YES(O(1), O(n^2)) 7.72/2.43 YES(O(1), O(n^2)) 7.72/2.47 7.72/2.47 7.72/2.47 7.72/2.47 7.72/2.47 7.72/2.47 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 7.72/2.47 7.72/2.47 7.72/2.47
7.72/2.47
7.72/2.47
7.72/2.47
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
7.72/2.47 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
7.72/2.47 
↳2 CdtProblem
7.72/2.47 
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
7.72/2.47 
↳4 CdtProblem
7.72/2.47 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
7.72/2.47 
↳6 CdtProblem
7.72/2.47 
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
7.72/2.47 
↳8 CdtProblem
7.72/2.47 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
7.72/2.47 
↳10 CdtProblem
7.72/2.47 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
7.72/2.47 
↳12 BOUNDS(O(1), O(1))
7.72/2.47
7.72/2.47 7.72/2.47
7.72/2.47
7.72/2.47

(0) Obligation:

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

eq(0, 0) → true 7.72/2.47
eq(0, s(X)) → false 7.72/2.47
eq(s(X), 0) → false 7.72/2.47
eq(s(X), s(Y)) → eq(X, Y) 7.72/2.47
rm(N, nil) → nil 7.72/2.47
rm(N, add(M, X)) → ifrm(eq(N, M), N, add(M, X)) 7.72/2.47
ifrm(true, N, add(M, X)) → rm(N, X) 7.72/2.47
ifrm(false, N, add(M, X)) → add(M, rm(N, X)) 7.72/2.47
purge(nil) → nil 7.72/2.47
purge(add(N, X)) → add(N, purge(rm(N, X)))

Rewrite Strategy: INNERMOST
7.72/2.47
7.72/2.47

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

Converted CpxTRS to CDT
7.72/2.47
7.72/2.47

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1) 7.72/2.47
rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
purge(nil) → nil 7.72/2.47
purge(add(z0, z1)) → add(z0, purge(rm(z0, z1)))
Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
S tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
K tuples:none
Defined Rule Symbols:

eq, rm, ifrm, purge

Defined Pair Symbols:

EQ, RM, IFRM, PURGE

Compound Symbols:

c3, c5, c6, c7, c9

7.72/2.47
7.72/2.47

(3) 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.

PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
We considered the (Usable) Rules:

rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1)
And the Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.72/2.47

POL(0) = 0    7.72/2.47
POL(EQ(x1, x2)) = 0    7.72/2.47
POL(IFRM(x1, x2, x3)) = 0    7.72/2.47
POL(PURGE(x1)) = x1    7.72/2.47
POL(RM(x1, x2)) = 0    7.72/2.47
POL(add(x1, x2)) = [1] + x2    7.72/2.47
POL(c3(x1)) = x1    7.72/2.47
POL(c5(x1, x2)) = x1 + x2    7.72/2.47
POL(c6(x1)) = x1    7.72/2.47
POL(c7(x1)) = x1    7.72/2.47
POL(c9(x1, x2)) = x1 + x2    7.72/2.47
POL(eq(x1, x2)) = 0    7.72/2.47
POL(false) = 0    7.72/2.47
POL(ifrm(x1, x2, x3)) = x3    7.72/2.47
POL(nil) = 0    7.72/2.47
POL(rm(x1, x2)) = x2    7.72/2.47
POL(s(x1)) = x1    7.72/2.47
POL(true) = 0   
7.72/2.47
7.72/2.47

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1) 7.72/2.47
rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
purge(nil) → nil 7.72/2.47
purge(add(z0, z1)) → add(z0, purge(rm(z0, z1)))
Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
S tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2))
K tuples:

PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
Defined Rule Symbols:

eq, rm, ifrm, purge

Defined Pair Symbols:

EQ, RM, IFRM, PURGE

Compound Symbols:

c3, c5, c6, c7, c9

7.72/2.47
7.72/2.47

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2))
We considered the (Usable) Rules:

rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1)
And the Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.72/2.47

POL(0) = 0    7.72/2.47
POL(EQ(x1, x2)) = 0    7.72/2.47
POL(IFRM(x1, x2, x3)) = [2]x3    7.72/2.47
POL(PURGE(x1)) = x12    7.72/2.47
POL(RM(x1, x2)) = [2]x2    7.72/2.47
POL(add(x1, x2)) = [1] + x2    7.72/2.47
POL(c3(x1)) = x1    7.72/2.47
POL(c5(x1, x2)) = x1 + x2    7.72/2.47
POL(c6(x1)) = x1    7.72/2.47
POL(c7(x1)) = x1    7.72/2.47
POL(c9(x1, x2)) = x1 + x2    7.72/2.47
POL(eq(x1, x2)) = 0    7.72/2.47
POL(false) = 0    7.72/2.47
POL(ifrm(x1, x2, x3)) = x3    7.72/2.47
POL(nil) = 0    7.72/2.47
POL(rm(x1, x2)) = x2    7.72/2.47
POL(s(x1)) = 0    7.72/2.47
POL(true) = 0   
7.72/2.47
7.72/2.47

