YES(O(1), O(n^1)) 6.34/2.06 YES(O(1), O(n^1)) 6.73/2.11 6.73/2.11 6.73/2.11 6.73/2.11 6.73/2.11 6.73/2.11 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.73/2.11 6.73/2.11 6.73/2.11
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6.73/2.11
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
6.73/2.11 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
6.73/2.11 
↳2 CdtProblem
6.73/2.11 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
6.73/2.11 
↳4 CdtProblem
6.73/2.11 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.73/2.11 
↳6 CdtProblem
6.73/2.11 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.73/2.11 
↳8 CdtProblem
6.73/2.11 
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
6.73/2.11 
↳10 CdtProblem
6.73/2.11 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.73/2.11 
↳12 CdtProblem
6.73/2.11 
↳13 CdtKnowledgeProof (⇔)
6.73/2.11 
↳14 BOUNDS(O(1), O(1))
6.73/2.11
6.73/2.11 6.73/2.11
6.73/2.11
6.73/2.11

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__and(true, X) → mark(X) 6.73/2.11
a__and(false, Y) → false 6.73/2.11
a__if(true, X, Y) → mark(X) 6.73/2.11
a__if(false, X, Y) → mark(Y) 6.73/2.11
a__add(0, X) → mark(X) 6.73/2.11
a__add(s(X), Y) → s(add(X, Y)) 6.73/2.11
a__first(0, X) → nil 6.73/2.11
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z)) 6.73/2.11
a__from(X) → cons(X, from(s(X))) 6.73/2.11
mark(and(X1, X2)) → a__and(mark(X1), X2) 6.73/2.11
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3) 6.73/2.11
mark(add(X1, X2)) → a__add(mark(X1), X2) 6.73/2.11
mark(first(X1, X2)) → a__first(mark(X1), mark(X2)) 6.73/2.11
mark(from(X)) → a__from(X) 6.73/2.11
mark(true) → true 6.73/2.11
mark(false) → false 6.73/2.11
mark(0) → 0 6.73/2.11
mark(s(X)) → s(X) 6.73/2.11
mark(nil) → nil 6.73/2.11
mark(cons(X1, X2)) → cons(X1, X2) 6.73/2.11
a__and(X1, X2) → and(X1, X2) 6.73/2.11
a__if(X1, X2, X3) → if(X1, X2, X3) 6.73/2.11
a__add(X1, X2) → add(X1, X2) 6.73/2.13
a__first(X1, X2) → first(X1, X2) 6.73/2.13
a__from(X) → from(X)

Rewrite Strategy: INNERMOST
6.73/2.13
6.73/2.13

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

Converted CpxTRS to CDT
6.73/2.13
6.73/2.13

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__and(true, z0) → mark(z0) 6.73/2.13
a__and(false, z0) → false 6.73/2.13
a__and(z0, z1) → and(z0, z1) 6.73/2.13
a__if(true, z0, z1) → mark(z0) 6.73/2.13
a__if(false, z0, z1) → mark(z1) 6.73/2.13
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.13
a__add(0, z0) → mark(z0) 6.73/2.13
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.13
a__add(z0, z1) → add(z0, z1) 6.73/2.13
a__first(0, z0) → nil 6.73/2.13
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.13
a__first(z0, z1) → first(z0, z1) 6.73/2.13
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.13
a__from(z0) → from(z0) 6.73/2.13
mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.13
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.13
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.13
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.13
mark(from(z0)) → a__from(z0) 6.73/2.13
mark(true) → true 6.73/2.13
mark(false) → false 6.73/2.13
mark(0) → 0 6.73/2.13
mark(s(z0)) → s(z0) 6.73/2.13
mark(nil) → nil 6.73/2.13
mark(cons(z0, z1)) → cons(z0, z1)
Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.13
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.13
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.13
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.13
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.13
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(first(z0, z1)) → c17(A__FIRST(mark(z0), mark(z1)), MARK(z0), MARK(z1)) 6.73/2.13
MARK(from(z0)) → c18(A__FROM(z0))
S tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.13
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.13
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.13
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.13
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.13
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(first(z0, z1)) → c17(A__FIRST(mark(z0), mark(z1)), MARK(z0), MARK(z1)) 6.73/2.13
MARK(from(z0)) → c18(A__FROM(z0))
K tuples:none
Defined Rule Symbols:

a__and, a__if, a__add, a__first, a__from, mark

Defined Pair Symbols:

