The Runtime Complexity (innermost) of the given CpxTRS could be proven to be 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, y) → cond2(gr(x, y), x, y)
cond2(true, x, y) → cond1(gr(add(x, y), 0), p(x), y)
cond2(false, x, y) → cond3(eq(x, y), x, y)
cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y)
cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
add(0, x) → x
add(s(x), y) → s(add(x, y))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
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, y) → cond2(gr(x, y), x, y) [1]
cond2(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1]
cond2(false, x, y) → cond3(eq(x, y), x, y) [1]
cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1]
cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y)) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
add(0, x) → x [1]
add(s(x), y) → s(add(x, y)) [1]
eq(0, 0) → true [1]
eq(0, s(x)) → false [1]
eq(s(x), 0) → false [1]
eq(s(x), s(y)) → eq(x, y) [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, y), x, y) [1]
cond2(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1]
cond2(false, x, y) → cond3(eq(x, y), x, y) [1]
cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1]
cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y)) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
add(0, x) → x [1]
add(s(x), y) → s(add(x, y)) [1]
eq(0, 0) → true [1]
eq(0, s(x)) → false [1]
eq(s(x), 0) → false [1]
eq(s(x), s(y)) → eq(x, y) [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 add :: 0:s → 0:s → 0:s 0 :: 0:s p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 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
cond3

(c) The following functions are completely defined:

gr
add
p
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(x, y), x, y) [1] cond2(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1] cond2(false, x, y) → cond3(eq(x, y), x, y) [1] cond3(true, x, y) → cond1(gr(add(x, y), 0), p(x), y) [1] cond3(false, x, y) → cond1(gr(add(x, y), 0), x, p(y)) [1] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] add(0, x) → x [1] add(s(x), y) → s(add(x, y)) [1] eq(0, 0) → true [1] eq(0, s(x)) → false [1] eq(s(x), 0) → false [1] eq(s(x), s(y)) → eq(x, y) [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 add :: 0:s → 0:s → 0:s 0 :: 0:s p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 eq :: 0:s → 0:s → true:false 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'), 0) → cond2(true, s(x'), 0) [2] cond1(true, s(x''), s(y')) → cond2(gr(x'', y'), s(x''), s(y')) [2] cond2(true, 0, y) → cond1(gr(y, 0), 0, y) [3] cond2(true, s(x1), y) → cond1(gr(s(add(x1, y)), 0), x1, y) [3] cond2(false, 0, 0) → cond3(true, 0, 0) [2] cond2(false, 0, s(x2)) → cond3(false, 0, s(x2)) [2] cond2(false, s(x3), 0) → cond3(false, s(x3), 0) [2] cond2(false, s(x4), s(y'')) → cond3(eq(x4, y''), s(x4), s(y'')) [2] cond3(true, 0, y) → cond1(gr(y, 0), 0, y) [3] cond3(true, s(x5), y) → cond1(gr(s(add(x5, y)), 0), x5, y) [3] cond3(false, 0, 0) → cond1(gr(0, 0), 0, 0) [3] cond3(false, 0, s(x7)) → cond1(gr(s(x7), 0), 0, x7) [3] cond3(false, s(x6), 0) → cond1(gr(s(add(x6, 0)), 0), s(x6), 0) [3] cond3(false, s(x6), s(x8)) → cond1(gr(s(add(x6, s(x8))), 0), s(x6), x8) [3] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] add(0, x) → x [1] add(s(x), y) → s(add(x, y)) [1] eq(0, 0) → true [1] eq(0, s(x)) → false [1] eq(s(x), 0) → false [1] eq(s(x), s(y)) → eq(x, y) [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 add :: 0:s → 0:s → 0:s 0 :: 0:s p :: 0:s → 0:s false :: true:false cond3 :: true:false → 0:s → 0:s → cond1:cond2:cond3 eq :: 0:s → 0:s → true:false 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:

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

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 0), 0), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 1 + (z'' - 1)), 0), 1 + (z' - 1), z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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

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

Found the following analysis order by SCC decomposition:

{ add } { eq } { p } { gr } { cond2, cond1, cond3 }

(14) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 0), 0), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 1 + (z'' - 1)), 0), 1 + (z' - 1), z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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: {add}, {eq}, {p}, {gr}, {cond2,cond1,cond3}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 0), 0), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 1 + (z'' - 1)), 0), 1 + (z' - 1), z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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: {add}, {eq}, {p}, {gr}, {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: ?, size: O(n1) [z + z']

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 }→ 1 + add(z - 1, z') :|: z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, z''), 0), z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 0), 0), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + add(z' - 1, 1 + (z'' - 1)), 0), 1 + (z' - 1), z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z']

(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:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z']

(21) 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

(22) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] eq: runtime: ?, size: O(1) [1]

(23) 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'

(24) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(eq(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] eq: runtime: O(n1) [1 + z'], size: O(1) [1]

(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:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] eq: runtime: O(n1) [1 + z'], size: O(1) [1]

(27) 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

(28) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] eq: runtime: O(n1) [1 + z'], size: O(1) [1] p: runtime: ?, size: O(n1) [z]

(29) 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

(30) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] eq: runtime: O(n1) [1 + z'], size: O(1) [1] p: runtime: O(1) [1], 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:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] eq: runtime: O(n1) [1 + z'], size: O(1) [1] p: runtime: O(1) [1], size: O(n1) [z]

(33) 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

(34) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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]

(35) 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'

(36) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s, 0), z' - 1, z'') :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(z'', 0), 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s', 0), z' - 1, z'') :|: s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s'', 0), 1 + (z' - 1), 0) :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 + z' }→ cond1(gr(1 + s1, 0), 1 + (z' - 1), z'' - 1) :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 3 }→ cond1(gr(1 + (z'' - 1), 0), 0, z'' - 1) :|: z'' - 1 >= 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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]

(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:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 + z'' }→ cond2(s5, 1 + (z' - 1), 1 + (z'' - 1)) :|: s5 >= 0, s5 <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond1(s6, 0, z'') :|: s6 >= 0, s6 <= 1, z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 4 + z' }→ cond1(s7, z' - 1, z'') :|: s7 >= 0, s7 <= 1, s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 4 }→ cond1(s10, 0, 0) :|: s10 >= 0, s10 <= 1, z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(s11, 0, z'' - 1) :|: s11 >= 0, s11 <= 1, z'' - 1 >= 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s12, 1 + (z' - 1), 0) :|: s12 >= 0, s12 <= 1, s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s13, 1 + (z' - 1), z'' - 1) :|: s13 >= 0, s13 <= 1, s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(s8, 0, z'') :|: s8 >= 0, s8 <= 1, z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s9, z' - 1, z'') :|: s9 >= 0, s9 <= 1, s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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) 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

(40) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 + z'' }→ cond2(s5, 1 + (z' - 1), 1 + (z'' - 1)) :|: s5 >= 0, s5 <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond1(s6, 0, z'') :|: s6 >= 0, s6 <= 1, z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 4 + z' }→ cond1(s7, z' - 1, z'') :|: s7 >= 0, s7 <= 1, s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 4 }→ cond1(s10, 0, 0) :|: s10 >= 0, s10 <= 1, z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(s11, 0, z'' - 1) :|: s11 >= 0, s11 <= 1, z'' - 1 >= 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s12, 1 + (z' - 1), 0) :|: s12 >= 0, s12 <= 1, s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s13, 1 + (z' - 1), z'' - 1) :|: s13 >= 0, s13 <= 1, s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(s8, 0, z'') :|: s8 >= 0, s8 <= 1, z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s9, z' - 1, z'') :|: s9 >= 0, s9 <= 1, s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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] cond3: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(42) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 2 + z'' }→ cond2(s5, 1 + (z' - 1), 1 + (z'' - 1)) :|: s5 >= 0, s5 <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 2 + z'' }→ cond3(s3, 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 0, z' - 1 >= 0
cond2(z, z', z'') -{ 4 }→ cond1(s6, 0, z'') :|: s6 >= 0, s6 <= 1, z = 1, z'' >= 0, z' = 0
cond2(z, z', z'') -{ 4 + z' }→ cond1(s7, z' - 1, z'') :|: s7 >= 0, s7 <= 1, s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
cond3(z, z', z'') -{ 4 }→ cond1(s10, 0, 0) :|: s10 >= 0, s10 <= 1, z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(s11, 0, z'' - 1) :|: s11 >= 0, s11 <= 1, z'' - 1 >= 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s12, 1 + (z' - 1), 0) :|: s12 >= 0, s12 <= 1, s'' >= 0, s'' <= 1 * (z' - 1) + 1 * 0, z'' = 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s13, 1 + (z' - 1), z'' - 1) :|: s13 >= 0, s13 <= 1, s1 >= 0, s1 <= 1 * (z' - 1) + 1 * (1 + (z'' - 1)), z'' - 1 >= 0, z' - 1 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(s8, 0, z'') :|: s8 >= 0, s8 <= 1, z = 1, z'' >= 0, z' = 0
cond3(z, z', z'') -{ 4 + z' }→ cond1(s9, z' - 1, z'') :|: s9 >= 0, s9 <= 1, s' >= 0, s' <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z = 1, z'' >= 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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: INF, size: O(1) [0] cond1: runtime: ?, size: O(1) [0] cond3: runtime: ?, size: O(1) [0]

