*** 1 Progress [(?,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(),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(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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(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(),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(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
0#() -> c_1()
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U12#(tt()) -> c_3()
U21#(tt()) -> c_4()
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U32#(tt()) -> c_6()
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U61#(tt()) -> c_10(0#())
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(X) -> c_13()
activate#(n__0()) -> c_14(0#())
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__0()) -> c_18()
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_22()
s#(X) -> c_23()
x#(X1,X2) -> c_24()
Weak DPs
and mark the set of starting terms.
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
0#() -> c_1()
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U12#(tt()) -> c_3()
U21#(tt()) -> c_4()
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U32#(tt()) -> c_6()
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U61#(tt()) -> c_10(0#())
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(X) -> c_13()
activate#(n__0()) -> c_14(0#())
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__0()) -> c_18()
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_22()
s#(X) -> c_23()
x#(X1,X2) -> c_24()
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(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/3,c_16/2,c_17/3,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0,c_24/0}
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:
UsableRules
Proof:
We replace rewrite rules by usable 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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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)
0#() -> c_1()
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U12#(tt()) -> c_3()
U21#(tt()) -> c_4()
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U32#(tt()) -> c_6()
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U61#(tt()) -> c_10(0#())
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(X) -> c_13()
activate#(n__0()) -> c_14(0#())
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__0()) -> c_18()
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_22()
s#(X) -> c_23()
x#(X1,X2) -> c_24()
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
0#() -> c_1()
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U12#(tt()) -> c_3()
U21#(tt()) -> c_4()
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U32#(tt()) -> c_6()
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U61#(tt()) -> c_10(0#())
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(X) -> c_13()
activate#(n__0()) -> c_14(0#())
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__0()) -> c_18()
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_22()
s#(X) -> c_23()
x#(X1,X2) -> c_24()
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/3,c_16/2,c_17/3,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3,4,6,13,18,22,23,24}
by application of
Pre({1,3,4,6,13,18,22,23,24}) = {2,5,7,8,9,10,11,12,14,15,16,17,19,20,21}.
Here rules are labelled as follows:
1: 0#() -> c_1()
2: U11#(tt(),V2) ->
c_2(U12#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
3: U12#(tt()) -> c_3()
4: U21#(tt()) -> c_4()
5: U31#(tt(),V2) ->
c_5(U32#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
6: U32#(tt()) -> c_6()
7: U41#(tt(),N) ->
c_7(activate#(N))
8: U51#(tt(),M,N) ->
c_8(U52#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
9: U52#(tt(),M,N) ->
c_9(s#(plus(activate(N)
,activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
10: U61#(tt()) -> c_10(0#())
11: U71#(tt(),M,N) ->
c_11(U72#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
12: U72#(tt(),M,N) ->
c_12(plus#(x(activate(N)
,activate(M))
,activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
13: activate#(X) -> c_13()
14: activate#(n__0()) -> c_14(0#())
15: activate#(n__plus(X1,X2)) ->
c_15(plus#(activate(X1)
,activate(X2))
,activate#(X1)
,activate#(X2))
16: activate#(n__s(X)) ->
c_16(s#(activate(X))
,activate#(X))
17: activate#(n__x(X1,X2)) ->
c_17(x#(activate(X1)
,activate(X2))
,activate#(X1)
,activate#(X2))
18: isNat#(n__0()) -> c_18()
19: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
20: isNat#(n__s(V1)) ->
c_20(U21#(isNat(activate(V1)))
,isNat#(activate(V1))
,activate#(V1))
21: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
22: plus#(X1,X2) -> c_22()
23: s#(X) -> c_23()
24: x#(X1,X2) -> c_24()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U61#(tt()) -> c_10(0#())
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(n__0()) -> c_14(0#())
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
0#() -> c_1()
U12#(tt()) -> c_3()
U21#(tt()) -> c_4()
U32#(tt()) -> c_6()
activate#(X) -> c_13()
isNat#(n__0()) -> c_18()
plus#(X1,X2) -> c_22()
s#(X) -> c_23()
x#(X1,X2) -> c_24()
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/3,c_16/2,c_17/3,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{6,9}
by application of
Pre({6,9}) = {1,2,3,4,5,7,8,10,11,12,13,14,15}.
Here rules are labelled as follows:
1: U11#(tt(),V2) ->
c_2(U12#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
2: U31#(tt(),V2) ->
c_5(U32#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
3: U41#(tt(),N) ->
c_7(activate#(N))
4: U51#(tt(),M,N) ->
c_8(U52#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
5: U52#(tt(),M,N) ->
c_9(s#(plus(activate(N)
,activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
6: U61#(tt()) -> c_10(0#())
7: U71#(tt(),M,N) ->
c_11(U72#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
8: U72#(tt(),M,N) ->
c_12(plus#(x(activate(N)
,activate(M))
,activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
9: activate#(n__0()) -> c_14(0#())
10: activate#(n__plus(X1,X2)) ->
c_15(plus#(activate(X1)
,activate(X2))
,activate#(X1)
,activate#(X2))
11: activate#(n__s(X)) ->
c_16(s#(activate(X))
,activate#(X))
12: activate#(n__x(X1,X2)) ->
c_17(x#(activate(X1)
,activate(X2))
,activate#(X1)
,activate#(X2))
13: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
14: isNat#(n__s(V1)) ->
c_20(U21#(isNat(activate(V1)))
,isNat#(activate(V1))
,activate#(V1))
15: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
16: 0#() -> c_1()
17: U12#(tt()) -> c_3()
18: U21#(tt()) -> c_4()
19: U32#(tt()) -> c_6()
20: activate#(X) -> c_13()
21: isNat#(n__0()) -> c_18()
22: plus#(X1,X2) -> c_22()
23: s#(X) -> c_23()
24: x#(X1,X2) -> c_24()
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
0#() -> c_1()
U12#(tt()) -> c_3()
U21#(tt()) -> c_4()
U32#(tt()) -> c_6()
U61#(tt()) -> c_10(0#())
activate#(X) -> c_13()
activate#(n__0()) -> c_14(0#())
isNat#(n__0()) -> c_18()
plus#(X1,X2) -> c_22()
s#(X) -> c_23()
x#(X1,X2) -> c_24()
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/3,c_16/2,c_17/3,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0,c_24/0}
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:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 isNat#(n__0()) -> c_18():21
-->_3 activate#(X) -> c_13():19
-->_1 U12#(tt()) -> c_3():15
2:S:U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 isNat#(n__0()) -> c_18():21
