*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(N,0()) -> U31(isNat(N))
x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N)
x(X1,X2) -> n__x(X1,X2)
Weak DP Rules:
Weak TRS Rules:
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,tt}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
x(N,0()) -> U31(isNat(N))
x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N)
All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Weak DP Rules:
Weak TRS Rules:
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [0]
p(U21) = [1] x2 + [1] x3 + [0]
p(U31) = [0]
p(U41) = [1] x2 + [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [0]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__x) = [1] x1 + [1] x2 + [0]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [0]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [7]
Following rules are strictly oriented:
x(X1,X2) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [0]
= s(plus(activate(N),activate(M)))
U31(tt()) = [0]
>= [0]
= 0()
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [7]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [7]
= x(X1,X2)
and(tt(),X) = [1] X + [0]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [0]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= 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 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [0]
p(U21) = [1] x2 + [1] x3 + [0]
p(U31) = [0]
p(U41) = [1] x2 + [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [0]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__x) = [1] x1 + [1] x2 + [5]
p(plus) = [1] x1 + [1] x2 + [3]
p(s) = [1] x1 + [5]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [5]
Following rules are strictly oriented:
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [5]
> [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [3]
> [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [5]
> [1] X + [0]
= n__s(X)
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [8]
= s(plus(activate(N),activate(M)))
U31(tt()) = [0]
>= [0]
= 0()
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [8]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [3]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [5]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= x(X1,X2)
and(tt(),X) = [1] X + [0]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [0]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= isNat(activate(V1))
x(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= n__x(X1,X2)
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:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
Weak DP Rules:
Weak TRS Rules:
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [0]
p(U21) = [1] x2 + [1] x3 + [0]
p(U31) = [3] x1 + [0]
p(U41) = [3] x1 + [1] x2 + [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [5]
p(isNat) = [1] x1 + [0]
p(n__0) = [0]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [3]
p(n__x) = [1] x1 + [1] x2 + [5]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [3]
p(tt) = [1]
p(x) = [1] x1 + [1] x2 + [5]
Following rules are strictly oriented:
U31(tt()) = [3]
> [0]
= 0()
and(tt(),X) = [1] X + [6]
> [1] X + [0]
= activate(X)
isNat(n__s(V1)) = [1] V1 + [3]
> [1] V1 + [0]
= isNat(activate(V1))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
U41(tt(),M,N) = [1] M + [2] N + [3]
>= [1] M + [2] N + [5]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [3]
>= [1] X + [3]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= x(X1,X2)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [0]
>= [1]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [5]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [1] V2 + [5]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [3]
>= [1] X + [3]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= n__x(X1,X2)
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:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
Weak DP Rules:
Weak TRS Rules:
U31(tt()) -> 0()
and(tt(),X) -> activate(X)
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [0]
p(U21) = [1] x2 + [1] x3 + [0]
p(U31) = [0]
p(U41) = [1] x2 + [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [5]
