*** 1 Progress [(O(1),O(n^1))] ***
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(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(N,0()) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(N,0()) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),M,N)
x(X1,X2) -> n__x(X1,X2)
Weak DP Rules:
Weak TRS Rules:
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
x(N,0()) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),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(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),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,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [3]
p(U11) = [1] x1 + [4] x2 + [0]
p(U12) = [1] x1 + [8]
p(U21) = [1] x1 + [8]
p(U31) = [1] x1 + [4] x2 + [0]
p(U32) = [1] x1 + [2]
p(U41) = [2] x1 + [1] x2 + [0]
p(U51) = [1] x1 + [1] x2 + [10] x3 + [7]
p(U52) = [2] x1 + [1] x2 + [1] x3 + [6]
p(U61) = [12]
p(U71) = [2] x1 + [1] x2 + [6] x3 + [15]
p(U72) = [1] x1 + [1] x2 + [2] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [4] x1 + [0]
p(n__0) = [3]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [3]
p(n__x) = [1] x1 + [1] x2 + [0]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [3]
p(tt) = [8]
p(x) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
U12(tt()) = [16]
> [8]
= tt()
U21(tt()) = [16]
> [8]
= tt()
U31(tt(),V2) = [4] V2 + [8]
> [4] V2 + [2]
= U32(isNat(activate(V2)))
U32(tt()) = [10]
> [8]
= tt()
U41(tt(),N) = [1] N + [16]
> [1] N + [0]
= activate(N)
U51(tt(),M,N) = [1] M + [10] N + [15]
> [1] M + [9] N + [6]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [1] M + [1] N + [22]
> [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
U61(tt()) = [12]
> [3]
= 0()
U71(tt(),M,N) = [1] M + [6] N + [31]
> [1] M + [6] N + [1]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [1] M + [2] N + [9]
> [1] M + [2] N + [0]
= plus(x(activate(N),activate(M))
,activate(N))
isNat(n__0()) = [12]
> [8]
= tt()
isNat(n__s(V1)) = [4] V1 + [12]
> [4] V1 + [8]
= U21(isNat(activate(V1)))
Following rules are (at-least) weakly oriented:
0() = [3]
>= [3]
= n__0()
U11(tt(),V2) = [4] V2 + [8]
>= [4] V2 + [8]
= U12(isNat(activate(V2)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [3]
>= [3]
= 0()
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 + [0]
>= [1] X1 + [1] X2 + [0]
= x(X1,X2)
isNat(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U31(isNat(activate(V1))
,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 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
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(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),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:
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
isNat(n__0()) -> tt()
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [2]
p(U11) = [1] x1 + [1] x2 + [1]
p(U12) = [1] x1 + [1]
p(U21) = [1] x1 + [2]
p(U31) = [1] x1 + [1] x2 + [2]
p(U32) = [1] x1 + [2]
p(U41) = [12] x1 + [1] x2 + [0]
p(U51) = [10] x1 + [1] x2 + [7] x3 + [10]
p(U52) = [4] x1 + [1] x2 + [3] x3 + [5]
p(U61) = [2] x1 + [1]
p(U71) = [9] x1 + [8] x2 + [11] x3 + [12]
p(U72) = [9] x1 + [2] x2 + [2] x3 + [1]
p(activate) = [1] x1 + [2]
p(isNat) = [1] x1 + [0]
p(n__0) = [2]
p(n__plus) = [1] x1 + [1] x2 + [5]
p(n__s) = [1] x1 + [4]
p(n__x) = [1] x1 + [1] x2 + [6]
p(plus) = [1] x1 + [1] x2 + [5]