-->_3 activate#(X) -> c_13():19
-->_1 U32#(tt()) -> c_6():17
3:S:U41#(tt(),N) -> c_7(activate#(N))
-->_1 activate#(n__0()) -> c_14(0#()):20
-->_1 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_1 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_1 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 activate#(X) -> c_13():19
4:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__0()) -> c_14(0#()):20
-->_4 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_5 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M)):5
-->_2 isNat#(n__0()) -> c_18():21
-->_5 activate#(X) -> c_13():19
-->_4 activate#(X) -> c_13():19
-->_3 activate#(X) -> c_13():19
5:S:U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
-->_4 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 s#(X) -> c_23():23
-->_2 plus#(X1,X2) -> c_22():22
-->_4 activate#(X) -> c_13():19
-->_3 activate#(X) -> c_13():19
6:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__0()) -> c_14(0#()):20
-->_4 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_5 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N)):7
-->_2 isNat#(n__0()) -> c_18():21
-->_5 activate#(X) -> c_13():19
-->_4 activate#(X) -> c_13():19
-->_3 activate#(X) -> c_13():19
7:S:U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__0()) -> c_14(0#()):20
-->_4 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_5 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 x#(X1,X2) -> c_24():24
-->_1 plus#(X1,X2) -> c_22():22
-->_5 activate#(X) -> c_13():19
-->_4 activate#(X) -> c_13():19
-->_3 activate#(X) -> c_13():19
8:S:activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_2 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_1 plus#(X1,X2) -> c_22():22
-->_3 activate#(X) -> c_13():19
-->_2 activate#(X) -> c_13():19
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
9:S:activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
-->_2 activate#(n__0()) -> c_14(0#()):20
-->_2 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_1 s#(X) -> c_23():23
-->_2 activate#(X) -> c_13():19
-->_2 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
10:S:activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 activate#(n__0()) -> c_14(0#()):20
-->_1 x#(X1,X2) -> c_24():24
-->_3 activate#(X) -> c_13():19
-->_2 activate#(X) -> c_13():19
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_2 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
11:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_4 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__0()) -> c_18():21
-->_4 activate#(X) -> c_13():19
-->_3 activate#(X) -> c_13():19
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1
12:S:isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__0()) -> c_18():21
-->_3 activate#(X) -> c_13():19
-->_1 U21#(tt()) -> c_4():16
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
13:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_4 activate#(n__0()) -> c_14(0#()):20
-->_3 activate#(n__0()) -> c_14(0#()):20
-->_2 isNat#(n__0()) -> c_18():21
-->_4 activate#(X) -> c_13():19
-->_3 activate#(X) -> c_13():19
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):2
14:W:0#() -> c_1()
15:W:U12#(tt()) -> c_3()
16:W:U21#(tt()) -> c_4()
17:W:U32#(tt()) -> c_6()
18:W:U61#(tt()) -> c_10(0#())
-->_1 0#() -> c_1():14
19:W:activate#(X) -> c_13()
20:W:activate#(n__0()) -> c_14(0#())
-->_1 0#() -> c_1():14
21:W:isNat#(n__0()) -> c_18()
22:W:plus#(X1,X2) -> c_22()
23:W:s#(X) -> c_23()
24:W:x#(X1,X2) -> c_24()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
18: U61#(tt()) -> c_10(0#())
15: U12#(tt()) -> c_3()
17: U32#(tt()) -> c_6()
22: plus#(X1,X2) -> c_22()
23: s#(X) -> c_23()
24: x#(X1,X2) -> c_24()
16: U21#(tt()) -> c_4()
19: activate#(X) -> c_13()
21: isNat#(n__0()) -> c_18()
20: activate#(n__0()) -> c_14(0#())
14: 0#() -> c_1()
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/3,c_3/0,c_4/0,c_5/3,c_6/0,c_7/1,c_8/5,c_9/4,c_10/1,c_11/5,c_12/5,c_13/0,c_14/1,c_15/3,c_16/2,c_17/3,c_18/0,c_19/4,c_20/3,c_21/4,c_22/0,c_23/0,c_24/0}
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:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
2:S:U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
3:S:U41#(tt(),N) -> c_7(activate#(N))
-->_1 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_1 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_1 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
4:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_5 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M)):5
5:S:U52#(tt(),M,N) -> c_9(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
6:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_5 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N)):7
7:S:U72#(tt(),M,N) -> c_12(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
8:S:activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_2 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
9:S:activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X))
-->_2 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
10:S:activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_2 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
11:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):1
12:S:isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
13:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_4 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(s#(activate(X)),activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):8
-->_1 U31#(tt(),V2) -> c_5(U32#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U41#(tt(),N) -> c_7(activate#(N))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
RemoveHeads
Proof:
Consider the dependency graph
1:S:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
2:S:U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
3:S:U41#(tt(),N) -> c_7(activate#(N))
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
4:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_5 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 U52#(tt(),M,N) -> c_9(activate#(N),activate#(M)):5
5:S:U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
6:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_5 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_5 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_5 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N)):7
7:S:U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
8:S:activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
9:S:activate#(n__s(X)) -> c_16(activate#(X))
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
10:S:activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
11:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1
12:S:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
13:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):13
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):12
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):11
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):10
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):9
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):8
-->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2)):2
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(3,U41#(tt(),N) -> c_7(activate#(N)))]
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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}
Problem (S)
Strict DP Rules:
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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}
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
We decompose the input problem according to the dependency graph into the upper component
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
and a lower component
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
Further, following extension rules are added to the lower component.
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{4,6}
by application of
Pre({4,6}) = {3,5}.
Here rules are labelled as follows:
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2))
,activate#(V2))
2: U31#(tt(),V2) ->
c_5(isNat#(activate(V2))
,activate#(V2))
3: U51#(tt(),M,N) ->
c_8(U52#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
4: U52#(tt(),M,N) ->
c_9(activate#(N),activate#(M))
5: U71#(tt(),M,N) ->
c_11(U72#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
6: U72#(tt(),M,N) ->
c_12(activate#(N)
,activate#(M)
,activate#(N))
7: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
8: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1))
,activate#(V1))
9: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
2:S:U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
3:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
-->_1 U52#(tt(),M,N) -> c_9(activate#(N),activate#(M)):8
4:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
-->_1 U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N)):9
5:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
-->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1
6:S:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
7:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
-->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2)):2
8:W:U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
9:W:U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: U72#(tt(),M,N) ->
c_12(activate#(N)
,activate#(M)
,activate#(N))
8: U52#(tt(),M,N) ->
c_9(activate#(N),activate#(M))
*** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
2:S:U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
3:S:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
4:S:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
5:S:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
-->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1
6:S:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
7:S:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):5
-->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
4: U71#(tt(),M,N) ->
c_11(isNat#(activate(N)))
6: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1)))
Consider the set of all dependency pairs
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2)))
2: U31#(tt(),V2) ->
c_5(isNat#(activate(V2)))
3: U51#(tt(),M,N) ->
c_8(isNat#(activate(N)))
4: U71#(tt(),M,N) ->
c_11(isNat#(activate(N)))
5: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
6: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1)))
7: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{4,6}
These cover all (indirect) predecessors of dependency pairs
{3,4,6}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1},
uargs(c_11) = {1},
uargs(c_19) = {1,2},
uargs(c_20) = {1},
uargs(c_21) = {1,2}
Following symbols are considered usable:
{0,activate,plus,s,x,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) = [4]
p(U11) = [3] x1 + [4] x2 + [1]
p(U12) = [1] x1 + [2]
p(U21) = [4]
p(U31) = [0]
p(U32) = [1] x1 + [5]
p(U41) = [4] x1 + [4] x2 + [1]
p(U51) = [4] x1 + [0]
p(U52) = [1] x1 + [1] x2 + [1] x3 + [1]
p(U61) = [1]
p(U71) = [1] x2 + [4]
p(U72) = [4] x1 + [1] x3 + [0]
p(activate) = [1] x1 + [0]
p(isNat) = [4]
p(n__0) = [4]
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 + [0]
p(s) = [1] x1 + [2]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [0]
p(0#) = [2]
p(U11#) = [4] x2 + [0]
p(U12#) = [0]
p(U21#) = [1]
p(U31#) = [4] x2 + [0]
p(U32#) = [1] x1 + [0]
p(U41#) = [4] x1 + [1]
p(U51#) = [4] x1 + [2] x2 + [4] x3 + [0]
p(U52#) = [2] x1 + [2] x3 + [0]
p(U61#) = [1] x1 + [2]
p(U71#) = [4] x3 + [1]
p(U72#) = [1] x3 + [1]
p(activate#) = [1]
p(isNat#) = [4] x1 + [0]
p(plus#) = [4] x1 + [2]
p(s#) = [4] x1 + [0]
p(x#) = [1] x2 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [4]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [2] x1 + [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x2 + [0]
p(c_10) = [4] x1 + [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [2] x3 + [0]
p(c_13) = [2]
p(c_14) = [2] x1 + [1]
p(c_15) = [2] x1 + [1]
p(c_16) = [2] x1 + [2]
p(c_17) = [2] x2 + [0]
p(c_18) = [4]
p(c_19) = [1] x1 + [1] x2 + [0]
p(c_20) = [1] x1 + [3]
p(c_21) = [1] x1 + [1] x2 + [0]
p(c_22) = [1]
p(c_23) = [2]
p(c_24) = [1]
Following rules are strictly oriented:
U71#(tt(),M,N) = [4] N + [1]
> [4] N + [0]
= c_11(isNat#(activate(N)))
isNat#(n__s(V1)) = [4] V1 + [8]
> [4] V1 + [3]
= c_20(isNat#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= c_2(isNat#(activate(V2)))
U31#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= c_5(isNat#(activate(V2)))
U51#(tt(),M,N) = [2] M + [4] N + [0]
>= [4] N + [0]
= c_8(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
0() = [4]
>= [4]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [4]
>= [4]
= 0()
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 + [2]
>= [1] X + [2]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= x(activate(X1),activate(X2))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: U31#(tt(),V2) ->
c_5(isNat#(activate(V2)))
Consider the set of all dependency pairs
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2)))
2: U31#(tt(),V2) ->
c_5(isNat#(activate(V2)))
3: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
4: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
5: U51#(tt(),M,N) ->
c_8(isNat#(activate(N)))
6: U71#(tt(),M,N) ->
c_11(isNat#(activate(N)))
7: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{2,5,6}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1},
uargs(c_11) = {1},
uargs(c_19) = {1,2},
uargs(c_20) = {1},
uargs(c_21) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,activate,isNat,plus,s,x,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) = [0]
p(U12) = [0]
p(U21) = [1]
p(U31) = [3]
p(U32) = [3]
p(U41) = [1] x1 + [4] x2 + [0]
p(U51) = [1]
p(U52) = [2] x1 + [1] x3 + [1]
p(U61) = [1] x1 + [0]
p(U71) = [1] x1 + [4] x2 + [0]
p(U72) = [1] x1 + [1] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [4]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [1]
p(n__x) = [1] x1 + [1] x2 + [2]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [2]
p(0#) = [0]
p(U11#) = [2] x1 + [4] x2 + [0]
p(U12#) = [2]
p(U21#) = [1] x1 + [2]
p(U31#) = [4] x2 + [7]
p(U32#) = [1] x1 + [0]
p(U41#) = [2] x1 + [1] x2 + [2]
p(U51#) = [2] x2 + [4] x3 + [1]
p(U52#) = [2] x1 + [1] x2 + [4] x3 + [1]
p(U61#) = [2]
p(U71#) = [4] x3 + [5]
p(U72#) = [1] x1 + [4] x2 + [1] x3 + [2]
p(activate#) = [2] x1 + [1]
p(isNat#) = [4] x1 + [0]
p(plus#) = [1] x1 + [1] x2 + [1]
p(s#) = [4] x1 + [1]
p(x#) = [2]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [4]
p(c_4) = [2]
p(c_5) = [1] x1 + [3]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [1]
p(c_10) = [2] x1 + [0]
p(c_11) = [1] x1 + [1]
p(c_12) = [4] x1 + [1] x2 + [2] x3 + [4]
p(c_13) = [1]
p(c_14) = [0]
p(c_15) = [1]
p(c_16) = [2] x1 + [4]
p(c_17) = [1] x2 + [0]
p(c_18) = [2]
p(c_19) = [1] x1 + [1] x2 + [0]
