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

0 CpxTRS
↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTypedWeightedTrs
↳5 CompletionProof (UPPER BOUND(ID), 0 ms)
↳6 CpxTypedWeightedCompleteTrs
↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 17 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳10 CpxRNTS
↳11 SimplificationProof (BOTH BOUNDS(ID, ID), 1 ms)
↳12 CpxRNTS
↳13 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CpxRNTS
↳15 IntTrsBoundProof (UPPER BOUND(ID), 399 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 61 ms)
↳18 CpxRNTS
↳19 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳20 CpxRNTS
↳21 IntTrsBoundProof (UPPER BOUND(ID), 175 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 65 ms)
↳24 CpxRNTS
↳25 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳26 CpxRNTS
↳27 IntTrsBoundProof (UPPER BOUND(ID), 160 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 111 ms)
↳30 CpxRNTS
↳31 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳32 CpxRNTS
↳33 IntTrsBoundProof (UPPER BOUND(ID), 258 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 120 ms)
↳36 CpxRNTS
↳37 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳38 CpxRNTS
↳39 IntTrsBoundProof (UPPER BOUND(ID), 2585 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 3605 ms)
↳42 CpxRNTS
↳43 FinalProof (⇔, 0 ms)
↳44 BOUNDS(1, n^2)

(0) 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:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) Obligation:

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


The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x) [1]
cond2(true, x) → cond1(neq(x, 0), div2(x)) [1]
cond2(false, x) → cond1(neq(x, 0), p(x)) [1]
neq(0, 0) → false [1]
neq(0, s(x)) → true [1]
neq(s(x), 0) → true [1]
neq(s(x), s(y)) → neq(x, y) [1]
even(0) → true [1]
even(s(0)) → false [1]
even(s(s(x))) → even(x) [1]
div2(0) → 0 [1]
div2(s(0)) → 0 [1]
div2(s(s(x))) → s(div2(x)) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x) [1]
cond2(true, x) → cond1(neq(x, 0), div2(x)) [1]
cond2(false, x) → cond1(neq(x, 0), p(x)) [1]
neq(0, 0) → false [1]
neq(0, s(x)) → true [1]
neq(s(x), 0) → true [1]
neq(s(x), s(y)) → neq(x, y) [1]
even(0) → true [1]
even(s(0)) → false [1]
even(s(s(x))) → even(x) [1]
div2(0) → 0 [1]
div2(s(0)) → 0 [1]
div2(s(s(x))) → s(div2(x)) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

The TRS has the following type information:
cond1 :: true:false → 0:s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0:s:y → cond1:cond2
even :: 0:s:y → true:false
neq :: 0:s:y → 0:s:y → true:false
0 :: 0:s:y
div2 :: 0:s:y → 0:s:y
false :: true:false
p :: 0:s:y → 0:s:y
s :: 0:s:y → 0:s:y
y :: 0:s:y

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:


cond1
cond2

(c) The following functions are completely defined:

neq
div2
p
even

Due to the following rules being added:

neq(v0, v1) → null_neq [0]
div2(v0) → null_div2 [0]
p(v0) → null_p [0]
even(v0) → null_even [0]

And the following fresh constants:

null_neq, null_div2, null_p, null_even, const

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

cond1(true, x) → cond2(even(x), x) [1]
cond2(true, x) → cond1(neq(x, 0), div2(x)) [1]
cond2(false, x) → cond1(neq(x, 0), p(x)) [1]
neq(0, 0) → false [1]
neq(0, s(x)) → true [1]
neq(s(x), 0) → true [1]
neq(s(x), s(y)) → neq(x, y) [1]
even(0) → true [1]
even(s(0)) → false [1]
even(s(s(x))) → even(x) [1]
div2(0) → 0 [1]
div2(s(0)) → 0 [1]
div2(s(s(x))) → s(div2(x)) [1]
p(0) → 0 [1]
p(s(x)) → x [1]
neq(v0, v1) → null_neq [0]
div2(v0) → null_div2 [0]
p(v0) → null_p [0]
even(v0) → null_even [0]

The TRS has the following type information:
cond1 :: true:false:null_neq:null_even → 0:s:y:null_div2:null_p → cond1:cond2
true :: true:false:null_neq:null_even
cond2 :: true:false:null_neq:null_even → 0:s:y:null_div2:null_p → cond1:cond2
even :: 0:s:y:null_div2:null_p → true:false:null_neq:null_even
neq :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p → true:false:null_neq:null_even
0 :: 0:s:y:null_div2:null_p
div2 :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p
false :: true:false:null_neq:null_even
p :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p
s :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p
y :: 0:s:y:null_div2:null_p
null_neq :: true:false:null_neq:null_even
null_div2 :: 0:s:y:null_div2:null_p
null_p :: 0:s:y:null_div2:null_p
null_even :: true:false:null_neq:null_even
const :: cond1:cond2