(43) RetryTechniqueProof (BOTH BOUNDS(ID, ID) transformation)

Performed narrowing of the following TRS rules:

cond1(true, s(x''), s(y')) → cond2(gr(x'', y'), s(x''), s(y')) [2]
cond3(false, 0, 0) → cond1(gr(0, 0), 0, 0) [3]
cond3(false, 0, s(x7)) → cond1(gr(s(x7), 0), 0, x7) [3]
cond2(true, 0, y) → cond1(gr(y, 0), 0, y) [3]
cond2(false, 0, s(x2)) → cond3(false, 0, s(x2)) [2]
cond2(false, 0, 0) → cond3(true, 0, 0) [2]
cond2(false, s(x4), s(y'')) → cond3(eq(x4, y''), s(x4), s(y'')) [2]
cond3(false, s(x6), s(x8)) → cond1(gr(s(add(x6, s(x8))), 0), s(x6), x8) [3]
cond1(true, s(x'), 0) → cond2(true, s(x'), 0) [2]
cond2(true, s(x1), y) → cond1(gr(s(add(x1, y)), 0), x1, y) [3]
cond1(true, 0, y) → cond2(false, 0, y) [2]
cond3(true, 0, y) → cond1(gr(y, 0), 0, y) [3]
cond2(false, s(x3), 0) → cond3(false, s(x3), 0) [2]
cond3(false, s(x6), 0) → cond1(gr(s(add(x6, 0)), 0), s(x6), 0) [3]
cond3(true, s(x5), y) → cond1(gr(s(add(x5, y)), 0), x5, y) [3]

And obtained the following new TRS rules:

cond1(true, s(0), s(y')) → cond2(false, s(0), s(y')) [3]
cond1(true, s(s(x)), s(0)) → cond2(true, s(s(x)), s(0)) [3]
cond1(true, s(s(x9)), s(s(y1))) → cond2(gr(x9, y1), s(s(x9)), s(s(y1))) [3]
cond3(false, 0, 0) → cond1(false, 0, 0) [4]
cond3(false, 0, s(x7)) → cond1(true, 0, x7) [4]
cond2(true, 0, 0) → cond1(false, 0, 0) [4]
cond2(true, 0, s(x10)) → cond1(true, 0, s(x10)) [4]
cond2(false, 0, s(x2)) → cond3(false, 0, s(x2)) [2]
cond2(false, 0, 0) → cond3(true, 0, 0) [2]
cond2(false, s(0), s(0)) → cond3(true, s(0), s(0)) [3]
cond2(false, s(0), s(s(x11))) → cond3(false, s(0), s(s(x11))) [3]
cond2(false, s(s(x12)), s(0)) → cond3(false, s(s(x12)), s(0)) [3]
cond2(false, s(s(x13)), s(s(y2))) → cond3(eq(x13, y2), s(s(x13)), s(s(y2))) [3]
cond3(false, s(0), s(x8)) → cond1(gr(s(s(x8)), 0), s(0), x8) [4]
cond3(false, s(s(x14)), s(x8)) → cond1(gr(s(s(add(x14, s(x8)))), 0), s(s(x14)), x8) [4]
cond1(true, s(x'), 0) → cond2(true, s(x'), 0) [2]
cond2(true, s(0), y) → cond1(gr(s(y), 0), 0, y) [4]
cond2(true, s(s(x15)), y) → cond1(gr(s(s(add(x15, y))), 0), s(x15), y) [4]
cond1(true, 0, y) → cond2(false, 0, y) [2]
cond3(true, 0, 0) → cond1(false, 0, 0) [4]
cond3(true, 0, s(x16)) → cond1(true, 0, s(x16)) [4]
cond2(false, s(x3), 0) → cond3(false, s(x3), 0) [2]
cond3(false, s(0), 0) → cond1(gr(s(0), 0), s(0), 0) [4]
cond3(false, s(s(x17)), 0) → cond1(gr(s(s(add(x17, 0))), 0), s(s(x17)), 0) [4]
cond3(true, s(0), y) → cond1(gr(s(y), 0), 0, y) [4]
cond3(true, s(s(x18)), y) → cond1(gr(s(s(add(x18, y))), 0), s(x18), y) [4]