p(c_20) = [1] x1 + [4]
p(c_21) = [1] x1 + [1] x2 + [1]
p(c_22) = [4]
p(c_23) = [1]
p(c_24) = [0]
Following rules are strictly oriented:
U31#(tt(),V2) = [4] V2 + [7]
> [4] V2 + [3]
= c_5(isNat#(activate(V2)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= c_2(isNat#(activate(V2)))
U51#(tt(),M,N) = [2] M + [4] N + [1]
>= [4] N + [0]
= c_8(isNat#(activate(N)))
U71#(tt(),M,N) = [4] N + [5]
>= [4] N + [1]
= c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [8]
= c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [4]
= c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [8]
= c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [0]
>= [0]
= U12(isNat(activate(V2)))
U12(tt()) = [0]
>= [0]
= tt()
U21(tt()) = [1]
>= [0]
= tt()
U31(tt(),V2) = [3]
>= [3]
= U32(isNat(activate(V2)))
U32(tt()) = [3]
>= [0]
= tt()
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 + [1]
>= [1] X + [1]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= x(activate(X1),activate(X2))
isNat(n__0()) = [4]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [4]
>= [0]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [4]
>= [1]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [4]
>= [3]
= U31(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 + [1]
>= [1] X + [1]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
3: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
Consider the set of all dependency pairs
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2)))
2: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
3: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
4: U31#(tt(),V2) ->
c_5(isNat#(activate(V2)))
5: U51#(tt(),M,N) ->
c_8(isNat#(activate(N)))
6: U71#(tt(),M,N) ->
c_11(isNat#(activate(N)))
7: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{3}
These cover all (indirect) predecessors of dependency pairs
{3,4,5,6}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1},
uargs(c_11) = {1},
uargs(c_19) = {1,2},
uargs(c_20) = {1},
uargs(c_21) = {1,2}
Following symbols are considered usable:
{0,activate,plus,s,x,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) = [1]
p(U11) = [1]
p(U12) = [2]
p(U21) = [1] x1 + [4]
p(U31) = [2] x2 + [0]
p(U32) = [2] x1 + [4]
p(U41) = [1] x1 + [0]
p(U51) = [1] x2 + [1] x3 + [2]
p(U52) = [1] x2 + [1]
p(U61) = [4] x1 + [1]
p(U71) = [0]
p(U72) = [1] x1 + [0]
p(activate) = [1] x1 + [0]
p(isNat) = [4]
p(n__0) = [1]
p(n__plus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [1]
p(n__x) = [1] x1 + [1] x2 + [2]
p(plus) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [2]
p(0#) = [0]
p(U11#) = [4] x2 + [0]
p(U12#) = [4] x1 + [4]
p(U21#) = [4]
p(U31#) = [4] x2 + [4]
p(U32#) = [1] x1 + [1]
p(U41#) = [0]
p(U51#) = [1] x1 + [4] x3 + [6]
p(U52#) = [2]
p(U61#) = [1] x1 + [4]
p(U71#) = [2] x1 + [4] x3 + [1]
p(U72#) = [4] x2 + [2]
p(activate#) = [2] x1 + [0]
p(isNat#) = [4] x1 + [0]
p(plus#) = [4]
p(s#) = [1] x1 + [0]
p(x#) = [1] x1 + [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [4]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [4]
p(c_9) = [1] x1 + [1]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x2 + [4]
p(c_13) = [2]
p(c_14) = [1] x1 + [1]
p(c_15) = [4] x1 + [1]
p(c_16) = [0]
p(c_17) = [1] x1 + [0]
p(c_18) = [0]
p(c_19) = [1] x1 + [1] x2 + [4]
p(c_20) = [1] x1 + [4]
p(c_21) = [1] x1 + [1] x2 + [0]
p(c_22) = [0]
p(c_23) = [4]
p(c_24) = [1]
Following rules are strictly oriented:
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8]
> [4] V1 + [4] V2 + [4]
= c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= c_2(isNat#(activate(V2)))
U31#(tt(),V2) = [4] V2 + [4]
>= [4] V2 + [4]
= c_5(isNat#(activate(V2)))
U51#(tt(),M,N) = [4] N + [6]
>= [4] N + [4]
= c_8(isNat#(activate(N)))
U71#(tt(),M,N) = [4] N + [1]
>= [4] N + [0]
= c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [4]
>= [4] V1 + [4] V2 + [4]
= c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [4]
= c_20(isNat#(activate(V1)))
0() = [1]
>= [1]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [1]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= x(activate(X1),activate(X2))
plus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
Consider the set of all dependency pairs
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2)))
2: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
3: U31#(tt(),V2) ->
c_5(isNat#(activate(V2)))
4: U51#(tt(),M,N) ->
c_8(isNat#(activate(N)))
5: U71#(tt(),M,N) ->
c_11(isNat#(activate(N)))
6: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1)))
7: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{1,2,4,5}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1},
uargs(c_11) = {1},
uargs(c_19) = {1,2},
uargs(c_20) = {1},
uargs(c_21) = {1,2}
Following symbols are considered usable:
{0,activate,plus,s,x,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) = [0]
p(U12) = [0]
p(U21) = [0]
p(U31) = [0]
p(U32) = [1] x1 + [0]
p(U41) = [0]
p(U51) = [0]
p(U52) = [0]
p(U61) = [0]
p(U71) = [0]
p(U72) = [0]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(n__0) = [0]
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) = [0]
p(x) = [1] x1 + [1] x2 + [3]
p(0#) = [0]
p(U11#) = [4] x2 + [7]
p(U12#) = [0]
p(U21#) = [0]
p(U31#) = [4] x2 + [1]
p(U32#) = [0]
p(U41#) = [0]
p(U51#) = [4] x3 + [2]
p(U52#) = [0]
p(U61#) = [0]
p(U71#) = [4] x3 + [7]
p(U72#) = [0]
p(activate#) = [0]
p(isNat#) = [4] x1 + [1]
p(plus#) = [0]
p(s#) = [0]
p(x#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [6]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [1]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [6]
p(c_12) = [1] x2 + [0]
p(c_13) = [0]
p(c_14) = [4] x1 + [0]
p(c_15) = [1] x1 + [0]
p(c_16) = [0]
p(c_17) = [1] x2 + [0]
p(c_18) = [0]
p(c_19) = [1] x1 + [1] x2 + [1]
p(c_20) = [1] x1 + [1]
p(c_21) = [1] x1 + [1] x2 + [5]
p(c_22) = [0]
p(c_23) = [0]
p(c_24) = [0]
Following rules are strictly oriented:
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [13]
> [4] V1 + [4] V2 + [9]
= c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [4] V2 + [7]
>= [4] V2 + [7]
= c_2(isNat#(activate(V2)))
U31#(tt(),V2) = [4] V2 + [1]
>= [4] V2 + [1]
= c_5(isNat#(activate(V2)))
U51#(tt(),M,N) = [4] N + [2]
>= [4] N + [2]
= c_8(isNat#(activate(N)))
U71#(tt(),M,N) = [4] N + [7]
>= [4] N + [7]
= c_11(isNat#(activate(N)))
isNat#(n__s(V1)) = [4] V1 + [9]
>= [4] V1 + [2]
= c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [13]
>= [4] V1 + [4] V2 + [7]
= c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
0() = [0]
>= [0]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= x(activate(X1),activate(X2))
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)
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:U11#(tt(),V2) -> c_2(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
2:W:U31#(tt(),V2) -> c_5(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
3:W:U51#(tt(),M,N) -> c_8(isNat#(activate(N)))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
4:W:U71#(tt(),M,N) -> c_11(isNat#(activate(N)))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
5:W:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
-->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2))):1
6:W:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
7:W:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1))):6
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
-->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: U71#(tt(),M,N) ->