A__AND, A__IF, A__ADD, MARK

Compound Symbols:

c, c3, c4, c6, c14, c15, c16, c17, c18

6.73/2.13
6.73/2.13

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

Removed 2 trailing tuple parts
6.73/2.13
6.73/2.13

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__and(true, z0) → mark(z0) 6.73/2.13
a__and(false, z0) → false 6.73/2.13
a__and(z0, z1) → and(z0, z1) 6.73/2.13
a__if(true, z0, z1) → mark(z0) 6.73/2.13
a__if(false, z0, z1) → mark(z1) 6.73/2.13
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.13
a__add(0, z0) → mark(z0) 6.73/2.13
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.13
a__add(z0, z1) → add(z0, z1) 6.73/2.13
a__first(0, z0) → nil 6.73/2.13
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.13
a__first(z0, z1) → first(z0, z1) 6.73/2.13
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.13
a__from(z0) → from(z0) 6.73/2.13
mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.13
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.13
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.13
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.13
mark(from(z0)) → a__from(z0) 6.73/2.13
mark(true) → true 6.73/2.13
mark(false) → false 6.73/2.13
mark(0) → 0 6.73/2.13
mark(s(z0)) → s(z0) 6.73/2.13
mark(nil) → nil 6.73/2.13
mark(cons(z0, z1)) → cons(z0, z1)
Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.13
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.13
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.13
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.13
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.13
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.13
MARK(from(z0)) → c18
S tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.13
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.13
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.13
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.13
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.13
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.13
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.13
MARK(from(z0)) → c18
K tuples:none
Defined Rule Symbols:

a__and, a__if, a__add, a__first, a__from, mark

Defined Pair Symbols:

A__AND, A__IF, A__ADD, MARK

Compound Symbols:

c, c3, c4, c6, c14, c15, c16, c17, c18

6.73/2.13
6.73/2.13

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MARK(from(z0)) → c18
6.73/2.15
6.73/2.15

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__and(true, z0) → mark(z0) 6.73/2.15
a__and(false, z0) → false 6.73/2.15
a__and(z0, z1) → and(z0, z1) 6.73/2.15
a__if(true, z0, z1) → mark(z0) 6.73/2.15
a__if(false, z0, z1) → mark(z1) 6.73/2.15
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.15
a__add(0, z0) → mark(z0) 6.73/2.15
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.15
a__add(z0, z1) → add(z0, z1) 6.73/2.15
a__first(0, z0) → nil 6.73/2.15
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.15
a__first(z0, z1) → first(z0, z1) 6.73/2.15
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.15
a__from(z0) → from(z0) 6.73/2.15
mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.15
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.15
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.15
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.15
mark(from(z0)) → a__from(z0) 6.73/2.15
mark(true) → true 6.73/2.15
mark(false) → false 6.73/2.15
mark(0) → 0 6.73/2.15
mark(s(z0)) → s(z0) 6.73/2.15
mark(nil) → nil 6.73/2.15
mark(cons(z0, z1)) → cons(z0, z1)
Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(from(z0)) → c18
S tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(from(z0)) → c18
K tuples:none
Defined Rule Symbols:

a__and, a__if, a__add, a__first, a__from, mark

Defined Pair Symbols:

A__AND, A__IF, A__ADD, MARK

Compound Symbols:

c, c3, c4, c6, c14, c15, c16, c17, c18

6.73/2.15
6.73/2.15

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1))
We considered the (Usable) Rules:

mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.15
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.15
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.15
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.15
mark(from(z0)) → a__from(z0) 6.73/2.15
mark(true) → true 6.73/2.15
mark(false) → false 6.73/2.15
mark(0) → 0 6.73/2.15
mark(s(z0)) → s(z0) 6.73/2.15
mark(nil) → nil 6.73/2.15
mark(cons(z0, z1)) → cons(z0, z1) 6.73/2.15
a__and(true, z0) → mark(z0) 6.73/2.15
a__and(false, z0) → false 6.73/2.15
a__and(z0, z1) → and(z0, z1) 6.73/2.15
a__if(true, z0, z1) → mark(z0) 6.73/2.15
a__if(false, z0, z1) → mark(z1) 6.73/2.15
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.15
a__add(0, z0) → mark(z0) 6.73/2.15
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.15
a__add(z0, z1) → add(z0, z1) 6.73/2.15
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.15
a__from(z0) → from(z0) 6.73/2.15
a__first(0, z0) → nil 6.73/2.15
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.15
a__first(z0, z1) → first(z0, z1)
And the Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(from(z0)) → c18
The order we found is given by the following interpretation:
Polynomial interpretation : 6.73/2.15

POL(0) = 0    6.73/2.15
POL(A__ADD(x1, x2)) = x2    6.73/2.15
POL(A__AND(x1, x2)) = x2    6.73/2.15
POL(A__IF(x1, x2, x3)) = x2 + x3    6.73/2.15
POL(MARK(x1)) = x1    6.73/2.15
POL(a__add(x1, x2)) = [3] + [3]x2    6.73/2.15
POL(a__and(x1, x2)) = [3] + [3]x2    6.73/2.15
POL(a__first(x1, x2)) = [3] + [3]x1 + [3]x2    6.73/2.15
POL(a__from(x1)) = [3] + [3]x1    6.73/2.15
POL(a__if(x1, x2, x3)) = [3] + [3]x2 + [3]x3    6.73/2.15
POL(add(x1, x2)) = x1 + x2    6.73/2.15
POL(and(x1, x2)) = [2] + x1 + x2    6.73/2.15
POL(c(x1)) = x1    6.73/2.15
POL(c14(x1, x2)) = x1 + x2    6.73/2.15
POL(c15(x1, x2)) = x1 + x2    6.73/2.15
POL(c16(x1, x2)) = x1 + x2    6.73/2.15
POL(c17(x1, x2)) = x1 + x2    6.73/2.15
POL(c18) = 0    6.73/2.15
POL(c3(x1)) = x1    6.73/2.15
POL(c4(x1)) = x1    6.73/2.15
POL(c6(x1)) = x1    6.73/2.15
POL(cons(x1, x2)) = x1    6.73/2.15
POL(false) = 0    6.73/2.15
POL(first(x1, x2)) = [2] + x1 + x2    6.73/2.15
POL(from(x1)) = 0    6.73/2.15
POL(if(x1, x2, x3)) = [1] + x1 + x2 + x3    6.73/2.15
POL(mark(x1)) = 0    6.73/2.15
POL(nil) = 0    6.73/2.15
POL(s(x1)) = 0    6.73/2.15
POL(true) = 0   
6.73/2.15
6.73/2.15

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__and(true, z0) → mark(z0) 6.73/2.15
a__and(false, z0) → false 6.73/2.15
a__and(z0, z1) → and(z0, z1) 6.73/2.15
a__if(true, z0, z1) → mark(z0) 6.73/2.15
a__if(false, z0, z1) → mark(z1) 6.73/2.15
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.15
a__add(0, z0) → mark(z0) 6.73/2.15
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.15
a__add(z0, z1) → add(z0, z1) 6.73/2.15
a__first(0, z0) → nil 6.73/2.15
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.15
a__first(z0, z1) → first(z0, z1) 6.73/2.15
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.15
a__from(z0) → from(z0) 6.73/2.15
mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.15
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.15
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.15
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.15
mark(from(z0)) → a__from(z0) 6.73/2.15
mark(true) → true 6.73/2.15
mark(false) → false 6.73/2.15
mark(0) → 0 6.73/2.15
mark(s(z0)) → s(z0) 6.73/2.15
mark(nil) → nil 6.73/2.15
mark(cons(z0, z1)) → cons(z0, z1)
Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(from(z0)) → c18
S tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(from(z0)) → c18
K tuples:

MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1))
Defined Rule Symbols:

a__and, a__if, a__add, a__first, a__from, mark

Defined Pair Symbols:

A__AND, A__IF, A__ADD, MARK

Compound Symbols:

c, c3, c4, c6, c14, c15, c16, c17, c18

6.73/2.15
6.73/2.15

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

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

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1))
6.73/2.15
6.73/2.15

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__and(true, z0) → mark(z0) 6.73/2.15
a__and(false, z0) → false 6.73/2.15
a__and(z0, z1) → and(z0, z1) 6.73/2.15
a__if(true, z0, z1) → mark(z0) 6.73/2.15
a__if(false, z0, z1) → mark(z1) 6.73/2.15
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.15
a__add(0, z0) → mark(z0) 6.73/2.15
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.15
a__add(z0, z1) → add(z0, z1) 6.73/2.15
a__first(0, z0) → nil 6.73/2.15
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.15
a__first(z0, z1) → first(z0, z1) 6.73/2.15
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.15
a__from(z0) → from(z0) 6.73/2.15
mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.15
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.15
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.15
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.15
mark(from(z0)) → a__from(z0) 6.73/2.15
mark(true) → true 6.73/2.15
mark(false) → false 6.73/2.15
mark(0) → 0 6.73/2.15
mark(s(z0)) → s(z0) 6.73/2.15
mark(nil) → nil 6.73/2.15
mark(cons(z0, z1)) → cons(z0, z1)
Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.15
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
MARK(from(z0)) → c18
S tuples:

A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.15
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(from(z0)) → c18
K tuples:

MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.15
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.15
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.15
A__AND(true, z0) → c(MARK(z0)) 6.73/2.15
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.15
A__IF(false, z0, z1) → c4(MARK(z1))
Defined Rule Symbols:

a__and, a__if, a__add, a__first, a__from, mark

Defined Pair Symbols:

A__AND, A__IF, A__ADD, MARK

Compound Symbols:

c, c3, c4, c6, c14, c15, c16, c17, c18

6.73/2.15
6.73/2.15

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0))
We considered the (Usable) Rules:

mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.15
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.15
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.15
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.15
mark(from(z0)) → a__from(z0) 6.73/2.15
mark(true) → true 6.73/2.15
mark(false) → false 6.73/2.15
mark(0) → 0 6.73/2.15
mark(s(z0)) → s(z0) 6.73/2.15
mark(nil) → nil 6.73/2.15
mark(cons(z0, z1)) → cons(z0, z1) 6.73/2.15
a__and(true, z0) → mark(z0) 6.73/2.15
a__and(false, z0) → false 6.73/2.15
a__and(z0, z1) → and(z0, z1) 6.73/2.15
a__if(true, z0, z1) → mark(z0) 6.73/2.15
a__if(false, z0, z1) → mark(z1) 6.73/2.15
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.15
a__add(0, z0) → mark(z0) 6.73/2.17
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.17
a__add(z0, z1) → add(z0, z1) 6.73/2.17
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.17
a__from(z0) → from(z0) 6.73/2.17
a__first(0, z0) → nil 6.73/2.17
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.17
a__first(z0, z1) → first(z0, z1)
And the Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.17
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.17
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.17
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.17
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.17
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.17
MARK(from(z0)) → c18
The order we found is given by the following interpretation:
Polynomial interpretation : 6.73/2.17

POL(0) = 0    6.73/2.17
POL(A__ADD(x1, x2)) = [2]x2    6.73/2.17
POL(A__AND(x1, x2)) = [2]x2    6.73/2.17
POL(A__IF(x1, x2, x3)) = [2]x2 + [2]x3    6.73/2.17
POL(MARK(x1)) = [2]x1    6.73/2.17
POL(a__add(x1, x2)) = [2] + x1 + x2    6.73/2.17
POL(a__and(x1, x2)) = [4] + x1 + x2    6.73/2.17
POL(a__first(x1, x2)) = [1] + x1 + x2    6.73/2.17
POL(a__from(x1)) = 0    6.73/2.17
POL(a__if(x1, x2, x3)) = x1 + x2 + x3    6.73/2.17
POL(add(x1, x2)) = [2] + x1 + x2    6.73/2.17
POL(and(x1, x2)) = [4] + x1 + x2    6.73/2.17
POL(c(x1)) = x1    6.73/2.17
POL(c14(x1, x2)) = x1 + x2    6.73/2.17
POL(c15(x1, x2)) = x1 + x2    6.73/2.17
POL(c16(x1, x2)) = x1 + x2    6.73/2.17
POL(c17(x1, x2)) = x1 + x2    6.73/2.17
POL(c18) = 0    6.73/2.17
POL(c3(x1)) = x1    6.73/2.17
POL(c4(x1)) = x1    6.73/2.17
POL(c6(x1)) = x1    6.73/2.17
POL(cons(x1, x2)) = 0    6.73/2.17
POL(false) = 0    6.73/2.17
POL(first(x1, x2)) = [1] + x1 + x2    6.73/2.17
POL(from(x1)) = 0    6.73/2.17
POL(if(x1, x2, x3)) = x1 + x2 + x3    6.73/2.17
POL(mark(x1)) = x1    6.73/2.17
POL(nil) = 0    6.73/2.17
POL(s(x1)) = 0    6.73/2.17
POL(true) = [2]   
6.73/2.17
6.73/2.17