Which were then size abstracted to RNTS rules to simplify the current SCC:

cond1(z, z', z'') -{ 3 }→ cond2(0, 1 + 0, 1 + y') :|: z = 1, y' >= 0, z' = 1 + 0, z'' = 1 + y'
cond1(z, z', z'') -{ 3 }→ cond2(1, 1 + (1 + x), 1 + 0) :|: z = 1, x >= 0, z' = 1 + (1 + x), z'' = 1 + 0
cond1(z, z', z'') -{ 3 }→ cond2(gr(x9, y1), 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, x7) :|: x7 >= 0, z'' = 1 + x7, z = 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x10) :|: z = 1, z'' = 1 + x10, x10 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + x2) :|: z'' = 1 + x2, z = 0, x2 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond3(1, 1 + 0, 1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + 0, 1 + (1 + x11)) :|: x11 >= 0, z' = 1 + 0, z = 0, z'' = 1 + (1 + x11)
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + (1 + x12), 1 + 0) :|: x12 >= 0, z = 0, z' = 1 + (1 + x12), z'' = 1 + 0
cond2(z, z', z'') -{ 3 }→ cond3(eq(x13, y2), 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + x8), 0), 1 + 0, x8) :|: x8 >= 0, z'' = 1 + x8, z' = 1 + 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x14, 1 + x8)), 0), 1 + (1 + x14), x8) :|: x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond2(z, z', z'') -{ 4 }→ cond1(gr(1 + y, 0), 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x15, y)), 0), 1 + x15, y) :|: z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + 0, 0), 1 + 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x17, 0)), 0), 1 + (1 + x17), 0) :|: z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + y, 0), 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x18, y)), 0), 1 + x18, y) :|: z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0

(44) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 3 }→ cond2(gr(x9, y1), 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond1(z, z', z'') -{ 3 }→ cond2(1, 1 + (1 + x), 1 + 0) :|: z = 1, x >= 0, z' = 1 + (1 + x), z'' = 1 + 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond1(z, z', z'') -{ 3 }→ cond2(0, 1 + 0, 1 + y') :|: z = 1, y' >= 0, z' = 1 + 0, z'' = 1 + y'
cond2(z, z', z'') -{ 3 }→ cond3(eq(x13, y2), 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond3(1, 1 + 0, 1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + x2) :|: z'' = 1 + x2, z = 0, x2 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + 0, 1 + (1 + x11)) :|: x11 >= 0, z' = 1 + 0, z = 0, z'' = 1 + (1 + x11)
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + (1 + x12), 1 + 0) :|: x12 >= 0, z = 0, z' = 1 + (1 + x12), z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond1(gr(1 + y, 0), 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x15, y)), 0), 1 + x15, y) :|: z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0
cond2(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x10) :|: z = 1, z'' = 1 + x10, x10 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + y, 0), 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + 0, 0), 1 + 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + x8), 0), 1 + 0, x8) :|: x8 >= 0, z'' = 1 + x8, z' = 1 + 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x14, 1 + x8)), 0), 1 + (1 + x14), x8) :|: x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x17, 0)), 0), 1 + (1 + x17), 0) :|: z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0
cond3(z, z', z'') -{ 4 }→ cond1(gr(1 + (1 + add(x18, y)), 0), 1 + x18, y) :|: z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, x7) :|: x7 >= 0, z'' = 1 + x7, z = 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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]

