YES(O(1), O(n^1)) 2.42/1.07 YES(O(1), O(n^1)) 2.81/1.12 2.81/1.12 2.81/1.12 2.81/1.12 2.81/1.12 2.81/1.12 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.81/1.12 2.81/1.12 2.81/1.12
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
2.81/1.12 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.81/1.12 
↳2 CdtProblem
2.81/1.12 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.81/1.12 
↳4 CdtProblem
2.81/1.12 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
2.81/1.12 
↳6 CdtProblem
2.81/1.12 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.81/1.12 
↳8 CdtProblem
2.81/1.12 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.81/1.12 
↳10 CdtProblem
2.81/1.12 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
2.81/1.12 
↳12 BOUNDS(O(1), O(1))
2.81/1.12
2.81/1.12 2.81/1.12
2.81/1.12
2.81/1.12

(0) Obligation:

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

del(.(x, .(y, z))) → f(=(x, y), x, y, z) 2.81/1.12
f(true, x, y, z) → del(.(y, z)) 2.81/1.12
f(false, x, y, z) → .(x, del(.(y, z))) 2.81/1.12
=(nil, nil) → true 2.81/1.12
=(.(x, y), nil) → false 2.81/1.12
=(nil, .(y, z)) → false 2.81/1.12
=(.(x, y), .(u, v)) → and(=(x, u), =(y, v))

Rewrite Strategy: INNERMOST
2.81/1.12
2.81/1.12

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

Converted CpxTRS to CDT
2.81/1.12
2.81/1.12

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2) 2.81/1.12
f(true, z0, z1, z2) → del(.(z1, z2)) 2.81/1.12
f(false, z0, z1, z2) → .(z0, del(.(z1, z2))) 2.81/1.12
=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6(='(z0, u), ='(z1, v))
S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6(='(z0, u), ='(z1, v))
K tuples:none
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F, ='

Compound Symbols:

c, c1, c2, c6

2.81/1.12
2.81/1.12

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

Removed 2 trailing tuple parts
2.81/1.12
2.81/1.12

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2) 2.81/1.12
f(true, z0, z1, z2) → del(.(z1, z2)) 2.81/1.12
f(false, z0, z1, z2) → .(z0, del(.(z1, z2))) 2.81/1.12
=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
K tuples:none
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F, ='

Compound Symbols:

c, c1, c2, c6

2.81/1.12
2.81/1.12

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

Removed 1 trailing nodes:

='(.(z0, z1), .(u, v)) → c6
2.81/1.12
2.81/1.12

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2) 2.81/1.12
f(true, z0, z1, z2) → del(.(z1, z2)) 2.81/1.12
f(false, z0, z1, z2) → .(z0, del(.(z1, z2))) 2.81/1.12
=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
K tuples:none
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F, ='

Compound Symbols:

c, c1, c2, c6

2.81/1.12
2.81/1.12

(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.

='(.(z0, z1), .(u, v)) → c6
We considered the (Usable) Rules:

=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
And the Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
The order we found is given by the following interpretation:
Polynomial interpretation : 2.81/1.12

POL(.(x1, x2)) = [2] + x2    2.81/1.12
POL(=(x1, x2)) = 0    2.81/1.12
POL(='(x1, x2)) = [2]    2.81/1.12
POL(DEL(x1)) = [2] + x1    2.81/1.12
POL(F(x1, x2, x3, x4)) = [4] + x4    2.81/1.12
POL(and(x1, x2)) = [3]    2.81/1.12
POL(c(x1, x2)) = x1 + x2    2.81/1.12
POL(c1(x1)) = x1    2.81/1.12
POL(c2(x1)) = x1    2.81/1.12
POL(c6) = 0    2.81/1.12
POL(false) = 0    2.81/1.12
POL(nil) = [3]    2.81/1.12
POL(true) = 0    2.81/1.12
POL(u) = 0    2.81/1.12
POL(v) = 0   
2.81/1.12
2.81/1.12

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2) 2.81/1.12
f(true, z0, z1, z2) → del(.(z1, z2)) 2.81/1.12
f(false, z0, z1, z2) → .(z0, del(.(z1, z2))) 2.81/1.12
=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))
K tuples:

='(.(z0, z1), .(u, v)) → c6
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F, ='

Compound Symbols:

c, c1, c2, c6

2.81/1.12
2.81/1.12

(9) 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.

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))
We considered the (Usable) Rules:

=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
And the Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
The order we found is given by the following interpretation:
Polynomial interpretation : 2.81/1.12

POL(.(x1, x2)) = [2] + x1 + x2    2.81/1.12
POL(=(x1, x2)) = [5] + [3]x1 + [5]x2    2.81/1.12
POL(='(x1, x2)) = x1    2.81/1.12
POL(DEL(x1)) = [2]x1    2.81/1.12
POL(F(x1, x2, x3, x4)) = [5] + [2]x3 + [2]x4    2.81/1.12
POL(and(x1, x2)) = [5] + x1 + x2    2.81/1.12
POL(c(x1, x2)) = x1 + x2    2.81/1.12
POL(c1(x1)) = x1    2.81/1.12
POL(c2(x1)) = x1    2.81/1.12
POL(c6) = 0    2.81/1.12
POL(false) = 0    2.81/1.12
POL(nil) = [1]    2.81/1.12
POL(true) = 0    2.81/1.12
POL(u) = [4]    2.81/1.12
POL(v) = [5]   
2.81/1.12
2.81/1.12

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2) 2.81/1.12
f(true, z0, z1, z2) → del(.(z1, z2)) 2.81/1.12
f(false, z0, z1, z2) → .(z0, del(.(z1, z2))) 2.81/1.12
=(nil, nil) → true 2.81/1.12
=(.(z0, z1), nil) → false 2.81/1.12
=(nil, .(z0, z1)) → false 2.81/1.12
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))
Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2))) 2.81/1.12
='(.(z0, z1), .(u, v)) → c6
S tuples:none
K tuples:

='(.(z0, z1), .(u, v)) → c6 2.81/1.12
DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) 2.81/1.12
F(true, z0, z1, z2) → c1(DEL(.(z1, z2))) 2.81/1.12
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F, ='

Compound Symbols:

c, c1, c2, c6

2.81/1.12
2.81/1.12

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

The set S is empty
2.81/1.12
2.81/1.12

(12) BOUNDS(O(1), O(1))

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2.81/1.12
2.81/1.17 EOF