p(s) = [1] x1 + [4]
p(tt) = [2]
p(x) = [1] x1 + [1] x2 + [8]
Following rules are strictly oriented:
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__0()) = [4]
> [2]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [5]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [6]
> [1] X + [4]
= s(X)
x(X1,X2) = [1] X1 + [1] X2 + [8]
> [1] X1 + [1] X2 + [6]
= n__x(X1,X2)
Following rules are (at-least) weakly oriented:
0() = [2]
>= [2]
= n__0()
U11(tt(),V2) = [1] V2 + [3]
>= [1] V2 + [3]
= U12(isNat(activate(V2)))
U12(tt()) = [3]
>= [2]
= tt()
U21(tt()) = [4]
>= [2]
= tt()
U31(tt(),V2) = [1] V2 + [4]
>= [1] V2 + [4]
= U32(isNat(activate(V2)))
U32(tt()) = [4]
>= [2]
= tt()
U41(tt(),N) = [1] N + [24]
>= [1] N + [2]
= activate(N)
U51(tt(),M,N) = [1] M + [7] N + [30]
>= [1] M + [7] N + [21]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [1] M + [3] N + [13]
>= [1] M + [1] N + [13]
= s(plus(activate(N),activate(M)))
U61(tt()) = [5]
>= [2]
= 0()
U71(tt(),M,N) = [8] M + [11] N + [30]
>= [2] M + [11] N + [27]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [2] M + [2] N + [19]
>= [1] M + [2] N + [19]
= plus(x(activate(N),activate(M))
,activate(N))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [8]
= x(X1,X2)
isNat(n__0()) = [2]
>= [2]
= 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 + [4]
>= [1] V1 + [4]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [6]
>= [1] V1 + [1] V2 + [6]
= U31(isNat(activate(V1))
,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 + [4]
= n__s(X)
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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)
isNat(n__0()) -> tt()
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [2]
p(U11) = [1] x1 + [4] x2 + [2]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [0]
p(U31) = [1] x1 + [4] x2 + [0]
p(U32) = [1] x1 + [1]
p(U41) = [1] x2 + [8]
p(U51) = [6] x1 + [8] x2 + [5] x3 + [0]
p(U52) = [1] x1 + [2] x2 + [1] x3 + [0]
p(U61) = [1] x1 + [12]
p(U71) = [4] x1 + [2] x2 + [12] x3 + [0]
p(U72) = [2] x1 + [1] x2 + [4] x3 + [2]
p(activate) = [1] x1 + [0]
p(isNat) = [4] x1 + [0]
p(n__0) = [2]
p(n__plus) = [1] x1 + [1] x2 + [1]
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) = [4]
p(x) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
U11(tt(),V2) = [4] V2 + [6]
> [4] V2 + [0]
= U12(isNat(activate(V2)))
isNat(n__plus(V1,V2)) = [4] V1 + [4] V2 + [4]
> [4] V1 + [4] V2 + [2]
= U11(isNat(activate(V1))
,activate(V2))
Following rules are (at-least) weakly oriented:
0() = [2]
>= [2]
= n__0()
U12(tt()) = [4]
>= [4]
= tt()
U21(tt()) = [4]
>= [4]
= tt()
U31(tt(),V2) = [4] V2 + [4]
>= [4] V2 + [1]
= U32(isNat(activate(V2)))
U32(tt()) = [5]
>= [4]
= tt()
U41(tt(),N) = [1] N + [8]
>= [1] N + [0]
= activate(N)
U51(tt(),M,N) = [8] M + [5] N + [24]
>= [2] M + [5] N + [0]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [2] M + [1] N + [4]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
U61(tt()) = [16]
>= [2]
= 0()
U71(tt(),M,N) = [2] M + [12] N + [16]
>= [1] M + [12] N + [2]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [1] M + [4] N + [10]
>= [1] M + [2] N + [1]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [2]
>= [2]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [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)
isNat(n__0()) = [8]
>= [4]
= tt()
isNat(n__s(V1)) = [4] V1 + [0]