c_11(isNat#(activate(N)))
3: U51#(tt(),M,N) ->
c_8(isNat#(activate(N)))
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2)))
5: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
7: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
6: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1)))
2: U31#(tt(),V2) ->
c_5(isNat#(activate(V2)))
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.1 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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/1,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/2,c_20/1,c_21/2,c_22/0,c_23/0,c_24/0}
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).
*** 1.1.1.1.1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: activate#(n__s(X)) ->
c_16(activate#(X))
Consider the set of all dependency pairs
1: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
2: activate#(n__s(X)) ->
c_16(activate#(X))
3: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
4: U11#(tt(),V2) -> activate#(V2)
5: U11#(tt(),V2) ->
isNat#(activate(V2))
6: U31#(tt(),V2) -> activate#(V2)
7: U31#(tt(),V2) ->
isNat#(activate(V2))
8: U51#(tt(),M,N) ->
U52#(isNat(activate(N))
,activate(M)
,activate(N))
9: U51#(tt(),M,N) -> activate#(M)
10: U51#(tt(),M,N) -> activate#(N)
11: U51#(tt(),M,N) ->
isNat#(activate(N))
12: U52#(tt(),M,N) -> activate#(M)
13: U52#(tt(),M,N) -> activate#(N)
14: U71#(tt(),M,N) ->
U72#(isNat(activate(N))
,activate(M)
,activate(N))
15: U71#(tt(),M,N) -> activate#(M)
16: U71#(tt(),M,N) -> activate#(N)
17: U71#(tt(),M,N) ->
isNat#(activate(N))
18: U72#(tt(),M,N) -> activate#(M)
19: U72#(tt(),M,N) -> activate#(N)
20: isNat#(n__plus(V1,V2)) ->
U11#(isNat(activate(V1))
,activate(V2))
21: isNat#(n__plus(V1,V2)) ->
activate#(V1)
22: isNat#(n__plus(V1,V2)) ->
activate#(V2)
23: isNat#(n__plus(V1,V2)) ->
isNat#(activate(V1))
24: isNat#(n__s(V1)) ->
activate#(V1)
25: isNat#(n__s(V1)) ->
isNat#(activate(V1))
26: isNat#(n__x(V1,V2)) ->
U31#(isNat(activate(V1))
,activate(V2))
27: isNat#(n__x(V1,V2)) ->
activate#(V1)
28: isNat#(n__x(V1,V2)) ->
activate#(V2)
29: isNat#(n__x(V1,V2)) ->
isNat#(activate(V1))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{2,8,9,10,11,12,13,14,15,16,17,18,19}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_15) = {1,2},
uargs(c_16) = {1},
uargs(c_17) = {1,2}
Following symbols are considered usable:
{0,activate,plus,s,x,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) = [0]
p(U12) = [0]
p(U21) = [0]
p(U31) = [0]
p(U32) = [0]
p(U41) = [0]
p(U51) = [0]
p(U52) = [0]
p(U61) = [4]
p(U71) = [4] x1 + [0]
p(U72) = [1] x1 + [4]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(n__x) = [1] x1 + [1] x2 + [0]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [0]
p(0#) = [0]
p(U11#) = [4] x2 + [0]
p(U12#) = [0]
p(U21#) = [1] x1 + [4]
p(U31#) = [4] x2 + [0]
p(U32#) = [1] x1 + [0]
p(U41#) = [0]
p(U51#) = [4] x1 + [3] x2 + [4] x3 + [2]
p(U52#) = [3] x2 + [4] x3 + [2]
p(U61#) = [4] x1 + [0]
p(U71#) = [4] x1 + [4] x2 + [7] x3 + [0]
p(U72#) = [4] x2 + [2] x3 + [0]
p(activate#) = [2] x1 + [0]
p(isNat#) = [4] x1 + [0]
p(plus#) = [1] x1 + [4] x2 + [1]
p(s#) = [1]
p(x#) = [1] x1 + [2] x2 + [0]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [1]
p(c_8) = [1] x1 + [1] x3 + [0]
p(c_9) = [1] x1 + [2] x2 + [1]
p(c_10) = [4] x1 + [0]
p(c_11) = [1] x1 + [2] x3 + [2] x5 + [1]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [4] x1 + [1]
p(c_15) = [1] x1 + [1] x2 + [0]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [1] x2 + [0]
p(c_18) = [1]
p(c_19) = [4]
p(c_20) = [1] x1 + [0]
p(c_21) = [4] x1 + [1] x2 + [0]
p(c_22) = [0]
p(c_23) = [0]
p(c_24) = [1]
Following rules are strictly oriented:
activate#(n__s(X)) = [2] X + [2]
> [2] X + [0]
= c_16(activate#(X))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [4] V2 + [0]
>= [2] V2 + [0]
= activate#(V2)
U11#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNat#(activate(V2))
U31#(tt(),V2) = [4] V2 + [0]
>= [2] V2 + [0]
= activate#(V2)
U31#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNat#(activate(V2))
U51#(tt(),M,N) = [3] M + [4] N + [2]
>= [3] M + [4] N + [2]
= U52#(isNat(activate(N))
,activate(M)
,activate(N))
U51#(tt(),M,N) = [3] M + [4] N + [2]
>= [2] M + [0]
= activate#(M)
U51#(tt(),M,N) = [3] M + [4] N + [2]
>= [2] N + [0]
= activate#(N)
U51#(tt(),M,N) = [3] M + [4] N + [2]
>= [4] N + [0]
= isNat#(activate(N))
U52#(tt(),M,N) = [3] M + [4] N + [2]
>= [2] M + [0]
= activate#(M)
U52#(tt(),M,N) = [3] M + [4] N + [2]
>= [2] N + [0]
= activate#(N)
U71#(tt(),M,N) = [4] M + [7] N + [0]
>= [4] M + [2] N + [0]
= U72#(isNat(activate(N))
,activate(M)
,activate(N))
U71#(tt(),M,N) = [4] M + [7] N + [0]
>= [2] M + [0]
= activate#(M)
U71#(tt(),M,N) = [4] M + [7] N + [0]
>= [2] N + [0]
= activate#(N)
U71#(tt(),M,N) = [4] M + [7] N + [0]
>= [4] N + [0]
= isNat#(activate(N))
U72#(tt(),M,N) = [4] M + [2] N + [0]
>= [2] M + [0]
= activate#(M)
U72#(tt(),M,N) = [4] M + [2] N + [0]
>= [2] N + [0]
= activate#(N)
activate#(n__plus(X1,X2)) = [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
= c_15(activate#(X1)
,activate#(X2))
activate#(n__x(X1,X2)) = [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
= c_17(activate#(X1)
,activate#(X2))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= U11#(isNat(activate(V1))
,activate(V2))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [2] V2 + [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= isNat#(activate(V1))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= U31#(isNat(activate(V1))
,activate(V2))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [2] V2 + [0]
= activate#(V2)
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
0() = [0]
>= [0]
= n__0()
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(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= x(activate(X1),activate(X2))
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_16(activate#(X))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_16(activate#(X))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
Consider the set of all dependency pairs
1: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
2: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
3: U11#(tt(),V2) -> activate#(V2)
4: U11#(tt(),V2) ->
isNat#(activate(V2))
5: U31#(tt(),V2) -> activate#(V2)
6: U31#(tt(),V2) ->
isNat#(activate(V2))
7: U51#(tt(),M,N) ->
U52#(isNat(activate(N))
,activate(M)
,activate(N))
8: U51#(tt(),M,N) -> activate#(M)
9: U51#(tt(),M,N) -> activate#(N)
10: U51#(tt(),M,N) ->
isNat#(activate(N))
11: U52#(tt(),M,N) -> activate#(M)
12: U52#(tt(),M,N) -> activate#(N)
13: U71#(tt(),M,N) ->
U72#(isNat(activate(N))
,activate(M)
,activate(N))
14: U71#(tt(),M,N) -> activate#(M)
15: U71#(tt(),M,N) -> activate#(N)
16: U71#(tt(),M,N) ->
isNat#(activate(N))
17: U72#(tt(),M,N) -> activate#(M)
18: U72#(tt(),M,N) -> activate#(N)
19: activate#(n__s(X)) ->
c_16(activate#(X))
20: isNat#(n__plus(V1,V2)) ->
U11#(isNat(activate(V1))
,activate(V2))
21: isNat#(n__plus(V1,V2)) ->
activate#(V1)
22: isNat#(n__plus(V1,V2)) ->
activate#(V2)
23: isNat#(n__plus(V1,V2)) ->
isNat#(activate(V1))
24: isNat#(n__s(V1)) ->
activate#(V1)
25: isNat#(n__s(V1)) ->
isNat#(activate(V1))
26: isNat#(n__x(V1,V2)) ->
U31#(isNat(activate(V1))
,activate(V2))
27: isNat#(n__x(V1,V2)) ->
activate#(V1)
28: isNat#(n__x(V1,V2)) ->
activate#(V2)
29: isNat#(n__x(V1,V2)) ->
