KoAT2 Proof Output

Initial Problem

Start: f0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: F
Locations: f0, f21, f7
Transitions:
t₀: f0(X₀, X₁, X₂, X₃, X₄) → f7(F, 0, 0, X₃, X₄) :|: 1+F ≤ 0
t₁: f0(X₀, X₁, X₂, X₃, X₄) → f7(F, 0, 0, X₃, X₄) :|: 1 ≤ F
t₂: f0(X₀, X₁, X₂, X₃, X₄) → f7(0, 1023, 0, X₃, X₄)
t₄: f7(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 1022 ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ X₄
t₅: f7(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₂ ∧ 1023 ≤ X₄
t₆: f7(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₂ ∧ 1+X₄ ≤ 0
t₃: f7(X₀, X₁, X₂, X₃, X₄) → f7(X₀, X₁, 1+X₂, 2+X₃, X₄) :|: X₂ ≤ X₁

Preprocessing

Eliminate variables [X₀; X₃] that do not contribute to the problem

Found invariant X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location f21

Found invariant X₁ ≤ 1024 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 2047 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ 1023+X₁ ∧ X₀ ≤ 1023 ∧ 0 ≤ X₀ for location f7

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁, X₂
Temp_Vars: F
Locations: f0, f21, f7
Transitions:
t₁₂: f0(X₀, X₁, X₂) → f7(0, 0, X₂) :|: 1+F ≤ 0
t₁₃: f0(X₀, X₁, X₂) → f7(0, 0, X₂) :|: 1 ≤ F
t₁₄: f0(X₀, X₁, X₂) → f7(1023, 0, X₂)
t₁₅: f7(X₀, X₁, X₂) → f21(X₀, X₁, X₂) :|: X₂ ≤ 1022 ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₀+X₁ ≤ 2047 ∧ X₁ ≤ 1024 ∧ X₀ ≤ 1023 ∧ X₀ ≤ 1023+X₁ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₁₆: f7(X₀, X₁, X₂) → f21(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 1023 ≤ X₂ ∧ X₀+X₁ ≤ 2047 ∧ X₁ ≤ 1024 ∧ X₀ ≤ 1023 ∧ X₀ ≤ 1023+X₁ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₁₇: f7(X₀, X₁, X₂) → f21(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ 0 ∧ X₀+X₁ ≤ 2047 ∧ X₁ ≤ 1024 ∧ X₀ ≤ 1023 ∧ X₀ ≤ 1023+X₁ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₁₈: f7(X₀, X₁, X₂) → f7(X₀, 1+X₁, X₂) :|: X₁ ≤ X₀ ∧ X₀+X₁ ≤ 2047 ∧ X₁ ≤ 1024 ∧ X₀ ≤ 1023 ∧ X₀ ≤ 1023+X₁ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁

MPRF for transition t₁₈: f7(X₀, X₁, X₂) → f7(X₀, 1+X₁, X₂) :|: X₁ ≤ X₀ ∧ X₀+X₁ ≤ 2047 ∧ X₁ ≤ 1024 ∧ X₀ ≤ 1023 ∧ X₀ ≤ 1023+X₁ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

3075 {O(1)}

MPRF:

• f7: [1025-X₁]

All Bounds

Timebounds

Overall timebound:3081 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 3075 {O(1)}

Costbounds

Overall costbound: 3081 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 3075 {O(1)}

Sizebounds

t₁₂, X₀: 0 {O(1)}
t₁₂, X₁: 0 {O(1)}
t₁₂, X₂: X₂ {O(n)}
t₁₃, X₀: 0 {O(1)}
t₁₃, X₁: 0 {O(1)}
t₁₃, X₂: X₂ {O(n)}
t₁₄, X₀: 1023 {O(1)}
t₁₄, X₁: 0 {O(1)}
t₁₄, X₂: X₂ {O(n)}
t₁₅, X₀: 1023 {O(1)}
t₁₅, X₁: 1024 {O(1)}
t₁₅, X₂: 1022 {O(1)}
t₁₆, X₀: 1023 {O(1)}
t₁₆, X₁: 1024 {O(1)}
t₁₆, X₂: 3⋅X₂ {O(n)}
t₁₇, X₀: 1023 {O(1)}
t₁₇, X₁: 1024 {O(1)}
t₁₇, X₂: 3⋅X₂ {O(n)}
t₁₈, X₀: 1023 {O(1)}
t₁₈, X₁: 1024 {O(1)}
t₁₈, X₂: 3⋅X₂ {O(n)}