KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ 2 ∧ X₂ ≤ 2+X₁ ∧ X₁+X₂ ≤ 5 ∧ X₂ ≤ 2+X₀ ∧ X₀+X₂ ≤ 5 ∧ 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ 3 ∧ X₁ ≤ 3+X₀ ∧ X₀+X₁ ≤ 6 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ X₀ ≤ 3 ∧ 0 ≤ X₀ for location f1

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: f0, f1
Transitions:
t₀: f0(X₀, X₁, X₂) → f1(X₀, X₁, 2) :|: X₀ ≤ 3 ∧ X₁ ≤ 3 ∧ 0 ≤ X₀ ∧ 0 ≤ X₁
t₁: f1(X₀, X₁, X₂) → f1(X₀, 1+X₁, X₂) :|: 1+2⋅X₁ ≤ X₀+X₂ ∧ X₀+X₁ ≤ 6 ∧ X₀+X₂ ≤ 5 ∧ X₁+X₂ ≤ 5 ∧ X₁ ≤ 3+X₀ ∧ X₀ ≤ 3 ∧ X₀ ≤ 3+X₁ ∧ X₁ ≤ 3 ∧ X₂ ≤ 2+X₀ ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ 2 ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₂: f1(X₀, X₁, X₂) → f1(X₀, X₁-1, X₂) :|: 2+X₀+X₂ ≤ 2⋅X₁ ∧ X₀+X₁ ≤ 6 ∧ X₀+X₂ ≤ 5 ∧ X₁+X₂ ≤ 5 ∧ X₁ ≤ 3+X₀ ∧ X₀ ≤ 3 ∧ X₀ ≤ 3+X₁ ∧ X₁ ≤ 3 ∧ X₂ ≤ 2+X₀ ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ 2 ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁

MPRF for transition t₁: f1(X₀, X₁, X₂) → f1(X₀, 1+X₁, X₂) :|: 1+2⋅X₁ ≤ X₀+X₂ ∧ X₀+X₁ ≤ 6 ∧ X₀+X₂ ≤ 5 ∧ X₁+X₂ ≤ 5 ∧ X₁ ≤ 3+X₀ ∧ X₀ ≤ 3 ∧ X₀ ≤ 3+X₁ ∧ X₁ ≤ 3 ∧ X₂ ≤ 2+X₀ ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ 2 ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

7 {O(1)}

MPRF:

• f1: [4-X₁]

MPRF for transition t₂: f1(X₀, X₁, X₂) → f1(X₀, X₁-1, X₂) :|: 2+X₀+X₂ ≤ 2⋅X₁ ∧ X₀+X₁ ≤ 6 ∧ X₀+X₂ ≤ 5 ∧ X₁+X₂ ≤ 5 ∧ X₁ ≤ 3+X₀ ∧ X₀ ≤ 3 ∧ X₀ ≤ 3+X₁ ∧ X₁ ≤ 3 ∧ X₂ ≤ 2+X₀ ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ 2 ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

4 {O(1)}

MPRF:

• f1: [1+X₁]

All Bounds

Timebounds

Overall timebound:12 {O(1)}
t₀: 1 {O(1)}
t₁: 7 {O(1)}
t₂: 4 {O(1)}

Costbounds

Overall costbound: 12 {O(1)}
t₀: 1 {O(1)}
t₁: 7 {O(1)}
t₂: 4 {O(1)}

Sizebounds

t₀, X₀: 3 {O(1)}
t₀, X₁: 3 {O(1)}
t₀, X₂: 2 {O(1)}
t₁, X₀: 3 {O(1)}
t₁, X₁: 3 {O(1)}
t₁, X₂: 2 {O(1)}
t₂, X₀: 2 {O(1)}
t₂, X₁: 2 {O(1)}
t₂, X₂: 2 {O(1)}