isNat#(activate(V1))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{2,7,8,9,10,11,12,13,14,15,16,17,18}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_16(activate#(X))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_15) = {1,2},
uargs(c_16) = {1},
uargs(c_17) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,activate,isNat,plus,s,x,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) = [0]
p(U12) = [0]
p(U21) = [0]
p(U31) = [2]
p(U32) = [1]
p(U41) = [0]
p(U51) = [0]
p(U52) = [0]
p(U61) = [0]
p(U71) = [0]
p(U72) = [2] x3 + [0]
p(activate) = [1] x1 + [0]
p(isNat) = [2]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(n__x) = [1] x1 + [1] x2 + [2]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [2]
p(0#) = [0]
p(U11#) = [4] x2 + [0]
p(U12#) = [0]
p(U21#) = [1] x1 + [0]
p(U31#) = [4] x1 + [4] x2 + [0]
p(U32#) = [0]
p(U41#) = [0]
p(U51#) = [6] x2 + [4] x3 + [0]
p(U52#) = [6] x2 + [4] x3 + [0]
p(U61#) = [0]
p(U71#) = [4] x2 + [4] x3 + [2]
p(U72#) = [1] x1 + [4] x2 + [4] x3 + [0]
p(activate#) = [4] x1 + [0]
p(isNat#) = [4] x1 + [0]
p(plus#) = [2] x1 + [1]
p(s#) = [1]
p(x#) = [1] x1 + [1]
p(c_1) = [0]
p(c_2) = [2] x2 + [0]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [4] x4 + [1] x5 + [1]
p(c_9) = [1]
p(c_10) = [2]
p(c_11) = [1] x5 + [4]
p(c_12) = [1] x2 + [4] x3 + [0]
p(c_13) = [0]
p(c_14) = [1] x1 + [2]
p(c_15) = [1] x1 + [1] x2 + [0]
p(c_16) = [1] x1 + [2]
p(c_17) = [1] x1 + [1] x2 + [3]
p(c_18) = [4]
p(c_19) = [1]
p(c_20) = [2] x1 + [1] x2 + [1]
p(c_21) = [1] x2 + [1] x3 + [4] x4 + [2]
p(c_22) = [1]
p(c_23) = [1]
p(c_24) = [0]
Following rules are strictly oriented:
activate#(n__x(X1,X2)) = [4] X1 + [4] X2 + [8]
> [4] X1 + [4] X2 + [3]
= c_17(activate#(X1)
,activate#(X2))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U11#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNat#(activate(V2))
U31#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U31#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNat#(activate(V2))
U51#(tt(),M,N) = [6] M + [4] N + [0]
>= [6] M + [4] N + [0]
= U52#(isNat(activate(N))
,activate(M)
,activate(N))
U51#(tt(),M,N) = [6] M + [4] N + [0]
>= [4] M + [0]
= activate#(M)
U51#(tt(),M,N) = [6] M + [4] N + [0]
>= [4] N + [0]
= activate#(N)
U51#(tt(),M,N) = [6] M + [4] N + [0]
>= [4] N + [0]
= isNat#(activate(N))
U52#(tt(),M,N) = [6] M + [4] N + [0]
>= [4] M + [0]
= activate#(M)
U52#(tt(),M,N) = [6] M + [4] N + [0]
>= [4] N + [0]
= activate#(N)
U71#(tt(),M,N) = [4] M + [4] N + [2]
>= [4] M + [4] N + [2]
= U72#(isNat(activate(N))
,activate(M)
,activate(N))
U71#(tt(),M,N) = [4] M + [4] N + [2]
>= [4] M + [0]
= activate#(M)
U71#(tt(),M,N) = [4] M + [4] N + [2]
>= [4] N + [0]
= activate#(N)
U71#(tt(),M,N) = [4] M + [4] N + [2]
>= [4] N + [0]
= isNat#(activate(N))
U72#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] M + [0]
= activate#(M)
U72#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] N + [0]
= activate#(N)
activate#(n__plus(X1,X2)) = [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
= c_15(activate#(X1)
,activate#(X2))
activate#(n__s(X)) = [4] X + [4]
>= [4] X + [2]
= c_16(activate#(X))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= U11#(isNat(activate(V1))
,activate(V2))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= activate#(V1)
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= isNat#(activate(V1))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V2 + [8]
= U31#(isNat(activate(V1))
,activate(V2))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= activate#(V1)
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V2 + [0]
= activate#(V2)
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= isNat#(activate(V1))
0() = [0]
>= [0]
= n__0()
U11(tt(),V2) = [0]
>= [0]
= U12(isNat(activate(V2)))
U12(tt()) = [0]
>= [0]
= tt()
U21(tt()) = [0]
>= [0]
= tt()
U31(tt(),V2) = [2]
>= [1]
= U32(isNat(activate(V2)))
U32(tt()) = [1]
>= [0]
= tt()
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(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= x(activate(X1),activate(X2))
isNat(n__0()) = [2]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [2]
>= [0]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [2]
>= [0]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [2]
>= [2]
= 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 + [1]
>= [1] X + [1]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
Consider the set of all dependency pairs
1: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
2: U11#(tt(),V2) -> activate#(V2)
3: U11#(tt(),V2) ->
isNat#(activate(V2))
4: U31#(tt(),V2) -> activate#(V2)
5: U31#(tt(),V2) ->
isNat#(activate(V2))
6: U51#(tt(),M,N) ->
U52#(isNat(activate(N))
,activate(M)
,activate(N))
7: U51#(tt(),M,N) -> activate#(M)
8: U51#(tt(),M,N) -> activate#(N)
9: U51#(tt(),M,N) ->
isNat#(activate(N))
10: U52#(tt(),M,N) -> activate#(M)
11: U52#(tt(),M,N) -> activate#(N)
12: U71#(tt(),M,N) ->
U72#(isNat(activate(N))
,activate(M)
,activate(N))
13: U71#(tt(),M,N) -> activate#(M)
14: U71#(tt(),M,N) -> activate#(N)
15: U71#(tt(),M,N) ->
isNat#(activate(N))
16: U72#(tt(),M,N) -> activate#(M)
17: U72#(tt(),M,N) -> activate#(N)
18: activate#(n__s(X)) ->
c_16(activate#(X))
19: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
20: isNat#(n__plus(V1,V2)) ->
U11#(isNat(activate(V1))
,activate(V2))
21: isNat#(n__plus(V1,V2)) ->
activate#(V1)
22: isNat#(n__plus(V1,V2)) ->
activate#(V2)
23: isNat#(n__plus(V1,V2)) ->
isNat#(activate(V1))
24: isNat#(n__s(V1)) ->
activate#(V1)
25: isNat#(n__s(V1)) ->
isNat#(activate(V1))
26: isNat#(n__x(V1,V2)) ->
U31#(isNat(activate(V1))
,activate(V2))
27: isNat#(n__x(V1,V2)) ->
activate#(V1)
28: isNat#(n__x(V1,V2)) ->
activate#(V2)
29: isNat#(n__x(V1,V2)) ->
isNat#(activate(V1))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,6,7,8,9,10,11,12,13,14,15,16,17}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.2.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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 first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_15) = {1,2},
uargs(c_16) = {1},
uargs(c_17) = {1,2}
Following symbols are considered usable:
{0,U11,U12,U21,U31,U32,activate,isNat,plus,s,x,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) = [7]
p(U11) = [1] x2 + [7]
p(U12) = [1] x1 + [0]
p(U21) = [1] x1 + [4]
p(U31) = [1] x2 + [2]
p(U32) = [2]
p(U41) = [1]
p(U51) = [1] x2 + [1] x3 + [1]
p(U52) = [1] x3 + [0]
p(U61) = [0]
p(U71) = [1] x2 + [1] x3 + [0]
p(U72) = [1] x2 + [0]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [4]
p(n__0) = [7]
p(n__plus) = [1] x1 + [1] x2 + [4]
p(n__s) = [1] x1 + [4]
p(n__x) = [1] x1 + [1] x2 + [1]
p(plus) = [1] x1 + [1] x2 + [4]
p(s) = [1] x1 + [4]
p(tt) = [2]
p(x) = [1] x1 + [1] x2 + [1]
p(0#) = [4]
p(U11#) = [2] x2 + [4]
p(U12#) = [4] x1 + [1]
p(U21#) = [4] x1 + [0]
p(U31#) = [1] x1 + [2] x2 + [0]
p(U32#) = [1] x1 + [1]
p(U41#) = [4]
p(U51#) = [4] x1 + [2] x2 + [4] x3 + [0]
p(U52#) = [1] x1 + [2] x2 + [2] x3 + [4]
p(U61#) = [4]
p(U71#) = [4] x1 + [2] x2 + [6] x3 + [0]
p(U72#) = [2] x1 + [2] x2 + [4] x3 + [0]
p(activate#) = [2] x1 + [0]
p(isNat#) = [2] x1 + [2]
p(plus#) = [1]
p(s#) = [1]
p(x#) = [2] x2 + [2]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [1]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [4] x1 + [1]
p(c_6) = [1]
p(c_7) = [4] x1 + [0]
p(c_8) = [4] x2 + [1] x3 + [1] x5 + [1]
p(c_9) = [1]
p(c_10) = [0]
p(c_11) = [4] x3 + [1]
p(c_12) = [0]
p(c_13) = [4]
p(c_14) = [2]
