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

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), 19 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳10 CpxRNTS
↳11 InliningProof (UPPER BOUND(ID), 257 ms)
↳12 CpxRNTS
↳13 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CpxRNTS
↳15 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 229 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 52 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 364 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 211 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 188 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 57 ms)
↳32 CpxRNTS
↳33 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 418 ms)
↳36 CpxRNTS
↳37 IntTrsBoundProof (UPPER BOUND(ID), 169 ms)
↳38 CpxRNTS
↳39 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 2278 ms)
↳42 CpxRNTS
↳43 IntTrsBoundProof (UPPER BOUND(ID), 790 ms)
↳44 CpxRNTS
↳45 FinalProof (⇔, 0 ms)
↳46 BOUNDS(1, n^1)

(0) Obligation:

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


The TRS R consists of the following rules:

cond1(true, x, y) → cond2(gr(y, 0), x, y)
cond2(true, x, y) → cond2(gr(y, 0), p(x), p(y))
cond2(false, x, y) → cond1(and(eq(x, y), gr(x, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

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^1).


The TRS R consists of the following rules:

cond1(true, x, y) → cond2(gr(y, 0), x, y) [1]
cond2(true, x, y) → cond2(gr(y, 0), p(x), p(y)) [1]
cond2(false, x, y) → cond1(and(eq(x, y), gr(x, 0)), x, y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]
eq(0, 0) → true [1]
eq(s(x), 0) → false [1]
eq(0, s(x)) → false [1]
eq(s(x), s(y)) → eq(x, y) [1]
and(true, true) → true [1]
and(false, x) → false [1]
and(x, false) → false [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, y) → cond2(gr(y, 0), x, y) [1]
cond2(true, x, y) → cond2(gr(y, 0), p(x), p(y)) [1]
cond2(false, x, y) → cond1(and(eq(x, y), gr(x, 0)), x, y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]
eq(0, 0) → true [1]
eq(s(x), 0) → false [1]
eq(0, s(x)) → false [1]
eq(s(x), s(y)) → eq(x, y) [1]
and(true, true) → true [1]
and(false, x) → false [1]
and(x, false) → false [1]

The TRS has the following type information:
cond1 :: true:false → 0:s → 0:s → cond1:cond2
true :: true:false
cond2 :: true:false → 0:s → 0:s → cond1:cond2
gr :: 0:s → 0:s → true:false
0 :: 0:s
p :: 0:s → 0:s
false :: true:false
and :: true:false → true:false → true:false
eq :: 0:s → 0:s → true:false
s :: 0:s → 0:s

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:

gr
p
and
eq

Due to the following rules being added:
none

And the following fresh constants:

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, y) → cond2(gr(y, 0), x, y) [1]
cond2(true, x, y) → cond2(gr(y, 0), p(x), p(y)) [1]
cond2(false, x, y) → cond1(and(eq(x, y), gr(x, 0)), x, y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]
eq(0, 0) → true [1]
eq(s(x), 0) → false [1]
eq(0, s(x)) → false [1]
eq(s(x), s(y)) → eq(x, y) [1]
and(true, true) → true [1]
and(false, x) → false [1]
and(x, false) → false [1]

The TRS has the following type information:
cond1 :: true:false → 0:s → 0:s → cond1:cond2
true :: true:false
cond2 :: true:false → 0:s → 0:s → cond1:cond2
gr :: 0:s → 0:s → true:false
0 :: 0:s
p :: 0:s → 0:s
false :: true:false
and :: true:false → true:false → true:false
eq :: 0:s → 0:s → true:false
s :: 0:s → 0:s
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, x, 0) → cond2(false, x, 0) [2]
cond1(true, x, s(x')) → cond2(true, x, s(x')) [2]
cond2(true, 0, 0) → cond2(false, 0, 0) [4]
cond2(true, s(x1), 0) → cond2(false, x1, 0) [4]
cond2(true, 0, s(x'')) → cond2(true, 0, x'') [4]
cond2(true, s(x2), s(x'')) → cond2(true, x2, x'') [4]
cond2(false, 0, 0) → cond1(and(true, false), 0, 0) [3]
cond2(false, s(x3), 0) → cond1(and(false, true), s(x3), 0) [3]
cond2(false, 0, s(x4)) → cond1(and(false, false), 0, s(x4)) [3]
cond2(false, s(x5), s(y')) → cond1(and(eq(x5, y'), true), s(x5), s(y')) [3]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]
eq(0, 0) → true [1]
eq(s(x), 0) → false [1]
eq(0, s(x)) → false [1]
eq(s(x), s(y)) → eq(x, y) [1]
and(true, true) → true [1]
and(false, x) → false [1]
and(x, false) → false [1]

