The Runtime Complexity (innermost) of the given CpxTRS could be proven to be 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(x, 0), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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^1).


The TRS R consists of the following rules:

cond1(true, x, y) → cond2(gr(x, 0), x, y) [1]
cond2(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
cond2(false, x, y) → cond3(gr(y, 0), x, y) [1]
cond3(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
cond3(false, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
or(false, false) → false [1]
or(true, x) → true [1]
or(x, true) → true [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, y) → cond2(gr(x, 0), x, y) [1]
cond2(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
cond2(false, x, y) → cond3(gr(y, 0), x, y) [1]
cond3(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
cond3(false, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
or(false, false) → false [1]
or(true, x) → true [1]
or(x, true) → true [1]
p(0) → 0 [1]
p(s(x)) → x [1]

The TRS has the following type information:

cond1 :: true:false → 0:s → 0:s → cond1:cond2:cond3 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2:cond3 gr :: 0:s → 0:s → true:false 0 :: 0:s or :: true:false → true:false → true:false p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 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
cond3

(c) The following functions are completely defined:

or
gr
p

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(x, 0), x, y) [1] cond2(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1] cond2(false, x, y) → cond3(gr(y, 0), x, y) [1] cond3(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1] cond3(false, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, y) [1] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] or(false, false) → false [1] or(true, x) → true [1] or(x, true) → true [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond1 :: true:false → 0:s → 0:s → cond1:cond2:cond3 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2:cond3 gr :: 0:s → 0:s → true:false 0 :: 0:s or :: true:false → true:false → true:false p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 s :: 0:s → 0:s const :: cond1:cond2:cond3 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, y) → cond2(false, 0, y) [2] cond1(true, s(x'), y) → cond2(true, s(x'), y) [2] cond2(true, 0, 0) → cond1(or(false, false), 0, 0) [4] cond2(true, 0, s(x1)) → cond1(or(false, true), 0, s(x1)) [4] cond2(true, s(x''), 0) → cond1(or(true, false), x'', 0) [4] cond2(true, s(x''), s(x2)) → cond1(or(true, true), x'', s(x2)) [4] cond2(false, x, 0) → cond3(false, x, 0) [2] cond2(false, x, s(x3)) → cond3(true, x, s(x3)) [2] cond3(true, 0, 0) → cond1(or(false, false), 0, 0) [4] cond3(true, 0, s(x5)) → cond1(or(false, true), 0, x5) [4] cond3(true, s(x4), 0) → cond1(or(true, false), s(x4), 0) [4] cond3(true, s(x4), s(x6)) → cond1(or(true, true), s(x4), x6) [4] cond3(false, 0, 0) → cond1(or(false, false), 0, 0) [3] cond3(false, 0, s(x8)) → cond1(or(false, true), 0, s(x8)) [3] cond3(false, s(x7), 0) → cond1(or(true, false), s(x7), 0) [3] cond3(false, s(x7), s(x9)) → cond1(or(true, true), s(x7), s(x9)) [3] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] or(false, false) → false [1] or(true, x) → true [1] or(x, true) → true [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond1 :: true:false → 0:s → 0:s → cond1:cond2:cond3 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2:cond3 gr :: 0:s → 0:s → true:false 0 :: 0:s or :: true:false → true:false → true:false p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 s :: 0:s → 0:s const :: cond1:cond2:cond3 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:

cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', y) :|: z' = 1 + x', z'' = y, z = 1, x' >= 0, y >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, x, 1 + x3) :|: z' = x, z'' = 1 + x3, x >= 0, z = 0, x3 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(0, x, 0) :|: z'' = 0, z' = x, x >= 0, z = 0
cond2(z, z', z'') -{ 4 }→ cond1(or(1, 1), x'', 1 + x2) :|: z' = 1 + x'', z = 1, z'' = 1 + x2, x'' >= 0, x2 >= 0
cond2(z, z', z'') -{ 4 }→ cond1(or(1, 0), x'', 0) :|: z'' = 0, z' = 1 + x'', z = 1, x'' >= 0
cond2(z, z', z'') -{ 4 }→ cond1(or(0, 1), 0, 1 + x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(or(0, 0), 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(or(1, 1), 1 + x4, x6) :|: x4 >= 0, z = 1, z' = 1 + x4, x6 >= 0, z'' = 1 + x6
cond3(z, z', z'') -{ 3 }→ cond1(or(1, 1), 1 + x7, 1 + x9) :|: z' = 1 + x7, x7 >= 0, z = 0, x9 >= 0, z'' = 1 + x9
cond3(z, z', z'') -{ 4 }→ cond1(or(1, 0), 1 + x4, 0) :|: z'' = 0, x4 >= 0, z = 1, z' = 1 + x4
cond3(z, z', z'') -{ 3 }→ cond1(or(1, 0), 1 + x7, 0) :|: z' = 1 + x7, z'' = 0, x7 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(or(0, 1), 0, x5) :|: x5 >= 0, z = 1, z'' = 1 + x5, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(or(0, 1), 0, 1 + x8) :|: x8 >= 0, z'' = 1 + x8, z = 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(or(0, 0), 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(or(0, 0), 0, 0) :|: z'' = 0, z = 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
or(z, z') -{ 1 }→ 1 :|: z' = x, z = 1, x >= 0
or(z, z') -{ 1 }→ 1 :|: x >= 0, z' = 1, z = x
or(z, z') -{ 1 }→ 0 :|: z = 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:

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', y) :|: z' = 1 + x', z'' = y, z = 1, x' >= 0, y >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, x, 1 + x3) :|: z' = x, z'' = 1 + x3, x >= 0, z = 0, x3 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(0, x, 0) :|: z'' = 0, z' = x, x >= 0, z = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, x'', 0) :|: z'' = 0, z' = 1 + x'', z = 1, x'' >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, x'', 1 + x2) :|: z' = 1 + x'', z = 1, z'' = 1 + x2, x'' >= 0, x2 >= 0, 1 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, 1 + x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, x5) :|: x5 >= 0, z = 1, z'' = 1 + x5, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x8) :|: x8 >= 0, z'' = 1 + x8, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + x4, x6) :|: x4 >= 0, z = 1, z' = 1 + x4, x6 >= 0, z'' = 1 + x6, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + x4, 0) :|: z'' = 0, x4 >= 0, z = 1, z' = 1 + x4, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + x7, 0) :|: z' = 1 + x7, z'' = 0, x7 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + x7, 1 + x9) :|: z' = 1 + x7, x7 >= 0, z = 0, x9 >= 0, z'' = 1 + x9, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = 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
or(z, z') -{ 1 }→ 1 :|: z' = x, z = 1, x >= 0
or(z, z') -{ 1 }→ 1 :|: x >= 0, z' = 1, z = x
or(z, z') -{ 1 }→ 0 :|: z = 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:

cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(0, z', 0) :|: z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 1 + (z'' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, z'' - 1) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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:

{ cond2, cond1, cond3 } { p } { gr } { or }

(16) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(0, z', 0) :|: z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 1 + (z'' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, z'' - 1) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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,cond3}, {p}, {gr}, {or}

(17) 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

Computed SIZE bound using CoFloCo for: cond3
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(18) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(0, z', 0) :|: z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 1 + (z'' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, z'' - 1) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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,cond3}, {p}, {gr}, {or}
Previous analysis results are:

cond2: runtime: ?, size: O(1) [0] cond1: runtime: ?, size: O(1) [0] cond3: runtime: ?, size: O(1) [0]

(19) 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: 48 + 7·z' + 9·z''

Computed RUNTIME bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 50 + 7·z' + 9·z''

Computed RUNTIME bound using CoFloCo for: cond3
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 55 + 7·z' + 9·z''

(20) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(0, z', 0) :|: z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, z' - 1, 1 + (z'' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, z'' - 1) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 5 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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}, {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0]

(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:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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}, {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0]

(23) 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

(24) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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}, {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: ?, size: O(n1) [z]

(25) 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

(26) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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}, {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z]

(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:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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}, {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z]

(29) 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

(30) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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}, {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: ?, size: O(1) [1]

(31) 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'

(32) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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: {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1]

(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:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
gr(z, z') -{ 1 + z' }→ s14 :|: s14 >= 0, s14 <= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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: {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1]

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

Computed SIZE bound using CoFloCo for: or
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:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
gr(z, z') -{ 1 + z' }→ s14 :|: s14 >= 0, s14 <= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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: {or}
Previous analysis results are:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1] or: runtime: ?, size: O(1) [1]

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

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 50 + 9·z'' }→ s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
cond1(z, z', z'') -{ 50 + 7·z' + 9·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
cond2(z, z', z'') -{ 57 + 7·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
cond2(z, z', z'') -{ 57 + 7·z' + 9·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 55 }→ s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond2(z, z', z'') -{ 55 + 9·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond2(z, z', z'') -{ 48 + 7·z' }→ s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
cond2(z, z', z'') -{ 48 + 7·z' + 9·z'' }→ s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 }→ s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
cond3(z, z', z'') -{ 54 + 9·z'' }→ s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 54 + 7·z' }→ s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 54 + 7·z' + 9·z'' }→ s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 55 }→ s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
cond3(z, z', z'') -{ 46 + 9·z'' }→ s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
cond3(z, z', z'') -{ 55 + 7·z' }→ s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
cond3(z, z', z'') -{ 46 + 7·z' + 9·z'' }→ s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
gr(z, z') -{ 1 + z' }→ s14 :|: s14 >= 0, s14 <= 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
or(z, z') -{ 1 }→ 1 :|: z = 1, z' >= 0
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1
or(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:

cond2: runtime: O(n1) [48 + 7·z' + 9·z''], size: O(1) [0] cond1: runtime: O(n1) [50 + 7·z' + 9·z''], size: O(1) [0] cond3: runtime: O(n1) [55 + 7·z' + 9·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1] or: runtime: O(1) [1], size: O(1) [1]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^1)