p(c_15) = [1] x1 + [1] x2 + [4]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [1] x2 + [2]
p(c_18) = [0]
p(c_19) = [1] x1 + [2] x2 + [0]
p(c_20) = [2] x2 + [1]
p(c_21) = [1] x1 + [1] x3 + [1] x4 + [0]
p(c_22) = [0]
p(c_23) = [0]
p(c_24) = [0]
Following rules are strictly oriented:
activate#(n__plus(X1,X2)) = [2] X1 + [2] X2 + [8]
> [2] X1 + [2] X2 + [4]
= c_15(activate#(X1)
,activate#(X2))
Following rules are (at-least) weakly oriented:
U11#(tt(),V2) = [2] V2 + [4]
>= [2] V2 + [0]
= activate#(V2)
U11#(tt(),V2) = [2] V2 + [4]
>= [2] V2 + [2]
= isNat#(activate(V2))
U31#(tt(),V2) = [2] V2 + [2]
>= [2] V2 + [0]
= activate#(V2)
U31#(tt(),V2) = [2] V2 + [2]
>= [2] V2 + [2]
= isNat#(activate(V2))
U51#(tt(),M,N) = [2] M + [4] N + [8]
>= [2] M + [3] N + [8]
= U52#(isNat(activate(N))
,activate(M)
,activate(N))
U51#(tt(),M,N) = [2] M + [4] N + [8]
>= [2] M + [0]
= activate#(M)
U51#(tt(),M,N) = [2] M + [4] N + [8]
>= [2] N + [0]
= activate#(N)
U51#(tt(),M,N) = [2] M + [4] N + [8]
>= [2] N + [2]
= isNat#(activate(N))
U52#(tt(),M,N) = [2] M + [2] N + [6]
>= [2] M + [0]
= activate#(M)
U52#(tt(),M,N) = [2] M + [2] N + [6]
>= [2] N + [0]
= activate#(N)
U71#(tt(),M,N) = [2] M + [6] N + [8]
>= [2] M + [6] N + [8]
= U72#(isNat(activate(N))
,activate(M)
,activate(N))
U71#(tt(),M,N) = [2] M + [6] N + [8]
>= [2] M + [0]
= activate#(M)
U71#(tt(),M,N) = [2] M + [6] N + [8]
>= [2] N + [0]
= activate#(N)
U71#(tt(),M,N) = [2] M + [6] N + [8]
>= [2] N + [2]
= isNat#(activate(N))
U72#(tt(),M,N) = [2] M + [4] N + [4]
>= [2] M + [0]
= activate#(M)
U72#(tt(),M,N) = [2] M + [4] N + [4]
>= [2] N + [0]
= activate#(N)
activate#(n__s(X)) = [2] X + [8]
>= [2] X + [0]
= c_16(activate#(X))
activate#(n__x(X1,X2)) = [2] X1 + [2] X2 + [2]
>= [2] X1 + [2] X2 + [2]
= c_17(activate#(X1)
,activate#(X2))
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [10]
>= [2] V2 + [4]
= U11#(isNat(activate(V1))
,activate(V2))
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [10]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [10]
>= [2] V2 + [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [10]
>= [2] V1 + [2]
= isNat#(activate(V1))
isNat#(n__s(V1)) = [2] V1 + [10]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__s(V1)) = [2] V1 + [10]
>= [2] V1 + [2]
= isNat#(activate(V1))
isNat#(n__x(V1,V2)) = [2] V1 + [2] V2 + [4]
>= [1] V1 + [2] V2 + [4]
= U31#(isNat(activate(V1))
,activate(V2))
isNat#(n__x(V1,V2)) = [2] V1 + [2] V2 + [4]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__x(V1,V2)) = [2] V1 + [2] V2 + [4]
>= [2] V2 + [0]
= activate#(V2)
isNat#(n__x(V1,V2)) = [2] V1 + [2] V2 + [4]
>= [2] V1 + [2]
= isNat#(activate(V1))
0() = [7]
>= [7]
= n__0()
U11(tt(),V2) = [1] V2 + [7]
>= [1] V2 + [4]
= U12(isNat(activate(V2)))
U12(tt()) = [2]
>= [2]
= tt()
U21(tt()) = [6]
>= [2]
= tt()
U31(tt(),V2) = [1] V2 + [2]
>= [2]
= U32(isNat(activate(V2)))
U32(tt()) = [2]
>= [2]
= tt()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [7]
>= [7]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [4]
>= [1] X + [4]
= s(activate(X))
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= x(activate(X1),activate(X2))
isNat(n__0()) = [11]
>= [2]
= tt()
isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [8]
>= [1] V2 + [7]
= U11(isNat(activate(V1))
,activate(V2))
isNat(n__s(V1)) = [1] V1 + [8]
>= [1] V1 + [8]
= U21(isNat(activate(V1)))
isNat(n__x(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V2 + [2]
= U31(isNat(activate(V1))
,activate(V2))
plus(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= n__plus(X1,X2)
s(X) = [1] X + [4]
>= [1] X + [4]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.2.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> activate#(V2)
U11#(tt(),V2) -> isNat#(activate(V2))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNat#(activate(V2))
U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
U51#(tt(),M,N) -> activate#(M)
U51#(tt(),M,N) -> activate#(N)
U51#(tt(),M,N) -> isNat#(activate(N))
U52#(tt(),M,N) -> activate#(M)
U52#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
U71#(tt(),M,N) -> activate#(M)
U71#(tt(),M,N) -> activate#(N)
U71#(tt(),M,N) -> isNat#(activate(N))
U72#(tt(),M,N) -> activate#(M)
U72#(tt(),M,N) -> activate#(N)
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNat#(activate(V1))
isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
isNat#(n__x(V1,V2)) -> activate#(V1)
isNat#(n__x(V1,V2)) -> activate#(V2)
isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:U11#(tt(),V2) -> activate#(V2)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
2:W:U11#(tt(),V2) -> isNat#(activate(V2))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
3:W:U31#(tt(),V2) -> activate#(V2)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
4:W:U31#(tt(),V2) -> isNat#(activate(V2))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
5:W:U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
-->_1 U52#(tt(),M,N) -> activate#(N):10
-->_1 U52#(tt(),M,N) -> activate#(M):9
6:W:U51#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
7:W:U51#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
8:W:U51#(tt(),M,N) -> isNat#(activate(N))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
9:W:U52#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
10:W:U52#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
11:W:U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
-->_1 U72#(tt(),M,N) -> activate#(N):16
-->_1 U72#(tt(),M,N) -> activate#(M):15
12:W:U71#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
13:W:U71#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
14:W:U71#(tt(),M,N) -> isNat#(activate(N))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
15:W:U72#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
16:W:U72#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
17:W:activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
18:W:activate#(n__s(X)) -> c_16(activate#(X))
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
19:W:activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
20:W:isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
-->_1 U11#(tt(),V2) -> isNat#(activate(V2)):2
-->_1 U11#(tt(),V2) -> activate#(V2):1
21:W:isNat#(n__plus(V1,V2)) -> activate#(V1)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
22:W:isNat#(n__plus(V1,V2)) -> activate#(V2)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
23:W:isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
24:W:isNat#(n__s(V1)) -> activate#(V1)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
25:W:isNat#(n__s(V1)) -> isNat#(activate(V1))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
26:W:isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
-->_1 U31#(tt(),V2) -> isNat#(activate(V2)):4
-->_1 U31#(tt(),V2) -> activate#(V2):3
27:W:isNat#(n__x(V1,V2)) -> activate#(V1)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
28:W:isNat#(n__x(V1,V2)) -> activate#(V2)
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):19
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):18
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):17
29:W:isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
-->_1 isNat#(n__x(V1,V2)) -> isNat#(activate(V1)):29
-->_1 isNat#(n__x(V1,V2)) -> activate#(V2):28
-->_1 isNat#(n__x(V1,V2)) -> activate#(V1):27
-->_1 isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)):26
-->_1 isNat#(n__s(V1)) -> isNat#(activate(V1)):25
-->_1 isNat#(n__s(V1)) -> activate#(V1):24
