate(V2)))
U21(tt()) = [4]
>= [4]
= tt()
U31(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [9]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
U42(tt(),M,N) = [1] M + [1] N + [9]
>= [1] M + [1] N + [9]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [7]
= s(activate(X))
isNat(n__0()) = [4]
>= [4]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [6]
>= [1] V1 + [1] V2 + [6]
= U11(isNat(activate(V1))
,activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [7]
>= [1] X + [2]
= n__s(X)
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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U21(tt()) -> tt()
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
activate(X) -> X
activate(n__s(X)) -> s(activate(X))
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U31(tt(),N) -> activate(N)
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [1] x2 + [5]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [2]
p(U31) = [5] x1 + [5] x2 + [4]
p(U41) = [1] x2 + [3] x3 + [0]
p(U42) = [1] x1 + [1] x2 + [1] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [1]
p(n__plus) = [1] x1 + [1] x2 + [5]
p(n__s) = [1] x1 + [2]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [2]
p(tt) = [1]
Following rules are strictly oriented:
U21(tt()) = [3]
> [1]
= tt()
Following rules are (at-least) weakly oriented:
0() = [0]
>= [1]
= n__0()
U11(tt(),V2) = [1] V2 + [6]
>= [1] V2 + [0]
= U12(isNat(activate(V2)))
U12(tt()) = [1]
>= [1]
= tt()
U31(tt(),N) = [5] N + [9]
>= [1] N + [0]
= activate(N)
U41(tt(),M,N) = [1] M + [3] N + [0]
>= [1] M + [2] N + [1]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
U42(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
isNat(n__0()) = [1]
>= [1]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [1] V2 + [5]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1] V1 + [2]
>= [1] V1 + [2]
= U21(isNat(activate(V1)))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [5]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
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(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
activate(X) -> X
activate(n__s(X)) -> s(activate(X))
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [1] x2 + [0]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [0]
p(U31) = [1] x2 + [1]
p(U41) = [1] x2 + [3] x3 + [5]
p(U42) = [1] x1 + [1] x2 + [1] x3 + [4]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [7]
p(n__s) = [1] x1 + [3]
p(plus) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [3]
p(tt) = [0]
Following rules are strictly oriented:
U41(tt(),M,N) = [1] M + [3] N + [5]
> [1] M + [2] N + [4]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [1] V2 + [0]
>= [1] V2 + [0]
= U12(isNat(activate(V2)))
U12(tt()) = [0]
>= [0]
= tt()
U21(tt()) = [0]
>= [0]
= tt()
U31(tt(),N) = [1] N + [1]
>= [1] N + [0]
= activate(N)
U42(tt(),M,N) = [1] M + [1] N + [4]
>= [1] M + [1] N + [4]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [1]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [3]
>= [1] X + [3]
= s(activate(X))
isNat(n__0()) = [0]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [7]
>= [1] V1 + [1] V2 + [0]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1] V1 + [3]
>= [1] V1 + [0]
= U21(isNat(activate(V1)))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [7]