Rewrite Strategy: INNERMOST

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

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

cond1(true, 0) → cond2(true, 0) [2]
cond1(true, s(0)) → cond2(false, s(0)) [2]
cond1(true, s(s(x'))) → cond2(even(x'), s(s(x'))) [2]
cond1(true, x) → cond2(null_even, x) [1]
cond2(true, 0) → cond1(false, 0) [3]
cond2(true, 0) → cond1(false, null_div2) [2]
cond2(true, s(0)) → cond1(true, 0) [3]
cond2(true, s(s(x1))) → cond1(true, s(div2(x1))) [3]
cond2(true, s(x'')) → cond1(true, null_div2) [2]
cond2(true, 0) → cond1(null_neq, 0) [2]
cond2(true, s(0)) → cond1(null_neq, 0) [2]
cond2(true, s(s(x2))) → cond1(null_neq, s(div2(x2))) [2]
cond2(true, x) → cond1(null_neq, null_div2) [1]
cond2(false, 0) → cond1(false, 0) [3]
cond2(false, 0) → cond1(false, null_p) [2]
cond2(false, s(x3)) → cond1(true, x3) [3]
cond2(false, s(x3)) → cond1(true, null_p) [2]
cond2(false, 0) → cond1(null_neq, 0) [2]
cond2(false, s(x4)) → cond1(null_neq, x4) [2]
cond2(false, x) → cond1(null_neq, null_p) [1]
neq(0, 0) → false [1]
neq(0, s(x)) → true [1]
neq(s(x), 0) → true [1]
neq(s(x), s(y)) → neq(x, y) [1]
even(0) → true [1]
even(s(0)) → false [1]
even(s(s(x))) → even(x) [1]
div2(0) → 0 [1]
div2(s(0)) → 0 [1]
div2(s(s(x))) → s(div2(x)) [1]
p(0) → 0 [1]
p(s(x)) → x [1]
neq(v0, v1) → null_neq [0]
div2(v0) → null_div2 [0]
p(v0) → null_p [0]
even(v0) → null_even [0]

The TRS has the following type information:
cond1 :: true:false:null_neq:null_even → 0:s:y:null_div2:null_p → cond1:cond2
true :: true:false:null_neq:null_even
cond2 :: true:false:null_neq:null_even → 0:s:y:null_div2:null_p → cond1:cond2
even :: 0:s:y:null_div2:null_p → true:false:null_neq:null_even
neq :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p → true:false:null_neq:null_even
0 :: 0:s:y:null_div2:null_p
div2 :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p
false :: true:false:null_neq:null_even
p :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p
s :: 0:s:y:null_div2:null_p → 0:s:y:null_div2:null_p
y :: 0:s:y:null_div2:null_p
null_neq :: true:false:null_neq:null_even
null_div2 :: 0:s:y:null_div2:null_p
null_p :: 0:s:y:null_div2:null_p
null_even :: true:false:null_neq:null_even
const :: cond1:cond2

Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

true => 2
0 => 0
false => 1
y => 1
null_neq => 0
null_div2 => 0
null_p => 0
null_even => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(x'), 1 + (1 + x')) :|: z = 2, z' = 1 + (1 + x'), x' >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, x) :|: z = 2, z' = x, x >= 0
cond2(z, z') -{ 3 }→ cond1(2, x3) :|: z = 1, z' = 1 + x3, x3 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' = 1 + x'', x'' >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' = 1 + x3, x3 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(x1)) :|: z = 2, x1 >= 0, z' = 1 + (1 + x1)
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, x4) :|: x4 >= 0, z = 1, z' = 1 + x4
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' = x, x >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z' = x, z = 1, x >= 0
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(x2)) :|: z = 2, z' = 1 + (1 + x2), x2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
div2(z) -{ 1 }→ 1 + div2(x) :|: x >= 0, z = 1 + (1 + x)
even(z) -{ 1 }→ even(x) :|: x >= 0, z = 1 + (1 + x)
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
neq(z, z') -{ 1 }→ neq(x, 1) :|: z' = 1 + 1, x >= 0, z = 1 + x
neq(z, z') -{ 1 }→ 2 :|: z' = 1 + x, x >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: x >= 0, z = 1 + x, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(12) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 1 }→ neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ neq }
{ p }
{ div2 }
{ even }
{ cond2, cond1 }

(14) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 1 }→ neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {neq}, {p}, {div2}, {even}, {cond2,cond1}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: neq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(16) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 1 }→ neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {neq}, {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: ?, size: O(1) [2]

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: neq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(18) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 1 }→ neq(z - 1, 1) :|: z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(20) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z

(22) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: ?, size: O(n1) [z]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(24) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(26) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: div2
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z