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__and(true, z0) → mark(z0) 6.73/2.17
a__and(false, z0) → false 6.73/2.17
a__and(z0, z1) → and(z0, z1) 6.73/2.17
a__if(true, z0, z1) → mark(z0) 6.73/2.17
a__if(false, z0, z1) → mark(z1) 6.73/2.17
a__if(z0, z1, z2) → if(z0, z1, z2) 6.73/2.17
a__add(0, z0) → mark(z0) 6.73/2.17
a__add(s(z0), z1) → s(add(z0, z1)) 6.73/2.17
a__add(z0, z1) → add(z0, z1) 6.73/2.17
a__first(0, z0) → nil 6.73/2.17
a__first(s(z0), cons(z1, z2)) → cons(z1, first(z0, z2)) 6.73/2.17
a__first(z0, z1) → first(z0, z1) 6.73/2.17
a__from(z0) → cons(z0, from(s(z0))) 6.73/2.17
a__from(z0) → from(z0) 6.73/2.17
mark(and(z0, z1)) → a__and(mark(z0), z1) 6.73/2.17
mark(if(z0, z1, z2)) → a__if(mark(z0), z1, z2) 6.73/2.17
mark(add(z0, z1)) → a__add(mark(z0), z1) 6.73/2.17
mark(first(z0, z1)) → a__first(mark(z0), mark(z1)) 6.73/2.17
mark(from(z0)) → a__from(z0) 6.73/2.17
mark(true) → true 6.73/2.17
mark(false) → false 6.73/2.17
mark(0) → 0 6.73/2.17
mark(s(z0)) → s(z0) 6.73/2.17
mark(nil) → nil 6.73/2.17
mark(cons(z0, z1)) → cons(z0, z1)
Tuples:

A__AND(true, z0) → c(MARK(z0)) 6.73/2.17
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.17
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.17
A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.17
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.17
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.17
MARK(from(z0)) → c18
S tuples:

A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.17
MARK(from(z0)) → c18
K tuples:

MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.17
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.17
A__AND(true, z0) → c(MARK(z0)) 6.73/2.17
A__IF(true, z0, z1) → c3(MARK(z0)) 6.73/2.17
A__IF(false, z0, z1) → c4(MARK(z1)) 6.73/2.17
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0))
Defined Rule Symbols:

a__and, a__if, a__add, a__first, a__from, mark

Defined Pair Symbols:

A__AND, A__IF, A__ADD, MARK

Compound Symbols:

c, c3, c4, c6, c14, c15, c16, c17, c18

6.73/2.17
6.73/2.17

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

A__ADD(0, z0) → c6(MARK(z0)) 6.73/2.17
MARK(from(z0)) → c18 6.73/2.17
MARK(and(z0, z1)) → c14(A__AND(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(if(z0, z1, z2)) → c15(A__IF(mark(z0), z1, z2), MARK(z0)) 6.73/2.17
MARK(add(z0, z1)) → c16(A__ADD(mark(z0), z1), MARK(z0)) 6.73/2.17
MARK(first(z0, z1)) → c17(MARK(z0), MARK(z1)) 6.73/2.17
MARK(from(z0)) → c18
Now S is empty
6.73/2.17
6.73/2.17

(14) BOUNDS(O(1), O(1))

6.73/2.17
6.73/2.17
7.12/2.22 EOF