KoAT2 Proof Output

Initial Problem

Start: f0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: G
Locations: f0, f12, f22, f24, f27
Transitions:
t₅: f0(X₀, X₁, X₂, X₃, X₄, X₅) → f12(4, G, 0, X₃, G, 4) :|: 1 ≤ G
t₀: f12(X₀, X₁, X₂, X₃, X₄, X₅) → f12(X₀-1, X₁-1, 1+X₂, G, X₄, X₅) :|: 1 ≤ G ∧ 1 ≤ X₀
t₁: f12(X₀, X₁, X₂, X₃, X₄, X₅) → f12(X₀-1, X₁, X₂, G, X₄, X₅) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ G ≤ 0
t₄: f12(X₀, X₁, X₂, X₃, X₄, X₅) → f22(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0
t₂: f22(X₀, X₁, X₂, X₃, X₄, X₅) → f22(X₀, X₁, X₂, X₃, X₄, X₅)
t₃: f24(X₀, X₁, X₂, X₃, X₄, X₅) → f27(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Cut unreachable locations [f24; f27] from the program graph

Eliminate variables [X₂; X₃; X₄; X₅] that do not contribute to the problem

Found invariant 0 ≤ 3+X₁ ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ X₀ ≤ 4 ∧ 0 ≤ X₀ for location f12

Found invariant 0 ≤ 3+X₁ ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location f22

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁
Temp_Vars: G
Locations: f0, f12, f22
Transitions:
t₁₂: f0(X₀, X₁) → f12(4, G) :|: 1 ≤ G
t₁₃: f12(X₀, X₁) → f12(X₀-1, X₁-1) :|: 1 ≤ G ∧ 1 ≤ X₀ ∧ X₀ ≤ 4 ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 0 ≤ 3+X₁ ∧ 0 ≤ X₀
t₁₄: f12(X₀, X₁) → f12(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ G ≤ 0 ∧ X₀ ≤ 4 ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 0 ≤ 3+X₁ ∧ 0 ≤ X₀
t₁₅: f12(X₀, X₁) → f22(X₀, X₁) :|: X₀ ≤ 0 ∧ X₀ ≤ 4 ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 0 ≤ 3+X₁ ∧ 0 ≤ X₀
t₁₆: f22(X₀, X₁) → f22(X₀, X₁) :|: 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 0 ≤ 3+X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0

MPRF for transition t₁₃: f12(X₀, X₁) → f12(X₀-1, X₁-1) :|: 1 ≤ G ∧ 1 ≤ X₀ ∧ X₀ ≤ 4 ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 0 ≤ 3+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

5 {O(1)}

MPRF:

• f12: [1+X₀]

MPRF for transition t₁₄: f12(X₀, X₁) → f12(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ G ≤ 0 ∧ X₀ ≤ 4 ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 0 ≤ 3+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

5 {O(1)}

MPRF:

• f12: [1+X₀]

Found invariant 0 ≤ 3+X₁ ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ X₀ ≤ 4 ∧ 0 ≤ X₀ for location f12

Found invariant 0 ≤ 3+X₁ ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location f22

Found invariant 0 ≤ 3+X₁ ∧ 0 ≤ 3+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location f22_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₁₂: 1 {O(1)}
t₁₃: 5 {O(1)}
t₁₄: 5 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₁₂: 1 {O(1)}
t₁₃: 5 {O(1)}
t₁₄: 5 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: inf {Infinity}

Sizebounds

t₁₂, X₀: 4 {O(1)}
t₁₃, X₀: 3 {O(1)}
t₁₄, X₀: 3 {O(1)}
t₁₄, X₁: 3 {O(1)}
t₁₅, X₀: 0 {O(1)}
t₁₆, X₀: 0 {O(1)}