(28) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {div2}, {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: ?, size: O(n1) [z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using PUBS for: div2
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + z

(30) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 2 }→ cond1(0, 1 + div2(z' - 2)) :|: z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ 1 }→ 1 + div2(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(32) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 2 + z' }→ cond1(2, 1 + s') :|: s' >= 0, s' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 1 + z' }→ cond1(0, 1 + s'') :|: s'' >= 0, s'' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: even
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(34) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 2 + z' }→ cond1(2, 1 + s') :|: s' >= 0, s' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 1 + z' }→ cond1(0, 1 + s'') :|: s'' >= 0, s'' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {even}, {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]
even: runtime: ?, size: O(1) [2]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using PUBS for: even
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + z

(36) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 2 }→ cond2(even(z' - 2), 1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 2 + z' }→ cond1(2, 1 + s') :|: s' >= 0, s' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 1 + z' }→ cond1(0, 1 + s'') :|: s'' >= 0, s'' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
even(z) -{ 1 }→ even(z - 2) :|: z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]
even: runtime: O(n1) [1 + z], size: O(1) [2]

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(38) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 1 + z' }→ cond2(s2, 1 + (1 + (z' - 2))) :|: s2 >= 0, s2 <= 2, z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 2 + z' }→ cond1(2, 1 + s') :|: s' >= 0, s' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 1 + z' }→ cond1(0, 1 + s'') :|: s'' >= 0, s'' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
even(z) -{ z }→ s3 :|: s3 >= 0, s3 <= 2, z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]
even: runtime: O(n1) [1 + z], size: O(1) [2]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(40) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 1 + z' }→ cond2(s2, 1 + (1 + (z' - 2))) :|: s2 >= 0, s2 <= 2, z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 2 + z' }→ cond1(2, 1 + s') :|: s' >= 0, s' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 1 + z' }→ cond1(0, 1 + s'') :|: s'' >= 0, s'' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
even(z) -{ z }→ s3 :|: s3 >= 0, s3 <= 2, z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]
even: runtime: O(n1) [1 + z], size: O(1) [2]
cond2: runtime: ?, size: O(1) [0]
cond1: runtime: ?, size: O(1) [0]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using KoAT for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n2) with polynomial bound: 42 + 63·z' + 6·z'2

Computed RUNTIME bound using PUBS for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n2) with polynomial bound: 113 + 64·z' + 6·z'2

(42) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z') -{ 1 + z' }→ cond2(s2, 1 + (1 + (z' - 2))) :|: s2 >= 0, s2 <= 2, z = 2, z' - 2 >= 0
cond1(z, z') -{ 2 }→ cond2(2, 0) :|: z = 2, z' = 0
cond1(z, z') -{ 2 }→ cond2(1, 1 + 0) :|: z = 2, z' = 1 + 0
cond1(z, z') -{ 1 }→ cond2(0, z') :|: z = 2, z' >= 0
cond2(z, z') -{ 3 }→ cond1(2, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 2, z' - 1 >= 0
cond2(z, z') -{ 2 }→ cond1(2, 0) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 3 }→ cond1(2, z' - 1) :|: z = 1, z' - 1 >= 0
cond2(z, z') -{ 2 + z' }→ cond1(2, 1 + s') :|: s' >= 0, s' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 3 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(1, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 2, z' = 1 + 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 2, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, 0) :|: z = 1, z' = 0
cond2(z, z') -{ 1 }→ cond1(0, 0) :|: z = 1, z' >= 0
cond2(z, z') -{ 2 }→ cond1(0, z' - 1) :|: z' - 1 >= 0, z = 1
cond2(z, z') -{ 1 + z' }→ cond1(0, 1 + s'') :|: s'' >= 0, s'' <= 1 * (z' - 2), z = 2, z' - 2 >= 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 0 }→ 0 :|: z >= 0
div2(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
even(z) -{ z }→ s3 :|: s3 >= 0, s3 <= 2, z - 2 >= 0
even(z) -{ 1 }→ 2 :|: z = 0
even(z) -{ 1 }→ 1 :|: z = 1 + 0
even(z) -{ 0 }→ 0 :|: z >= 0
neq(z, z') -{ 3 }→ s :|: s >= 0, s <= 2, z' = 1 + 1, z - 1 >= 0
neq(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0
neq(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0
neq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
neq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:
neq: runtime: O(1) [2], size: O(1) [2]
p: runtime: O(1) [1], size: O(n1) [z]
div2: runtime: O(n1) [1 + z], size: O(n1) [z]
even: runtime: O(n1) [1 + z], size: O(1) [2]
cond2: runtime: O(n2) [42 + 63·z' + 6·z'2], size: O(1) [0]
cond1: runtime: O(n2) [113 + 64·z' + 6·z'2], size: O(1) [0]

(43) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(44) BOUNDS(1, n^2)