-->_1 isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)):23
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):21
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)):20
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
14: U71#(tt(),M,N) ->
isNat#(activate(N))
13: U71#(tt(),M,N) -> activate#(N)
12: U71#(tt(),M,N) -> activate#(M)
11: U71#(tt(),M,N) ->
U72#(isNat(activate(N))
,activate(M)
,activate(N))
15: U72#(tt(),M,N) -> activate#(M)
16: U72#(tt(),M,N) -> activate#(N)
8: U51#(tt(),M,N) ->
isNat#(activate(N))
7: U51#(tt(),M,N) -> activate#(N)
6: U51#(tt(),M,N) -> activate#(M)
5: U51#(tt(),M,N) ->
U52#(isNat(activate(N))
,activate(M)
,activate(N))
9: U52#(tt(),M,N) -> activate#(M)
10: U52#(tt(),M,N) -> activate#(N)
2: U11#(tt(),V2) ->
isNat#(activate(V2))
20: isNat#(n__plus(V1,V2)) ->
U11#(isNat(activate(V1))
,activate(V2))
29: isNat#(n__x(V1,V2)) ->
isNat#(activate(V1))
25: isNat#(n__s(V1)) ->
isNat#(activate(V1))
23: isNat#(n__plus(V1,V2)) ->
isNat#(activate(V1))
4: U31#(tt(),V2) ->
isNat#(activate(V2))
26: isNat#(n__x(V1,V2)) ->
U31#(isNat(activate(V1))
,activate(V2))
3: U31#(tt(),V2) -> activate#(V2)
21: isNat#(n__plus(V1,V2)) ->
activate#(V1)
22: isNat#(n__plus(V1,V2)) ->
activate#(V2)
24: isNat#(n__s(V1)) ->
activate#(V1)
27: isNat#(n__x(V1,V2)) ->
activate#(V1)
28: isNat#(n__x(V1,V2)) ->
activate#(V2)
1: U11#(tt(),V2) -> activate#(V2)
19: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
18: activate#(n__s(X)) ->
c_16(activate#(X))
17: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
*** 1.1.1.1.1.1.1.1.1.1.2.2.2.2.1 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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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).
*** 1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,4}
by application of
Pre({2,4}) = {1,3}.
Here rules are labelled as follows:
1: U51#(tt(),M,N) ->
c_8(U52#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
2: U52#(tt(),M,N) ->
c_9(activate#(N),activate#(M))
3: U71#(tt(),M,N) ->
c_11(U72#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
4: U72#(tt(),M,N) ->
c_12(activate#(N)
,activate#(M)
,activate#(N))
5: U11#(tt(),V2) ->
c_2(isNat#(activate(V2))
,activate#(V2))
6: U31#(tt(),V2) ->
c_5(isNat#(activate(V2))
,activate#(V2))
7: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
8: activate#(n__s(X)) ->
c_16(activate#(X))
9: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
10: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
11: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1))
,activate#(V1))
12: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
*** 1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,2}
by application of
Pre({1,2}) = {}.
Here rules are labelled as follows:
1: U51#(tt(),M,N) ->
c_8(U52#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
2: U71#(tt(),M,N) ->
c_11(U72#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
3: U11#(tt(),V2) ->
c_2(isNat#(activate(V2))
,activate#(V2))
4: U31#(tt(),V2) ->
c_5(isNat#(activate(V2))
,activate#(V2))
5: U52#(tt(),M,N) ->
c_9(activate#(N),activate#(M))
6: U72#(tt(),M,N) ->
c_12(activate#(N)
,activate#(M)
,activate#(N))
7: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
8: activate#(n__s(X)) ->
c_16(activate#(X))
9: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
10: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
11: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1))
,activate#(V1))
12: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
*** 1.1.1.1.1.1.1.1.1.2.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_16(activate#(X))
activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
2:W:U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
3:W:U51#(tt(),M,N) -> c_8(U52#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_5 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_5 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_5 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 U52#(tt(),M,N) -> c_9(activate#(N),activate#(M)):4
4:W:U52#(tt(),M,N) -> c_9(activate#(N),activate#(M))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
5:W:U71#(tt(),M,N) -> c_11(U72#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_5 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_5 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_5 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N)):6
6:W:U72#(tt(),M,N) -> c_12(activate#(N),activate#(M),activate#(N))
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
7:W:activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
8:W:activate#(n__s(X)) -> c_16(activate#(X))
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
9:W:activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_1 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_1 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
10:W:isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 U11#(tt(),V2) -> c_2(isNat#(activate(V2)),activate#(V2)):1
11:W:isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_1 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_2 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_2 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_2 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
12:W:isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_21(U31#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):12
-->_2 isNat#(n__s(V1)) -> c_20(isNat#(activate(V1)),activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_19(U11#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):10
-->_4 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_3 activate#(n__x(X1,X2)) -> c_17(activate#(X1),activate#(X2)):9
-->_4 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_3 activate#(n__s(X)) -> c_16(activate#(X)):8
-->_4 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_3 activate#(n__plus(X1,X2)) -> c_15(activate#(X1),activate#(X2)):7
-->_1 U31#(tt(),V2) -> c_5(isNat#(activate(V2)),activate#(V2)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: U71#(tt(),M,N) ->
c_11(U72#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
6: U72#(tt(),M,N) ->
c_12(activate#(N)
,activate#(M)
,activate#(N))
3: U51#(tt(),M,N) ->
c_8(U52#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
4: U52#(tt(),M,N) ->
c_9(activate#(N),activate#(M))
1: U11#(tt(),V2) ->
c_2(isNat#(activate(V2))
,activate#(V2))
10: isNat#(n__plus(V1,V2)) ->
c_19(U11#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
12: isNat#(n__x(V1,V2)) ->
c_21(U31#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
11: isNat#(n__s(V1)) ->
c_20(isNat#(activate(V1))
,activate#(V1))
2: U31#(tt(),V2) ->
c_5(isNat#(activate(V2))
,activate#(V2))
9: activate#(n__x(X1,X2)) ->
c_17(activate#(X1)
,activate#(X2))
8: activate#(n__s(X)) ->
c_16(activate#(X))
7: activate#(n__plus(X1,X2)) ->
c_15(activate#(X1)
,activate#(X2))
*** 1.1.1.1.1.1.1.1.1.2.1.1.1 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()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(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)))
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/2,0#/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,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/5,c_9/2,c_10/1,c_11/5,c_12/3,c_13/0,c_14/1,c_15/2,c_16/1,c_17/2,c_18/0,c_19/4,c_20/2,c_21/4,c_22/0,c_23/0,c_24/0}
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).