= n__plus(X1,X2)
s(X) = [1] X + [3]
>= [1] X + [3]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
activate(X) -> X
activate(n__s(X)) -> s(activate(X))
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [5]
p(U11) = [1] x1 + [1] x2 + [0]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [0]
p(U31) = [1] x1 + [5] x2 + [5]
p(U41) = [2] x1 + [1] x2 + [4] x3 + [0]
p(U42) = [1] x1 + [1] x2 + [1] x3 + [5]
p(activate) = [1] x1 + [1]
p(isNat) = [1] x1 + [0]
p(n__0) = [4]
p(n__plus) = [1] x1 + [1] x2 + [7]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [5]
p(s) = [1] x1 + [2]
p(tt) = [4]
Following rules are strictly oriented:
0() = [5]
> [4]
= n__0()
activate(X) = [1] X + [1]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
U11(tt(),V2) = [1] V2 + [4]
>= [1] V2 + [1]
= U12(isNat(activate(V2)))
U12(tt()) = [4]
>= [4]
= tt()
U21(tt()) = [4]
>= [4]
= tt()
U31(tt(),N) = [5] N + [9]
>= [1] N + [1]
= activate(N)
U41(tt(),M,N) = [1] M + [4] N + [8]
>= [1] M + [2] N + [8]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
U42(tt(),M,N) = [1] M + [1] N + [9]
>= [1] M + [1] N + [9]
= s(plus(activate(N),activate(M)))
activate(n__0()) = [5]
>= [5]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [7]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [3]
= s(activate(X))
isNat(n__0()) = [4]
>= [4]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [7]
>= [1] V1 + [1] V2 + [2]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1] V1 + [1]
>= [1] V1 + [1]
= U21(isNat(activate(V1)))
plus(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [7]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [1]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__s(X)) -> s(activate(X))
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__s,tt}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}
TcT has computed the following interpretation:
p(0) = [4]
[0]
p(U11) = [1 0] x1 + [1 1] x2 + [2]
[0 0] [0 4] [0]
p(U12) = [1 0] x1 + [3]
[0 0] [0]
p(U21) = [1 0] x1 + [1]
[0 0] [0]
p(U31) = [1 4] x1 + [2 4] x2 + [6]
[2 0] [2 1] [2]
p(U41) = [0 0] x1 + [1 5] x2 + [2
5] x3 + [2]
[2 2] [4 4] [4
4] [2]
p(U42) = [1 0] x1 + [1 4] x2 + [1
3] x3 + [2]
[3 0] [0 1] [0
1] [3]
p(activate) = [1 1] x1 + [0]
[0 1] [0]
p(isNat) = [1 0] x1 + [0]
[0 4] [5]
p(n__0) = [4]
[0]
p(n__plus) = [1 1] x1 + [1 2] x2 + [2]
[0 1] [0 1] [0]
p(n__s) = [1 1] x1 + [4]
[0 1] [2]
p(plus) = [1 1] x1 + [1 2] x2 + [2]
[0 1] [0 1] [0]
p(s) = [1 1] x1 + [4]
[0 1] [2]
p(tt) = [4]
[0]
Following rules are strictly oriented:
activate(n__s(X)) = [1 2] X + [6]
[0 1] [2]
> [1 2] X + [4]
[0 1] [2]
= s(activate(X))
Following rules are (at-least) weakly oriented:
0() = [4]
[0]
>= [4]
[0]
= n__0()
U11(tt(),V2) = [1 1] V2 + [6]
[0 4] [0]
>= [1 1] V2 + [3]
[0 0] [0]
= U12(isNat(activate(V2)))
U12(tt()) = [7]
[0]
>= [4]
[0]
= tt()
U21(tt()) = [5]
[0]
>= [4]
[0]
= tt()
U31(tt(),N) = [2 4] N + [10]
[2 1] [10]
>= [1 1] N + [0]
[0 1] [0]
= activate(N)
U41(tt(),M,N) = [1 5] M + [2 5] N + [2]
[4 4] [4 4] [10]
>= [1 5] M + [2 5] N + [2]
[0 1] [3 4] [3]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
U42(tt(),M,N) = [1 4] M + [1 3] N + [6]
[0 1] [0 1] [15]
>= [1 4] M + [1 3] N + [6]
[0 1] [0 1] [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1 1] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [4]