p(n__0) = [0]
p(n__isNat) = [1] x1 + [6]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [4]
p(n__x) = [1] x1 + [1] x2 + [6]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [5]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [7]
Following rules are strictly oriented:
activate(n__isNat(X)) = [1] X + [6]
> [1] X + [5]
= isNat(X)
isNat(n__0()) = [5]
> [0]
= tt()
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [7]
= s(plus(activate(N),activate(M)))
U31(tt()) = [0]
>= [0]
= 0()
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [9]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [2]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [4]
>= [1] X + [5]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [7]
= x(X1,X2)
and(tt(),X) = [1] X + [0]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [5]
>= [1] X + [6]
= n__isNat(X)
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [1] V2 + [11]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [9]
>= [1] V1 + [5]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [11]
>= [1] V1 + [1] V2 + [11]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [5]
>= [1] X + [4]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [6]
= n__x(X1,X2)
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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(X) -> n__isNat(X)
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
Weak DP Rules:
Weak TRS Rules:
U31(tt()) -> 0()
activate(n__isNat(X)) -> isNat(X)
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [0]
p(U21) = [1] x2 + [1] x3 + [0]
p(U31) = [0]
p(U41) = [1] x2 + [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [2]
p(n__0) = [0]
p(n__isNat) = [1] x1 + [2]
p(n__plus) = [1] x1 + [1] x2 + [6]
p(n__s) = [1] x1 + [0]
p(n__x) = [1] x1 + [1] x2 + [4]
p(plus) = [1] x1 + [1] x2 + [6]
p(s) = [1] x1 + [3]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [5]
Following rules are strictly oriented:
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8]
> [1] V1 + [1] V2 + [4]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [9]
= s(plus(activate(N),activate(M)))
U31(tt()) = [0]
>= [0]
= 0()
U41(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [11]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [2]
>= [1] X + [2]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [6]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [3]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [5]
= x(X1,X2)
and(tt(),X) = [1] X + [0]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [2]
>= [1] X + [2]
= n__isNat(X)
isNat(n__0()) = [2]
>= [0]
= tt()
isNat(n__s(V1)) = [1] V1 + [2]
>= [1] V1 + [2]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [6]
>= [1] V1 + [1] V2 + [4]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [6]
= n__plus(X1,X2)
s(X) = [1] X + [3]
>= [1] X + [0]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [4]
= n__x(X1,X2)
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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(X) -> n__isNat(X)
Weak DP Rules:
Weak TRS Rules:
U31(tt()) -> 0()
activate(n__isNat(X)) -> isNat(X)
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [1]
p(U21) = [2] x2 + [5] x3 + [0]
p(U31) = [0]
p(U41) = [1] x2 + [2] x3 + [3]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [5]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [7]
p(n__s) = [1] x1 + [0]
p(n__x) = [1] x1 + [1] x2 + [4]
p(plus) = [1] x1 + [1] x2 + [7]
p(s) = [1] x1 + [0]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [5]
Following rules are strictly oriented:
U11(tt(),N) = [1] N + [1]
> [1] N + [0]
= activate(N)
activate(n__0()) = [5]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
0() = [0]
>= [5]
= n__0()
U21(tt(),M,N) = [2] M + [5] N + [0]
>= [1] M + [1] N + [7]
= s(plus(activate(N),activate(M)))
U31(tt()) = [0]
>= [0]
= 0()
U41(tt(),M,N) = [1] M + [2] N + [3]
>= [1] M + [2] N + [12]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [5]
= x(X1,X2)
and(tt(),X) = [1] X + [0]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [5]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [7]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [4]
= n__x(X1,X2)
