KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0
t₃: l2(X₀, X₁, X₂) → l2(X₀+(X₂)², X₁+1, X₂+1) :|: X₀ < (X₁)²
t₄: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: (X₁)² ≤ X₀
t₅: l3(X₀, X₁, X₂) → l3(X₀-1, X₁, X₂) :|: 0 < X₀

Preprocessing

Found invariant 1 ≤ X₀ for location l2

Found invariant 1 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀
t₃: l2(X₀, X₁, X₂) → l2(X₀+(X₂)², X₁+1, X₂+1) :|: X₀ < (X₁)² ∧ 1 ≤ X₀
t₄: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: (X₁)² ≤ X₀ ∧ 1 ≤ X₀
t₅: l3(X₀, X₁, X₂) → l3(X₀-1, X₁, X₂) :|: 0 < X₀ ∧ 0 ≤ X₀

MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

TWN: t₃: l2→l2

cycle: [t₃: l2→l2]
loop: (X₀ < (X₁)²,(X₀,X₁,X₂) -> (X₀+(X₂)²,X₁+1,X₂+1)
order: [X₂; X₀; X₁]
closed-form:

X₂: X₂ + [[n != 0]] * n^1
X₀: X₀ + [[n != 0]] * (X₂)² * n^1 + [[n != 0, n != 1]] * 1/3 * n^3 + [[n != 0, n != 1]] * X₂-1/2 * n^2 + [[n != 0, n != 1]] * 1/6-X₂ * n^1
X₁: X₁ + [[n != 0]] * n^1

Termination: true
Formula:

2 < 0
∨ 6⋅X₂ < 9 ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 6⋅(X₂)²+1 < 6⋅X₂+12⋅X₁ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 6⋅X₂ ≤ 9 ∧ 9 ≤ 6⋅X₂
∨ 6⋅X₀ < 6⋅(X₁)² ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 6⋅X₂ ≤ 9 ∧ 9 ≤ 6⋅X₂ ∧ 6⋅(X₂)²+1 ≤ 6⋅X₂+12⋅X₁ ∧ 6⋅X₂+12⋅X₁ ≤ 6⋅(X₂)²+1

Stabilization-Threshold for: X₀ < (X₁)²

alphas_abs: 9+6⋅X₀+12⋅X₁+6⋅(X₁)²+6⋅X₂+6⋅(X₂)²
M: 0
N: 3
Bound: 12⋅X₁⋅X₁+12⋅X₂⋅X₂+12⋅X₀+12⋅X₂+24⋅X₁+22 {O(n^2)}

TWN - Lifting for t₃: l2→l2 of 12⋅X₁⋅X₁+12⋅X₂⋅X₂+12⋅X₀+12⋅X₂+24⋅X₁+24 {O(n^2)}

relevant size-bounds w.r.t. t₂:
X₀: 2⋅X₂⋅X₂+2⋅X₀+2⋅X₂ {O(n^2)}
X₁: 2⋅X₁ {O(n)}
X₂: 2⋅X₂ {O(n)}
Runtime-bound of t₂: 1 {O(1)}
Results in: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₁+48⋅X₂+24 {O(n^2)}

Analysing control-flow refined program

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Found invariant 1 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l3

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Found invariant 1 ≤ X₀ for location l2

Found invariant 0 ≤ X₀ for location n_l3___2

Found invariant 1 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location n_l3___1

Found invariant 1 ≤ X₀ for location l3

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₁+48⋅X₂+24 {O(n^2)}
t₄: 1 {O(1)}
t₅: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₁+48⋅X₂+24 {O(n^2)}
t₄: 1 {O(1)}
t₅: inf {Infinity}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 2⋅X₂⋅X₂+2⋅X₂+X₀ {O(n^2)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₂⋅X₂+2⋅X₀+2⋅X₂ {O(n^2)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₁: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₂+50⋅X₁+24 {O(n^2)}
t₃, X₂: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₁+50⋅X₂+24 {O(n^2)}
t₄, X₁: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₂+52⋅X₁+24 {O(n^2)}
t₄, X₂: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₁+52⋅X₂+24 {O(n^2)}
t₅, X₁: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₂+52⋅X₁+24 {O(n^2)}
t₅, X₂: 48⋅X₁⋅X₁+72⋅X₂⋅X₂+24⋅X₀+48⋅X₁+52⋅X₂+24 {O(n^2)}