(45) InliningProof (UPPER BOUND(ID) transformation)

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

gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 + z' }→ s14 :|: s14 >= 0, s14 <= 1, z - 1 >= 0, z' - 1 >= 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1, z - 1 >= 0, z' - 1 >= 0
eq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
eq(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
eq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

(46) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 4 + y1 }→ cond2(s14, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, s14 >= 0, s14 <= 1, x9 - 1 >= 0, y1 - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond1(z, z', z'') -{ 3 }→ cond2(1, 1 + (1 + x), 1 + 0) :|: z = 1, x >= 0, z' = 1 + (1 + x), z'' = 1 + 0
cond1(z, z', z'') -{ 4 }→ cond2(1, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 - 1 >= 0, y1 = 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond1(z, z', z'') -{ 3 }→ cond2(0, 1 + 0, 1 + y') :|: z = 1, y' >= 0, z' = 1 + 0, z'' = 1 + y'
cond1(z, z', z'') -{ 4 }→ cond2(0, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 = 0
cond2(z, z', z'') -{ 4 + y2 }→ cond3(s4, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, s4 >= 0, s4 <= 1, x13 - 1 >= 0, y2 - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond3(1, 1 + 0, 1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(1, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 = 0, y2 = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + x2) :|: z'' = 1 + x2, z = 0, x2 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + 0, 1 + (1 + x11)) :|: x11 >= 0, z' = 1 + 0, z = 0, z'' = 1 + (1 + x11)
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + (1 + x12), 1 + 0) :|: x12 >= 0, z = 0, z' = 1 + (1 + x12), z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, y2 - 1 >= 0, x13 = 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 - 1 >= 0, y2 = 0
cond2(z, z', z'') -{ 5 }→ cond1(gr(1 + (1 + y), 0), 1 + x15, y) :|: z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, x15 = 0
cond2(z, z', z'') -{ 5 + x15 }→ cond1(gr(1 + (1 + (1 + s2)), 0), 1 + x15, y) :|: z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, s2 >= 0, s2 <= 1 * (x15 - 1) + 1 * y, x15 - 1 >= 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x10) :|: z = 1, z'' = 1 + x10, x10 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(gr(1 + (1 + y), 0), 1 + x18, y) :|: z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, x18 = 0
cond3(z, z', z'') -{ 5 }→ cond1(gr(1 + (1 + 0), 0), 1 + (1 + x17), 0) :|: z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, 0 >= 0, x17 = 0
cond3(z, z', z'') -{ 5 + x18 }→ cond1(gr(1 + (1 + (1 + s2)), 0), 1 + x18, y) :|: z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, s2 >= 0, s2 <= 1 * (x18 - 1) + 1 * y, x18 - 1 >= 0
cond3(z, z', z'') -{ 5 + x14 }→ cond1(gr(1 + (1 + (1 + s2)), 0), 1 + (1 + x14), x8) :|: x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x14 - 1) + 1 * (1 + x8), x14 - 1 >= 0, 1 + x8 >= 0
cond3(z, z', z'') -{ 5 + x17 }→ cond1(gr(1 + (1 + (1 + s2)), 0), 1 + (1 + x17), 0) :|: z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x17 - 1) + 1 * 0, x17 - 1 >= 0, 0 >= 0
cond3(z, z', z'') -{ 5 }→ cond1(gr(1 + (1 + (1 + x8)), 0), 1 + (1 + x14), x8) :|: x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, 1 + x8 >= 0, x14 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, x7) :|: x7 >= 0, z'' = 1 + x7, z = 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, x8) :|: x8 >= 0, z'' = 1 + x8, z' = 1 + 0, z = 0, 1 + (1 + x8) - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0, 1 + 0 - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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]