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^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(X) -> n__isNat(X)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),N) -> activate(N)
U31(tt()) -> 0()
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [3] x1 + [1] x2 + [0]
p(U21) = [2] x1 + [1] x2 + [2] x3 + [1]
p(U31) = [0]
p(U41) = [4] x2 + [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [5]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [2]
p(n__x) = [1] x1 + [1] x2 + [0]
p(plus) = [1] x1 + [1] x2 + [4]
p(s) = [1] x1 + [4]
p(tt) = [5]
p(x) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
U21(tt(),M,N) = [1] M + [2] N + [11]
> [1] M + [1] N + [8]
= s(plus(activate(N),activate(M)))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [5]
= n__0()
U11(tt(),N) = [1] N + [15]
>= [1] N + [0]
= activate(N)
U31(tt()) = [0]
>= [0]
= 0()
U41(tt(),M,N) = [4] M + [2] N + [0]
>= [1] M + [2] N + [4]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [5]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [4]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [2]
>= [1] X + [4]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= x(X1,X2)
and(tt(),X) = [1] X + [5]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [5]
>= [5]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [2]
>= [1] V1 + [0]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [4]
>= [1] X + [2]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(X) -> n__isNat(X)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(U11) = [4] x2 + [0]
p(U21) = [5] x1 + [1] x2 + [4] x3 + [4]
p(U31) = [6] x1 + [4]
p(U41) = [4] x1 + [4] x2 + [2] x3 + [4]
p(activate) = [1] x1 + [0]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [0]
p(n__0) = [6]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__x) = [1] x1 + [1] x2 + [0]
p(plus) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [0]
p(tt) = [1]
p(x) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
U41(tt(),M,N) = [4] M + [2] N + [8]
> [1] M + [2] N + [1]
= plus(x(activate(N),activate(M))
,activate(N))
Following rules are (at-least) weakly oriented:
0() = [2]
>= [6]
= n__0()
U11(tt(),N) = [4] N + [0]
>= [1] N + [0]
= activate(N)
U21(tt(),M,N) = [1] M + [4] N + [9]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
U31(tt()) = [10]
>= [2]
= 0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [6]
>= [2]
= 0()
activate(n__isNat(X)) = [1] X + [0]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [1]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= x(X1,X2)
and(tt(),X) = [1] X + [1]
>= [1] X + [0]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [6]
>= [1]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
activate(X) -> X
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(X) -> n__isNat(X)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
and(tt(),X) -> activate(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [3] x1 + [1] x2 + [0]
p(U21) = [3] x1 + [1] x2 + [3] x3 + [1]
p(U31) = [1] x1 + [0]
p(U41) = [3] x1 + [2] x2 + [3] x3 + [3]
p(activate) = [1] x1 + [2]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [2]
p(n__0) = [4]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [5]
p(n__s) = [1] x1 + [2]
p(n__x) = [1] x1 + [1] x2 + [4]
p(plus) = [1] x1 + [1] x2 + [5]
p(s) = [1] x1 + [4]
p(tt) = [4]
p(x) = [1] x1 + [1] x2 + [4]
Following rules are strictly oriented:
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [5]
= plus(X1,X2)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [6]
> [1] X1 + [1] X2 + [4]
= x(X1,X2)
isNat(X) = [1] X + [2]
> [1] X + [0]
= n__isNat(X)
Following rules are (at-least) weakly oriented:
0() = [0]
>= [4]
= n__0()
U11(tt(),N) = [1] N + [12]
>= [1] N + [2]
= activate(N)
U21(tt(),M,N) = [1] M + [3] N + [13]
>= [1] M + [1] N + [13]
= s(plus(activate(N),activate(M)))
U31(tt()) = [4]
>= [0]
= 0()
U41(tt(),M,N) = [2] M + [3] N + [15]
>= [1] M + [2] N + [15]
= plus(x(activate(N),activate(M))
,activate(N))
activate(n__0()) = [6]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [2]
>= [1] X + [2]
= isNat(X)
activate(n__s(X)) = [1] X + [4]
>= [1] X + [4]
= s(X)
and(tt(),X) = [1] X + [4]
>= [1] X + [2]
