The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).

0 CpxRelTRS
↳1 RelTrsToTrsProof (UPPER BOUND(ID), 0 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 8 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 103 ms)
↳10 CdtProblem
↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 63 ms)
↳14 CdtProblem
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Rewrite Strategy: INNERMOST

(1) RelTrsToTrsProof (UPPER BOUND(ID) transformation)

transformed relative TRS to TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

member(x', Cons(x, xs)) → member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) → False
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x, xs) → member(x, xs)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
member[Ite][True][Ite](False, x', Cons(x, xs)) → member(x', xs)
member[Ite][True][Ite](True, x, xs) → True

Rewrite Strategy: INNERMOST

(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

member(z0, Cons(z1, z2)) → member[Ite][True][Ite](!EQ(z0, z1), z0, Cons(z1, z2))
member(z0, Nil) → False
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → member(z0, z1)
!EQ(S(z0), S(z1)) → !EQ(z0, z1)
!EQ(0, S(z0)) → False
!EQ(S(z0), 0) → False
!EQ(0, 0) → True
member[Ite][True][Ite](False, z0, Cons(z1, z2)) → member(z0, z2)
member[Ite][True][Ite](True, z0, z1) → True
Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
MEMBER(z0, Nil) → c1
NOTEMPTY(Cons(z0, z1)) → c2
NOTEMPTY(Nil) → c3
GOAL(z0, z1) → c4(MEMBER(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
!EQ'(0, S(z0)) → c6
!EQ'(S(z0), 0) → c7
!EQ'(0, 0) → c8
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
MEMBER[ITE][TRUE][ITE](True, z0, z1) → c10
S tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
MEMBER(z0, Nil) → c1
NOTEMPTY(Cons(z0, z1)) → c2
NOTEMPTY(Nil) → c3
GOAL(z0, z1) → c4(MEMBER(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
!EQ'(0, S(z0)) → c6
!EQ'(S(z0), 0) → c7
!EQ'(0, 0) → c8
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
MEMBER[ITE][TRUE][ITE](True, z0, z1) → c10
K tuples:none
Defined Rule Symbols:

member, notEmpty, goal, !EQ, member[Ite][True][Ite]

Defined Pair Symbols:

MEMBER, NOTEMPTY, GOAL, !EQ', MEMBER[ITE][TRUE][ITE]

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0, z1) → c4(MEMBER(z0, z1))
Removed 7 trailing nodes:

!EQ'(0, 0) → c8
!EQ'(0, S(z0)) → c6
NOTEMPTY(Nil) → c3
MEMBER[ITE][TRUE][ITE](True, z0, z1) → c10
!EQ'(S(z0), 0) → c7
MEMBER(z0, Nil) → c1
NOTEMPTY(Cons(z0, z1)) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

member(z0, Cons(z1, z2)) → member[Ite][True][Ite](!EQ(z0, z1), z0, Cons(z1, z2))
member(z0, Nil) → False
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → member(z0, z1)
!EQ(S(z0), S(z1)) → !EQ(z0, z1)
!EQ(0, S(z0)) → False
!EQ(S(z0), 0) → False
!EQ(0, 0) → True
member[Ite][True][Ite](False, z0, Cons(z1, z2)) → member(z0, z2)
member[Ite][True][Ite](True, z0, z1) → True
Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
S tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
K tuples:none
Defined Rule Symbols:

member, notEmpty, goal, !EQ, member[Ite][True][Ite]

Defined Pair Symbols:

MEMBER, !EQ', MEMBER[ITE][TRUE][ITE]

Compound Symbols:

c, c5, c9

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

member(z0, Cons(z1, z2)) → member[Ite][True][Ite](!EQ(z0, z1), z0, Cons(z1, z2))
member(z0, Nil) → False
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → member(z0, z1)
member[Ite][True][Ite](False, z0, Cons(z1, z2)) → member(z0, z2)
member[Ite][True][Ite](True, z0, z1) → True

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

!EQ(S(z0), S(z1)) → !EQ(z0, z1)
!EQ(0, S(z0)) → False
!EQ(S(z0), 0) → False
!EQ(0, 0) → True
Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
S tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
K tuples:none
Defined Rule Symbols:

!EQ

Defined Pair Symbols:

MEMBER, !EQ', MEMBER[ITE][TRUE][ITE]

Compound Symbols:

c, c5, c9

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(!EQ(x1, x2)) = 0   
POL(!EQ'(x1, x2)) = 0   
POL(0) = 0   
POL(Cons(x1, x2)) = [1] + x2   
POL(False) = 0   
POL(MEMBER(x1, x2)) = [1] + x2   
POL(MEMBER[ITE][TRUE][ITE](x1, x2, x3)) = [1] + x3   
POL(S(x1)) = 0   
POL(True) = 0   
POL(c(x1, x2)) = x1 + x2   
POL(c5(x1)) = x1   
POL(c9(x1)) = x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

!EQ(S(z0), S(z1)) → !EQ(z0, z1)
!EQ(0, S(z0)) → False
!EQ(S(z0), 0) → False
!EQ(0, 0) → True
Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
S tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
K tuples:

MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
Defined Rule Symbols:

!EQ

Defined Pair Symbols:

MEMBER, !EQ', MEMBER[ITE][TRUE][ITE]

Compound Symbols:

c, c5, c9

(11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

!EQ(S(z0), S(z1)) → !EQ(z0, z1)
!EQ(0, S(z0)) → False
!EQ(S(z0), 0) → False
!EQ(0, 0) → True
Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
S tuples:

!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
K tuples:

MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
Defined Rule Symbols:

!EQ

Defined Pair Symbols:

MEMBER, !EQ', MEMBER[ITE][TRUE][ITE]

Compound Symbols:

c, c5, c9

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(!EQ(x1, x2)) = [2]x12   
POL(!EQ'(x1, x2)) = x2   
POL(0) = 0   
POL(Cons(x1, x2)) = [2] + x1 + x2   
POL(False) = 0   
POL(MEMBER(x1, x2)) = [1] + x1 + x2 + [2]x22 + [2]x1·x2   
POL(MEMBER[ITE][TRUE][ITE](x1, x2, x3)) = [2]x32 + [2]x2·x3   
POL(S(x1)) = [1] + x1   
POL(True) = [1]   
POL(c(x1, x2)) = x1 + x2   
POL(c5(x1)) = x1   
POL(c9(x1)) = x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

!EQ(S(z0), S(z1)) → !EQ(z0, z1)
!EQ(0, S(z0)) → False
!EQ(S(z0), 0) → False
!EQ(0, 0) → True
Tuples:

MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
S tuples:none
K tuples:

MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) → c9(MEMBER(z0, z2))
MEMBER(z0, Cons(z1, z2)) → c(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
!EQ'(S(z0), S(z1)) → c5(!EQ'(z0, z1))
Defined Rule Symbols:

!EQ

Defined Pair Symbols:

MEMBER, !EQ', MEMBER[ITE][TRUE][ITE]

Compound Symbols:

c, c5, c9

(15) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(16) BOUNDS(1, 1)