(47) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(48) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 4 + y1 }→ cond2(s14, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, s14 >= 0, s14 <= 1, x9 - 1 >= 0, y1 - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond1(z, z', z'') -{ 3 }→ cond2(1, 1 + (1 + x), 1 + 0) :|: z = 1, x >= 0, z' = 1 + (1 + x), z'' = 1 + 0
cond1(z, z', z'') -{ 4 }→ cond2(1, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 - 1 >= 0, y1 = 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond1(z, z', z'') -{ 3 }→ cond2(0, 1 + 0, 1 + y') :|: z = 1, y' >= 0, z' = 1 + 0, z'' = 1 + y'
cond1(z, z', z'') -{ 4 }→ cond2(0, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 = 0
cond2(z, z', z'') -{ 4 + y2 }→ cond3(s4, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, s4 >= 0, s4 <= 1, x13 - 1 >= 0, y2 - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond3(1, 1 + 0, 1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(1, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 = 0, y2 = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + x2) :|: z'' = 1 + x2, z = 0, x2 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + 0, 1 + (1 + x11)) :|: x11 >= 0, z' = 1 + 0, z = 0, z'' = 1 + (1 + x11)
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + (1 + x12), 1 + 0) :|: x12 >= 0, z = 0, z' = 1 + (1 + x12), z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, y2 - 1 >= 0, x13 = 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 - 1 >= 0, y2 = 0
cond2(z, z', z'') -{ 6 + x15 }→ cond1(s, 1 + x15, y) :|: s >= 0, s <= 1, z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, s2 >= 0, s2 <= 1 * (x15 - 1) + 1 * y, x15 - 1 >= 0
cond2(z, z', z'') -{ 6 }→ cond1(s', 1 + x15, y) :|: s' >= 0, s' <= 1, z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, x15 = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x10) :|: z = 1, z'' = 1 + x10, x10 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 6 + x17 }→ cond1(s'', 1 + (1 + x17), 0) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x17 - 1) + 1 * 0, x17 - 1 >= 0, 0 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s1, 1 + (1 + x17), 0) :|: s1 >= 0, s1 <= 1, z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, 0 >= 0, x17 = 0
cond3(z, z', z'') -{ 6 + x14 }→ cond1(s3, 1 + (1 + x14), x8) :|: s3 >= 0, s3 <= 1, x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x14 - 1) + 1 * (1 + x8), x14 - 1 >= 0, 1 + x8 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s5, 1 + (1 + x14), x8) :|: s5 >= 0, s5 <= 1, x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, 1 + x8 >= 0, x14 = 0
cond3(z, z', z'') -{ 6 + x18 }→ cond1(s6, 1 + x18, y) :|: s6 >= 0, s6 <= 1, z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, s2 >= 0, s2 <= 1 * (x18 - 1) + 1 * y, x18 - 1 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s7, 1 + x18, y) :|: s7 >= 0, s7 <= 1, z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, x18 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, x7) :|: x7 >= 0, z'' = 1 + x7, z = 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, x8) :|: x8 >= 0, z'' = 1 + x8, z' = 1 + 0, z = 0, 1 + (1 + x8) - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0, 1 + 0 - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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]

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT 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