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1) 7.72/2.47
rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
purge(nil) → nil 7.72/2.47
purge(add(z0, z1)) → add(z0, purge(rm(z0, z1)))
Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
S tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1))
K tuples:

PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2))
Defined Rule Symbols:

eq, rm, ifrm, purge

Defined Pair Symbols:

EQ, RM, IFRM, PURGE

Compound Symbols:

c3, c5, c6, c7, c9

7.72/2.47
7.72/2.47

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

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

RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2))
7.72/2.47
7.72/2.47

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1) 7.72/2.47
rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
purge(nil) → nil 7.72/2.47
purge(add(z0, z1)) → add(z0, purge(rm(z0, z1)))
Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
S tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1))
K tuples:

PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1))
Defined Rule Symbols:

eq, rm, ifrm, purge

Defined Pair Symbols:

EQ, RM, IFRM, PURGE

Compound Symbols:

c3, c5, c6, c7, c9

7.72/2.47
7.72/2.47

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(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)) → c3(EQ(z0, z1))
We considered the (Usable) Rules:

rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1)
And the Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.72/2.47

POL(0) = 0    7.72/2.47
POL(EQ(x1, x2)) = x1    7.72/2.47
POL(IFRM(x1, x2, x3)) = x2·x3    7.72/2.47
POL(PURGE(x1)) = x12    7.72/2.47
POL(RM(x1, x2)) = x1 + x1·x2    7.72/2.47
POL(add(x1, x2)) = [1] + x1 + x2    7.72/2.47
POL(c3(x1)) = x1    7.72/2.47
POL(c5(x1, x2)) = x1 + x2    7.72/2.47
POL(c6(x1)) = x1    7.72/2.47
POL(c7(x1)) = x1    7.72/2.47
POL(c9(x1, x2)) = x1 + x2    7.72/2.47
POL(eq(x1, x2)) = 0    7.72/2.47
POL(false) = 0    7.72/2.47
POL(ifrm(x1, x2, x3)) = x3    7.72/2.47
POL(nil) = 0    7.72/2.47
POL(rm(x1, x2)) = x2    7.72/2.47
POL(s(x1)) = [2] + x1    7.72/2.47
POL(true) = 0   
7.72/2.47
7.72/2.47

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

eq(0, 0) → true 7.72/2.47
eq(0, s(z0)) → false 7.72/2.47
eq(s(z0), 0) → false 7.72/2.47
eq(s(z0), s(z1)) → eq(z0, z1) 7.72/2.47
rm(z0, nil) → nil 7.72/2.47
rm(z0, add(z1, z2)) → ifrm(eq(z0, z1), z0, add(z1, z2)) 7.72/2.47
ifrm(true, z0, add(z1, z2)) → rm(z0, z2) 7.72/2.47
ifrm(false, z0, add(z1, z2)) → add(z1, rm(z0, z2)) 7.72/2.47
purge(nil) → nil 7.72/2.47
purge(add(z0, z1)) → add(z0, purge(rm(z0, z1)))
Tuples:

EQ(s(z0), s(z1)) → c3(EQ(z0, z1)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1))
S tuples:none
K tuples:

PURGE(add(z0, z1)) → c9(PURGE(rm(z0, z1)), RM(z0, z1)) 7.72/2.47
IFRM(true, z0, add(z1, z2)) → c6(RM(z0, z2)) 7.72/2.47
IFRM(false, z0, add(z1, z2)) → c7(RM(z0, z2)) 7.72/2.47
RM(z0, add(z1, z2)) → c5(IFRM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) 7.72/2.47
EQ(s(z0), s(z1)) → c3(EQ(z0, z1))
Defined Rule Symbols:

eq, rm, ifrm, purge

Defined Pair Symbols:

EQ, RM, IFRM, PURGE

Compound Symbols:

c3, c5, c6, c7, c9

7.72/2.47
7.72/2.47

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

The set S is empty
7.72/2.47
7.72/2.47

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

7.72/2.47
7.72/2.47
8.17/2.53 EOF