The TRS has the following type information:
cond1 :: true:false → 0:s → 0:s → cond1:cond2
true :: true:false
cond2 :: true:false → 0:s → 0:s → cond1:cond2
gr :: 0:s → 0:s → true:false
0 :: 0:s
p :: 0:s → 0:s
false :: true:false
and :: true:false → true:false → true:false
eq :: 0:s → 0:s → true:false
s :: 0:s → 0:s
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 => 1
0 => 0
false => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, x, 1 + x') :|: z' = x, z = 1, z'' = 1 + x', x >= 0, x' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, x2, x'') :|: z' = 1 + x2, z = 1, z'' = 1 + x'', x'' >= 0, x2 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, x1, 0) :|: z'' = 0, x1 >= 0, z = 1, z' = 1 + x1
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(x5, y'), 1), 1 + x5, 1 + y') :|: x5 >= 0, z' = 1 + x5, y' >= 0, z = 0, z'' = 1 + y'
cond2(z, z', z'') -{ 3 }→ cond1(and(1, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(and(0, 1), 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(and(0, 0), 0, 1 + x4) :|: x4 >= 0, z'' = 1 + x4, z = 0, z' = 0
eq(z, z') -{ 1 }→ eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' = 1 + x, x >= 0, z = 0
gr(z, z') -{ 1 }→ gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
gr(z, z') -{ 1 }→ 1 :|: x >= 0, z = 1 + x, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 0 :|: z = 0

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, x, 1 + x') :|: z' = x, z = 1, z'' = 1 + x', x >= 0, x' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, x2, x'') :|: z' = 1 + x2, z = 1, z'' = 1 + x'', x'' >= 0, x2 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, x1, 0) :|: z'' = 0, x1 >= 0, z = 1, z' = 1 + x1
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(x5, y'), 1), 1 + x5, 1 + y') :|: x5 >= 0, z' = 1 + x5, y' >= 0, z = 0, z'' = 1 + y'
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + x4) :|: x4 >= 0, z'' = 1 + x4, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' = 1 + x, x >= 0, z = 0
gr(z, z') -{ 1 }→ gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
gr(z, z') -{ 1 }→ 1 :|: x >= 0, z = 1 + x, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 0 :|: z = 0

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ and }
{ eq }
{ p }
{ gr }
{ cond2, cond1 }

(16) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {and}, {eq}, {p}, {gr}, {cond2,cond1}

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


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

(18) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {and}, {eq}, {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: ?, size: O(1) [1]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(20) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {eq}, {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {eq}, {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]

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


Computed SIZE bound using CoFloCo for: eq
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:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {eq}, {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: ?, size: O(1) [1]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(26) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 3 }→ cond1(and(eq(z' - 1, z'' - 1), 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 }→ eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]

(29) 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

(30) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: ?, size: O(n1) [z]

(31) 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

(32) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]

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


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

(36) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]
gr: runtime: ?, size: O(1) [1]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(38) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]
gr: runtime: O(n1) [1 + z'], size: O(1) [1]

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(40) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 + z' }→ s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]
gr: runtime: O(n1) [1 + z'], size: O(1) [1]

(41) 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

(42) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 + z' }→ s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]
gr: runtime: O(n1) [1 + z'], size: O(1) [1]
cond2: runtime: ?, size: O(1) [0]
cond1: runtime: ?, size: O(1) [0]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 17 + 8·z' + 5·z''

Computed RUNTIME bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 19 + 8·z' + 5·z''

(44) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 4 }→ cond2(1, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(1, z' - 1, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond2(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond2(0, z' - 1, 0) :|: z'' = 0, z' - 1 >= 0, z = 1
cond2(z, z', z'') -{ 4 + z'' }→ cond1(s'', 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, x >= 0, 1 = x, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0, 1 = x, x >= 0, 0 = 0
eq(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
gr(z, z') -{ 1 + z' }→ s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:
and: runtime: O(1) [1], size: O(1) [1]
eq: runtime: O(n1) [1 + z'], size: O(1) [1]
p: runtime: O(1) [1], size: O(n1) [z]
gr: runtime: O(n1) [1 + z'], size: O(1) [1]
cond2: runtime: O(n1) [17 + 8·z' + 5·z''], size: O(1) [0]
cond1: runtime: O(n1) [19 + 8·z' + 5·z''], size: O(1) [0]

(45) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(46) BOUNDS(1, n^1)