(50) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 4 + y1 }→ cond2(s14, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, s14 >= 0, s14 <= 1, x9 - 1 >= 0, y1 - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond1(z, z', z'') -{ 3 }→ cond2(1, 1 + (1 + x), 1 + 0) :|: z = 1, x >= 0, z' = 1 + (1 + x), z'' = 1 + 0
cond1(z, z', z'') -{ 4 }→ cond2(1, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 - 1 >= 0, y1 = 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond1(z, z', z'') -{ 3 }→ cond2(0, 1 + 0, 1 + y') :|: z = 1, y' >= 0, z' = 1 + 0, z'' = 1 + y'
cond1(z, z', z'') -{ 4 }→ cond2(0, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 = 0
cond2(z, z', z'') -{ 4 + y2 }→ cond3(s4, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, s4 >= 0, s4 <= 1, x13 - 1 >= 0, y2 - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond3(1, 1 + 0, 1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(1, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 = 0, y2 = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + x2) :|: z'' = 1 + x2, z = 0, x2 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + 0, 1 + (1 + x11)) :|: x11 >= 0, z' = 1 + 0, z = 0, z'' = 1 + (1 + x11)
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + (1 + x12), 1 + 0) :|: x12 >= 0, z = 0, z' = 1 + (1 + x12), z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, y2 - 1 >= 0, x13 = 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 - 1 >= 0, y2 = 0
cond2(z, z', z'') -{ 6 + x15 }→ cond1(s, 1 + x15, y) :|: s >= 0, s <= 1, z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, s2 >= 0, s2 <= 1 * (x15 - 1) + 1 * y, x15 - 1 >= 0
cond2(z, z', z'') -{ 6 }→ cond1(s', 1 + x15, y) :|: s' >= 0, s' <= 1, z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, x15 = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x10) :|: z = 1, z'' = 1 + x10, x10 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 6 + x17 }→ cond1(s'', 1 + (1 + x17), 0) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x17 - 1) + 1 * 0, x17 - 1 >= 0, 0 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s1, 1 + (1 + x17), 0) :|: s1 >= 0, s1 <= 1, z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, 0 >= 0, x17 = 0
cond3(z, z', z'') -{ 6 + x14 }→ cond1(s3, 1 + (1 + x14), x8) :|: s3 >= 0, s3 <= 1, x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x14 - 1) + 1 * (1 + x8), x14 - 1 >= 0, 1 + x8 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s5, 1 + (1 + x14), x8) :|: s5 >= 0, s5 <= 1, x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, 1 + x8 >= 0, x14 = 0
cond3(z, z', z'') -{ 6 + x18 }→ cond1(s6, 1 + x18, y) :|: s6 >= 0, s6 <= 1, z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, s2 >= 0, s2 <= 1 * (x18 - 1) + 1 * y, x18 - 1 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s7, 1 + x18, y) :|: s7 >= 0, s7 <= 1, z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, x18 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, x7) :|: x7 >= 0, z'' = 1 + x7, z = 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, x8) :|: x8 >= 0, z'' = 1 + x8, z' = 1 + 0, z = 0, 1 + (1 + x8) - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0, 1 + 0 - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1,cond3}
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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] cond3: runtime: ?, size: O(1) [0]

(51) 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: 463 + 411·z' + 15·z'·z'' + 6·z'² + 215·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: 3405 + 519·z' + 15·z'·z'' + 6·z'² + 324·z'' + 6·z''²

Computed RUNTIME bound using PUBS for: cond3
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5975 + 583·z' + 15·z'·z'' + 6·z'² + 396·z'' + 6·z''²