= activate(X)
isNat(n__0()) = [6]
>= [4]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [7]
>= [1] V1 + [1] V2 + [6]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [4]
>= [1] V1 + [4]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [6]
>= [1] V1 + [1] V2 + [6]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= n__plus(X1,X2)
s(X) = [1] X + [4]
>= [1] X + [2]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= n__x(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
activate(n__s(X)) -> s(X)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [2] x1 + [2] x2 + [0]
p(U21) = [3] x1 + [4] x2 + [2] x3 + [0]
p(U31) = [1] x1 + [2]
p(U41) = [2] x1 + [1] x2 + [4] x3 + [6]
p(activate) = [1] x1 + [1]
p(and) = [1] x1 + [1] x2 + [3]
p(isNat) = [1] x1 + [0]
p(n__0) = [4]
p(n__isNat) = [1] x1 + [0]
p(n__plus) = [1] x1 + [1] x2 + [5]
p(n__s) = [1] x1 + [1]
p(n__x) = [1] x1 + [1] x2 + [5]
p(plus) = [1] x1 + [1] x2 + [5]
p(s) = [1] x1 + [1]
p(tt) = [4]
p(x) = [1] x1 + [1] x2 + [6]
Following rules are strictly oriented:
activate(n__s(X)) = [1] X + [2]
> [1] X + [1]
= s(X)
Following rules are (at-least) weakly oriented:
0() = [0]
>= [4]
= n__0()
U11(tt(),N) = [2] N + [8]
>= [1] N + [1]
= activate(N)
U21(tt(),M,N) = [4] M + [2] N + [12]
>= [1] M + [1] N + [8]
= s(plus(activate(N),activate(M)))
U31(tt()) = [6]
>= [0]
= 0()
U41(tt(),M,N) = [1] M + [4] N + [14]
>= [1] M + [2] N + [14]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__0()) = [5]
>= [0]
= 0()
activate(n__isNat(X)) = [1] X + [1]
>= [1] X + [0]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [5]
= plus(X1,X2)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [6]
= x(X1,X2)
and(tt(),X) = [1] X + [7]
>= [1] X + [1]
= activate(X)
isNat(X) = [1] X + [0]
>= [1] X + [0]
= n__isNat(X)
isNat(n__0()) = [4]
>= [4]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [1] V2 + [5]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [1]
>= [1] V1 + [1]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [1] V2 + [5]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [5]
= n__x(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
Weak DP Rules:
Weak TRS Rules:
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,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(and) = {1,2},
uargs(isNat) = {1},
uargs(n__isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [1] x1 + [1] x2 + [0]
p(U21) = [5] x1 + [1] x2 + [1] x3 + [4]
p(U31) = [1]
p(U41) = [4] x1 + [1] x2 + [2] x3 + [7]
p(activate) = [1] x1 + [1]
p(and) = [1] x1 + [1] x2 + [0]
p(isNat) = [1] x1 + [1]
p(n__0) = [0]
p(n__isNat) = [1] x1 + [1]
p(n__plus) = [1] x1 + [1] x2 + [3]
p(n__s) = [1] x1 + [2]
p(n__x) = [1] x1 + [1] x2 + [3]
p(plus) = [1] x1 + [1] x2 + [4]
p(s) = [1] x1 + [3]
p(tt) = [1]
p(x) = [1] x1 + [1] x2 + [4]
Following rules are strictly oriented:
0() = [1]
> [0]
= n__0()
Following rules are (at-least) weakly oriented:
U11(tt(),N) = [1] N + [1]
>= [1] N + [1]
= activate(N)
U21(tt(),M,N) = [1] M + [1] N + [9]
>= [1] M + [1] N + [9]
= s(plus(activate(N),activate(M)))
U31(tt()) = [1]
>= [1]
= 0()
U41(tt(),M,N) = [1] M + [2] N + [11]
>= [1] M + [2] N + [11]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [1]
= 0()
activate(n__isNat(X)) = [1] X + [2]
>= [1] X + [1]
= isNat(X)
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [3]
>= [1] X + [3]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= x(X1,X2)
and(tt(),X) = [1] X + [1]
>= [1] X + [1]
= activate(X)
isNat(X) = [1] X + [1]
>= [1] X + [1]
= n__isNat(X)
isNat(n__0()) = [1]
>= [1]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [4]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
isNat(n__s(V1)) = [1] V1 + [3]
>= [1] V1 + [2]
= isNat(activate(V1))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [4]
= and(isNat(activate(V1))
,n__isNat(activate(V2)))
plus(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [3]
= n__plus(X1,X2)
s(X) = [1] X + [3]
>= [1] X + [2]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [3]
= n__x(X1,X2)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U21,U31,U41,activate,and,isNat,plus,s,x}/{n__0,n__isNat,n__plus,n__s,n__x,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).