>= [4] V1 + [0]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U31(isNat(activate(V1))
,activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= 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)
*** 1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
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] x1 + [1] x2 + [0]
p(U32) = [1] x1 + [0]
p(U41) = [1] x2 + [9]
p(U51) = [1] x1 + [1] x2 + [14] x3 + [4]
p(U52) = [1] x1 + [1] x2 + [13] x3 + [2]
p(U61) = [9] x1 + [5]
p(U71) = [10] x1 + [8] x2 + [5] x3 + [3]
p(U72) = [1] x1 + [4] x2 + [4] x3 + [8]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [2]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__x) = [1] x1 + [1] x2 + [1]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [0]
p(tt) = [2]
p(x) = [1] x1 + [1] x2 + [1]
Following rules are strictly oriented:
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [3]
> [1] V1 + [1] V2 + [2]
= U31(isNat(activate(V1))
,activate(V2))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [1] V2 + [2]
>= [1] V2 + [2]
= U12(isNat(activate(V2)))
U12(tt()) = [2]
>= [2]
= tt()
U21(tt()) = [2]
>= [2]
= tt()
U31(tt(),V2) = [1] V2 + [2]
>= [1] V2 + [2]
= U32(isNat(activate(V2)))
U32(tt()) = [2]
>= [2]
= tt()
U41(tt(),N) = [1] N + [9]
>= [1] N + [0]
= activate(N)
U51(tt(),M,N) = [1] M + [14] N + [6]
>= [1] M + [14] N + [4]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [1] M + [13] N + [4]
>= [1] M + [1] N + [0]
= s(plus(activate(N),activate(M)))
U61(tt()) = [23]
>= [0]
= 0()
U71(tt(),M,N) = [8] M + [5] N + [23]
>= [4] M + [5] N + [10]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [4] M + [4] N + [10]
>= [1] M + [2] N + [1]
= 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 + [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 + [1]
>= [1] X1 + [1] X2 + [1]
= x(X1,X2)
isNat(n__0()) = [2]
>= [2]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [2]
>= [1] V1 + [1] V2 + [2]
= 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 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__x(X1,X2)
*** 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__x(X1,X2)) -> x(X1,X2)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [3]
p(U11) = [1] x1 + [8] x2 + [8]
p(U12) = [1] x1 + [3]
p(U21) = [1] x1 + [2]
p(U31) = [1] x1 + [8] x2 + [8]
p(U32) = [1] x1 + [8]
p(U41) = [1] x2 + [5]
p(U51) = [1] x1 + [1] x2 + [10] x3 + [4]
p(U52) = [1] x1 + [1] x2 + [2] x3 + [1]
p(U61) = [3] x1 + [2]
p(U71) = [2] x1 + [12] x2 + [10] x3 + [15]
p(U72) = [1] x1 + [12] x2 + [2] x3 + [8]
p(activate) = [1] x1 + [1]
p(isNat) = [8] x1 + [0]
p(n__0) = [2]
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 + [3]
p(s) = [1] x1 + [2]
p(tt) = [8]
p(x) = [1] x1 + [1] x2 + [3]
Following rules are strictly oriented:
0() = [3]
> [2]
= n__0()
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [4]
> [1] X1 + [1] X2 + [3]
= x(X1,X2)
Following rules are (at-least) weakly oriented:
U11(tt(),V2) = [8] V2 + [16]
>= [8] V2 + [11]
= U12(isNat(activate(V2)))
U12(tt()) = [11]
>= [8]
= tt()
U21(tt()) = [10]
>= [8]
= tt()
U31(tt(),V2) = [8] V2 + [16]
>= [8] V2 + [16]
= U32(isNat(activate(V2)))
U32(tt()) = [16]
>= [8]
= tt()
U41(tt(),N) = [1] N + [5]
>= [1] N + [1]
= activate(N)
U51(tt(),M,N) = [1] M + [10] N + [12]
>= [1] M + [10] N + [12]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [1] M + [2] N + [9]
