KoAT2 Proof Output

Initial Problem

Start: f12
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: G, H
Locations: f10, f11, f12, f5, f7, f8
Transitions:
t₀: f12(X₀, X₁, X₂, X₃, X₄, X₅) → f5(400, 0, 0, X₃, X₄, X₅)
t₂: f5(X₀, X₁, X₂, X₃, X₄, X₅) → f10(X₀, X₁, X₂, X₂, X₂, X₅) :|: 1 ≤ X₂
t₃: f5(X₀, X₁, X₂, X₃, X₄, X₅) → f10(X₀, X₁, X₂, X₂, X₂, X₅) :|: 1+X₂ ≤ 0
t₁: f5(X₀, X₁, X₂, X₃, X₄, X₅) → f11(X₀, X₁, 0, 0, 0, X₅) :|: X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₆: f5(X₀, X₁, X₂, X₃, X₄, X₅) → f5(X₀, 1+X₁, G, H, X₄, G) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₄: f7(X₀, X₁, X₂, X₃, X₄, X₅) → f5(X₀, 1+X₁, G, H, X₄, G)
t₅: f8(X₀, X₁, X₂, X₃, X₄, X₅) → f5(X₀, 1+X₁, G, H, X₄, G)

Preprocessing

Cut unreachable locations [f7; f8] from the program graph

Eliminate variables [H; X₃; X₄; X₅] that do not contribute to the problem

Found invariant 0 ≤ X₁ ∧ 400 ≤ X₀+X₁ ∧ X₀ ≤ 400+X₁ ∧ X₀ ≤ 400 ∧ 400 ≤ X₀ for location f10

Found invariant X₂ ≤ 0 ∧ 400+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 400 ∧ 400+X₂ ≤ X₀ ∧ X₀+X₂ ≤ 400 ∧ 0 ≤ X₂ ∧ 400 ≤ X₁+X₂ ∧ X₁ ≤ 400+X₂ ∧ 400 ≤ X₀+X₂ ∧ X₀ ≤ 400+X₂ ∧ X₁ ≤ 400 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 800 ∧ 400 ≤ X₁ ∧ 800 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 400 ∧ 400 ≤ X₀ for location f11

Found invariant X₁ ≤ 400 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 800 ∧ 0 ≤ X₁ ∧ 400 ≤ X₀+X₁ ∧ X₀ ≤ 400+X₁ ∧ X₀ ≤ 400 ∧ 400 ≤ X₀ for location f5

Problem after Preprocessing

Start: f12
Program_Vars: X₀, X₁, X₂
Temp_Vars: G
Locations: f10, f11, f12, f5
Transitions:
t₁₂: f12(X₀, X₁, X₂) → f5(400, 0, 0)
t₁₃: f5(X₀, X₁, X₂) → f10(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ X₀+X₁ ≤ 800 ∧ X₀ ≤ 400 ∧ X₀ ≤ 400+X₁ ∧ X₁ ≤ 400 ∧ 400 ≤ X₀ ∧ 400 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₄: f5(X₀, X₁, X₂) → f10(X₀, X₁, X₂) :|: 1+X₂ ≤ 0 ∧ X₀+X₁ ≤ 800 ∧ X₀ ≤ 400 ∧ X₀ ≤ 400+X₁ ∧ X₁ ≤ 400 ∧ 400 ≤ X₀ ∧ 400 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₅: f5(X₀, X₁, X₂) → f11(X₀, X₁, 0) :|: X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₀+X₁ ≤ 800 ∧ X₀ ≤ 400 ∧ X₀ ≤ 400+X₁ ∧ X₁ ≤ 400 ∧ 400 ≤ X₀ ∧ 400 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₆: f5(X₀, X₁, X₂) → f5(X₀, 1+X₁, G) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₀+X₁ ≤ 800 ∧ X₀ ≤ 400 ∧ X₀ ≤ 400+X₁ ∧ X₁ ≤ 400 ∧ 400 ≤ X₀ ∧ 400 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁

MPRF for transition t₁₆: f5(X₀, X₁, X₂) → f5(X₀, 1+X₁, G) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₀+X₁ ≤ 800 ∧ X₀ ≤ 400 ∧ X₀ ≤ 400+X₁ ∧ X₁ ≤ 400 ∧ 400 ≤ X₀ ∧ 400 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

401 {O(1)}

MPRF:

• f5: [401-X₁]

All Bounds

Timebounds

Overall timebound:405 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 401 {O(1)}

Costbounds

Overall costbound: 405 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 401 {O(1)}

Sizebounds

t₁₂, X₀: 400 {O(1)}
t₁₂, X₁: 0 {O(1)}
t₁₂, X₂: 0 {O(1)}
t₁₃, X₀: 400 {O(1)}
t₁₃, X₁: 400 {O(1)}
t₁₄, X₀: 400 {O(1)}
t₁₄, X₁: 400 {O(1)}
t₁₅, X₀: 400 {O(1)}
t₁₅, X₁: 400 {O(1)}
t₁₅, X₂: 0 {O(1)}
t₁₆, X₀: 400 {O(1)}
t₁₆, X₁: 400 {O(1)}