(52) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add(z, z') -{ 1 + z }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
cond1(z, z', z'') -{ 4 + y1 }→ cond2(s14, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, s14 >= 0, s14 <= 1, x9 - 1 >= 0, y1 - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond1(z, z', z'') -{ 3 }→ cond2(1, 1 + (1 + x), 1 + 0) :|: z = 1, x >= 0, z' = 1 + (1 + x), z'' = 1 + 0
cond1(z, z', z'') -{ 4 }→ cond2(1, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 - 1 >= 0, y1 = 0
cond1(z, z', z'') -{ 2 }→ cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
cond1(z, z', z'') -{ 3 }→ cond2(0, 1 + 0, 1 + y') :|: z = 1, y' >= 0, z' = 1 + 0, z'' = 1 + y'
cond1(z, z', z'') -{ 4 }→ cond2(0, 1 + (1 + x9), 1 + (1 + y1)) :|: y1 >= 0, z = 1, z'' = 1 + (1 + y1), z' = 1 + (1 + x9), x9 >= 0, x9 = 0
cond2(z, z', z'') -{ 4 + y2 }→ cond3(s4, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, s4 >= 0, s4 <= 1, x13 - 1 >= 0, y2 - 1 >= 0
cond2(z, z', z'') -{ 2 }→ cond3(1, 0, 0) :|: z'' = 0, z = 0, z' = 0
cond2(z, z', z'') -{ 3 }→ cond3(1, 1 + 0, 1 + 0) :|: z' = 1 + 0, z = 0, z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(1, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 = 0, y2 = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 0, 1 + x2) :|: z'' = 1 + x2, z = 0, x2 >= 0, z' = 0
cond2(z, z', z'') -{ 2 }→ cond3(0, 1 + x3, 0) :|: z'' = 0, z' = 1 + x3, z = 0, x3 >= 0
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + 0, 1 + (1 + x11)) :|: x11 >= 0, z' = 1 + 0, z = 0, z'' = 1 + (1 + x11)
cond2(z, z', z'') -{ 3 }→ cond3(0, 1 + (1 + x12), 1 + 0) :|: x12 >= 0, z = 0, z' = 1 + (1 + x12), z'' = 1 + 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, y2 - 1 >= 0, x13 = 0
cond2(z, z', z'') -{ 4 }→ cond3(0, 1 + (1 + x13), 1 + (1 + y2)) :|: x13 >= 0, z' = 1 + (1 + x13), z'' = 1 + (1 + y2), y2 >= 0, z = 0, x13 - 1 >= 0, y2 = 0
cond2(z, z', z'') -{ 6 + x15 }→ cond1(s, 1 + x15, y) :|: s >= 0, s <= 1, z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, s2 >= 0, s2 <= 1 * (x15 - 1) + 1 * y, x15 - 1 >= 0
cond2(z, z', z'') -{ 6 }→ cond1(s', 1 + x15, y) :|: s' >= 0, s' <= 1, z' = 1 + (1 + x15), z'' = y, z = 1, y >= 0, x15 >= 0, x15 = 0
cond2(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond2(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x10) :|: z = 1, z'' = 1 + x10, x10 >= 0, z' = 0
cond2(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 6 + x17 }→ cond1(s'', 1 + (1 + x17), 0) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x17 - 1) + 1 * 0, x17 - 1 >= 0, 0 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s1, 1 + (1 + x17), 0) :|: s1 >= 0, s1 <= 1, z' = 1 + (1 + x17), z'' = 0, x17 >= 0, z = 0, 0 >= 0, x17 = 0
cond3(z, z', z'') -{ 6 + x14 }→ cond1(s3, 1 + (1 + x14), x8) :|: s3 >= 0, s3 <= 1, x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, s2 >= 0, s2 <= 1 * (x14 - 1) + 1 * (1 + x8), x14 - 1 >= 0, 1 + x8 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s5, 1 + (1 + x14), x8) :|: s5 >= 0, s5 <= 1, x8 >= 0, z' = 1 + (1 + x14), z'' = 1 + x8, x14 >= 0, z = 0, 1 + x8 >= 0, x14 = 0
cond3(z, z', z'') -{ 6 + x18 }→ cond1(s6, 1 + x18, y) :|: s6 >= 0, s6 <= 1, z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, s2 >= 0, s2 <= 1 * (x18 - 1) + 1 * y, x18 - 1 >= 0
cond3(z, z', z'') -{ 6 }→ cond1(s7, 1 + x18, y) :|: s7 >= 0, s7 <= 1, z' = 1 + (1 + x18), z'' = y, z = 1, y >= 0, x18 >= 0, x18 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, x7) :|: x7 >= 0, z'' = 1 + x7, z = 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + 0, 1 + y - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(1, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, x8) :|: x8 >= 0, z'' = 1 + x8, z' = 1 + 0, z = 0, 1 + (1 + x8) - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 5 }→ cond1(1, 1 + 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0, 1 + 0 - 1 >= 0, 0 = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0
cond3(z, z', z'') -{ 4 }→ cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0
eq(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 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' }→ 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
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

add: runtime: O(n1) [1 + z], size: O(n1) [z + z'] 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(n2) [463 + 411·z' + 15·z'·z'' + 6·z'2 + 215·z'' + 6·z''2], size: O(1) [0] cond1: runtime: O(n2) [3405 + 519·z' + 15·z'·z'' + 6·z'2 + 324·z'' + 6·z''2], size: O(1) [0] cond3: runtime: O(n2) [5975 + 583·z' + 15·z'·z'' + 6·z'2 + 396·z'' + 6·z''2], size: O(1) [0]

(53) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(54) BOUNDS(1, n^2)