[0]
>= [4]
[0]
= 0()
activate(n__plus(X1,X2)) = [1 2] X1 + [1 3] X2 + [2]
[0 1] [0 1] [0]
>= [1 2] X1 + [1 3] X2 + [2]
[0 1] [0 1] [0]
= plus(activate(X1),activate(X2))
isNat(n__0()) = [4]
[5]
>= [4]
[0]
= tt()
isNat(n__plus(V1,V2)) = [1 1] V1 + [1 2] V2 + [2]
[0 4] [0 4] [5]
>= [1 1] V1 + [1 2] V2 + [2]
[0 0] [0 4] [0]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1 1] V1 + [4]
[0 4] [13]
>= [1 1] V1 + [1]
[0 0] [0]
= U21(isNat(activate(V1)))
plus(X1,X2) = [1 1] X1 + [1 2] X2 + [2]
[0 1] [0 1] [0]
>= [1 1] X1 + [1 2] X2 + [2]
[0 1] [0 1] [0]
= n__plus(X1,X2)
s(X) = [1 1] X + [4]
[0 1] [2]
>= [1 1] X + [4]
[0 1] [2]
= n__s(X)
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(X1,X2) -> n__plus(X1,X2)
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__s,tt}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, 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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U42) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(U11) = [1 0] x1 + [2 4] x2 + [0]
[0 0] [0 0] [0]
p(U12) = [1 1] x1 + [0]
[0 0] [0]
p(U21) = [1 0] x1 + [0]
[0 1] [4]
p(U31) = [1 0] x1 + [1 1] x2 + [0]
[1 0] [4 2] [3]
p(U41) = [2 1] x1 + [1 7] x2 + [3
7] x3 + [5]
[2 0] [4 2] [4
7] [4]
p(U42) = [1 0] x1 + [1 6] x2 + [1
4] x3 + [3]
[2 0] [0 1] [0
2] [2]
p(activate) = [1 1] x1 + [0]
[0 1] [0]
p(isNat) = [2 0] x1 + [4]
[0 2] [0]
p(n__0) = [0]
[0]
p(n__plus) = [1 1] x1 + [1 3] x2 + [0]
[0 1] [0 1] [2]
p(n__s) = [1 2] x1 + [0]
[0 1] [4]
p(plus) = [1 1] x1 + [1 3] x2 + [1]
[0 1] [0 1] [2]
p(s) = [1 2] x1 + [0]
[0 1] [4]
p(tt) = [4]
[0]
Following rules are strictly oriented:
plus(X1,X2) = [1 1] X1 + [1 3] X2 + [1]
[0 1] [0 1] [2]
> [1 1] X1 + [1 3] X2 + [0]
[0 1] [0 1] [2]
= n__plus(X1,X2)
Following rules are (at-least) weakly oriented:
0() = [0]
[0]
>= [0]
[0]
= n__0()
U11(tt(),V2) = [2 4] V2 + [4]
[0 0] [0]
>= [2 4] V2 + [4]
[0 0] [0]
= U12(isNat(activate(V2)))
U12(tt()) = [4]
[0]
>= [4]
[0]
= tt()
U21(tt()) = [4]
[4]
>= [4]
[0]
= tt()
U31(tt(),N) = [1 1] N + [4]
[4 2] [7]
>= [1 1] N + [0]
[0 1] [0]
= activate(N)
U41(tt(),M,N) = [1 7] M + [3 7] N + [13]
[4 2] [4 7] [12]
>= [1 7] M + [3 7] N + [7]
[0 1] [4 6] [10]
= U42(isNat(activate(N))
,activate(M)
,activate(N))
U42(tt(),M,N) = [1 6] M + [1 4] N + [7]
[0 1] [0 2] [10]
>= [1 6] M + [1 4] N + [5]
[0 1] [0 1] [6]
= s(plus(activate(N),activate(M)))
activate(X) = [1 1] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [0]
[0]
>= [0]
[0]
= 0()
activate(n__plus(X1,X2)) = [1 2] X1 + [1 4] X2 + [2]
[0 1] [0 1] [2]
>= [1 2] X1 + [1 4] X2 + [1]
[0 1] [0 1] [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1 3] X + [4]
[0 1] [4]
>= [1 3] X + [0]
[0 1] [4]
= s(activate(X))
isNat(n__0()) = [4]
[0]
>= [4]
[0]
= tt()
isNat(n__plus(V1,V2)) = [2 2] V1 + [2 6] V2 + [4]
[0 2] [0 2] [4]
>= [2 2] V1 + [2 6] V2 + [4]
[0 0] [0 0] [0]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [2 4] V1 + [4]
[0 2] [8]
>= [2 2] V1 + [4]
[0 2] [4]
= U21(isNat(activate(V1)))
s(X) = [1 2] X + [0]
[0 1] [4]
>= [1 2] X + [0]
[0 1] [4]
= n__s(X)
*** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),N) -> activate(N)
U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus,s}/{n__0,n__plus,n__s,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).