>= [1] M + [1] N + [7]
= s(plus(activate(N),activate(M)))
U61(tt()) = [26]
>= [3]
= 0()
U71(tt(),M,N) = [12] M + [10] N + [31]
>= [12] M + [10] N + [30]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [12] M + [2] N + [16]
>= [1] M + [2] N + [9]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__0()) = [3]
>= [3]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [3]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [3]
>= [1] X + [2]
= s(X)
isNat(n__0()) = [16]
>= [8]
= tt()
isNat(n__plus(V1,V2)) = [8] V1 + [8] V2 + [24]
>= [8] V1 + [8] V2 + [24]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [8] V1 + [16]
>= [8] V1 + [10]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [8] V1 + [8] V2 + [24]
>= [8] V1 + [8] V2 + [24]
= U31(isNat(activate(V1))
,activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [13]
p(U11) = [1] x1 + [2] x2 + [0]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [4]
p(U31) = [1] x1 + [2] x2 + [0]
p(U32) = [1] x1 + [0]
p(U41) = [6] x1 + [1] x2 + [0]
p(U51) = [6] x1 + [1] x2 + [6] x3 + [0]
p(U52) = [2] x1 + [1] x2 + [2] x3 + [7]
p(U61) = [4] x1 + [0]
p(U71) = [7] x1 + [8] x2 + [6] x3 + [1]
p(U72) = [1] x1 + [1] x2 + [4] x3 + [14]
p(activate) = [1] x1 + [2]
p(isNat) = [2] x1 + [0]
p(n__0) = [11]
p(n__plus) = [1] x1 + [1] x2 + [4]
p(n__s) = [1] x1 + [4]
p(n__x) = [1] x1 + [1] x2 + [6]
p(plus) = [1] x1 + [1] x2 + [6]
p(s) = [1] x1 + [4]
p(tt) = [4]
p(x) = [1] x1 + [1] x2 + [6]
Following rules are strictly oriented:
plus(X1,X2) = [1] X1 + [1] X2 + [6]
> [1] X1 + [1] X2 + [4]
= n__plus(X1,X2)
Following rules are (at-least) weakly oriented:
0() = [13]
>= [11]
= n__0()
U11(tt(),V2) = [2] V2 + [4]
>= [2] V2 + [4]
= U12(isNat(activate(V2)))
U12(tt()) = [4]
>= [4]
= tt()
U21(tt()) = [8]
>= [4]
= tt()
U31(tt(),V2) = [2] V2 + [4]
>= [2] V2 + [4]
= U32(isNat(activate(V2)))
U32(tt()) = [4]
>= [4]
= tt()
U41(tt(),N) = [1] N + [24]
>= [1] N + [2]
= activate(N)
U51(tt(),M,N) = [1] M + [6] N + [24]
>= [1] M + [6] N + [21]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [1] M + [2] N + [15]
>= [1] M + [1] N + [14]
= s(plus(activate(N),activate(M)))
U61(tt()) = [16]
>= [13]
= 0()
U71(tt(),M,N) = [8] M + [6] N + [29]
>= [1] M + [6] N + [28]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [1] M + [4] N + [18]
>= [1] M + [2] N + [18]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__0()) = [13]
>= [13]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [6]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [6]
>= [1] X + [4]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [6]
= x(X1,X2)
isNat(n__0()) = [22]
>= [4]
= tt()
isNat(n__plus(V1,V2)) = [2] V1 + [2] V2 + [8]
>= [2] V1 + [2] V2 + [8]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [2] V1 + [8]
>= [2] V1 + [8]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [2] V1 + [2] V2 + [12]
>= [2] V1 + [2] V2 + [8]
= U31(isNat(activate(V1))
,activate(V2))
s(X) = [1] X + [4]
>= [1] X + [4]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [6]
>= [1] X1 + [1] X2 + [6]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
s(X) -> n__s(X)
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V2) -> U12(isNat(activate(V2)))
U12(tt()) -> tt()
U21(tt()) -> tt()
U31(tt(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
x(X1,X2) -> n__x(X1,X2)
Signature:
{0/0,U11/2,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,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(U11) = {1,2},
uargs(U12) = {1},
uargs(U21) = {1},
uargs(U31) = {1,2},
uargs(U32) = {1},
uargs(U52) = {1,2,3},
uargs(U72) = {1,2,3},
uargs(isNat) = {1},
uargs(plus) = {1,2},
uargs(s) = {1},
uargs(x) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}
TcT has computed the following interpretation:
p(0) = [3]
p(U11) = [1] x1 + [1] x2 + [7]
p(U12) = [1] x1 + [2]
p(U21) = [1] x1 + [3]
p(U31) = [1] x1 + [1] x2 + [0]
p(U32) = [1] x1 + [1]
p(U41) = [2] x2 + [1]
p(U51) = [2] x1 + [4] x2 + [7] x3 + [13]
p(U52) = [6] x1 + [4] x2 + [1] x3 + [2]
p(U61) = [8] x1 + [2]
p(U71) = [8] x1 + [1] x2 + [10] x3 + [4]
p(U72) = [2] x1 + [1] x2 + [8] x3 + [10]
p(activate) = [1] x1 + [1]
p(isNat) = [1] x1 + [1]
p(n__0) = [2]
p(n__plus) = [1] x1 + [1] x2 + [9]
p(n__s) = [1] x1 + [6]
p(n__x) = [1] x1 + [1] x2 + [2]
p(plus) = [1] x1 + [1] x2 + [10]
p(s) = [1] x1 + [7]
p(tt) = [3]
p(x) = [1] x1 + [1] x2 + [3]
Following rules are strictly oriented:
s(X) = [1] X + [7]
> [1] X + [6]
= n__s(X)
Following rules are (at-least) weakly oriented:
0() = [3]
>= [2]
= n__0()
U11(tt(),V2) = [1] V2 + [10]
>= [1] V2 + [4]
= U12(isNat(activate(V2)))
U12(tt()) = [5]
>= [3]
= tt()
U21(tt()) = [6]
>= [3]
= tt()
U31(tt(),V2) = [1] V2 + [3]
>= [1] V2 + [3]
= U32(isNat(activate(V2)))
U32(tt()) = [4]
>= [3]
= tt()
U41(tt(),N) = [2] N + [1]
>= [1] N + [1]
= activate(N)
U51(tt(),M,N) = [4] M + [7] N + [19]
>= [4] M + [7] N + [19]
= U52(isNat(activate(N))
,activate(M)
,activate(N))
U52(tt(),M,N) = [4] M + [1] N + [20]
>= [1] M + [1] N + [19]
= s(plus(activate(N),activate(M)))
U61(tt()) = [26]
>= [3]
= 0()
U71(tt(),M,N) = [1] M + [10] N + [28]
>= [1] M + [10] N + [23]
= U72(isNat(activate(N))
,activate(M)
,activate(N))
U72(tt(),M,N) = [1] M + [8] N + [16]
>= [1] M + [2] N + [16]
= plus(x(activate(N),activate(M))
,activate(N))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__0()) = [3]
>= [3]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [10]
>= [1] X1 + [1] X2 + [10]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [7]
>= [1] X + [7]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= x(X1,X2)
isNat(n__0()) = [3]
>= [3]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [10]
>= [1] V1 + [1] V2 + [10]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1] V1 + [7]
>= [1] V1 + [5]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [3]
= U31(isNat(activate(V1))
,activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [10]
>= [1] X1 + [1] X2 + [9]
= n__plus(X1,X2)
x(X1,X2) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [2]
= n__x(X1,X2)
*** 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(),V2) -> U32(isNat(activate(V2)))
U32(tt()) -> tt()
U41(tt(),N) -> activate(N)
U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N))
U52(tt(),M,N) -> s(plus(activate(N),activate(M)))
U61(tt()) -> 0()
U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N))
U72(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(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),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,U12/1,U21/1,U31/2,U32/1,U41/2,U51/3,U52/3,U61/1,U71/3,U72/3,activate/1,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {0,U11,U12,U21,U31,U32,U41,U51,U52,U61,U71,U72,activate,isNat,plus,s,x}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).