KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location f24

Found invariant X₀ ≤ 6 ∧ 1 ≤ X₀ for location f10

Found invariant X₂ ≤ 6 ∧ X₂ ≤ 5+X₁ ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ 5+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 4+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 5 ∧ X₁ ≤ 4+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location f27

Found invariant X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ X₀+X₁ ≤ 11 ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 5 ∧ 1 ≤ X₀ for location f13

Found invariant 1 ≤ X₀ for location f21

Found invariant 6 ≤ X₀ for location f39

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: f0, f10, f13, f21, f24, f27, f39
Transitions:
t₀: f0(X₀, X₁, X₂) → f10(1, X₁, X₂)
t₁: f10(X₀, X₁, X₂) → f13(X₀, 1, X₂) :|: X₀ ≤ 5 ∧ X₀ ≤ 6 ∧ 1 ≤ X₀
t₁₀: f10(X₀, X₁, X₂) → f21(1, X₁, X₂) :|: 6 ≤ X₀ ∧ X₀ ≤ 6 ∧ 1 ≤ X₀
t₉: f13(X₀, X₁, X₂) → f10(1+X₀, X₁, X₂) :|: 6 ≤ X₁ ∧ X₀+X₁ ≤ 11 ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ X₀ ≤ 5 ∧ X₀ ≤ 4+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁
t₂: f13(X₀, X₁, X₂) → f13(X₀, 1+X₁, X₂) :|: X₁ ≤ 5 ∧ X₀+X₁ ≤ 11 ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ X₀ ≤ 5 ∧ X₀ ≤ 4+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁
t₃: f21(X₀, X₁, X₂) → f24(X₀, 1, X₂) :|: X₀ ≤ 5 ∧ 1 ≤ X₀
t₈: f21(X₀, X₁, X₂) → f39(X₀, X₁, X₂) :|: 6 ≤ X₀ ∧ 1 ≤ X₀
t₇: f24(X₀, X₁, X₂) → f21(1+X₀, X₁, X₂) :|: 6 ≤ X₁ ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁
t₄: f24(X₀, X₁, X₂) → f27(X₀, X₁, 1) :|: X₁ ≤ 5 ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁
t₆: f27(X₀, X₁, X₂) → f24(X₀, 1+X₁, X₂) :|: 6 ≤ X₂ ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ 6 ∧ X₂ ≤ 5+X₀ ∧ X₂ ≤ 5+X₁ ∧ X₁ ≤ 5 ∧ X₁ ≤ 4+X₀ ∧ X₁ ≤ 4+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂
t₅: f27(X₀, X₁, X₂) → f27(X₀, X₁, 1+X₂) :|: X₂ ≤ 5 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ 6 ∧ X₂ ≤ 5+X₀ ∧ X₂ ≤ 5+X₁ ∧ X₁ ≤ 5 ∧ X₁ ≤ 4+X₀ ∧ X₁ ≤ 4+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂

MPRF for transition t₁: f10(X₀, X₁, X₂) → f13(X₀, 1, X₂) :|: X₀ ≤ 5 ∧ X₀ ≤ 6 ∧ 1 ≤ X₀ of depth 1:

new bound:

7 {O(1)}

MPRF:

• f10: [6-X₀]
• f13: [5-X₀]

MPRF for transition t₂: f13(X₀, X₁, X₂) → f13(X₀, 1+X₁, X₂) :|: X₁ ≤ 5 ∧ X₀+X₁ ≤ 11 ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ X₀ ≤ 5 ∧ X₀ ≤ 4+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

35 {O(1)}

MPRF:

• f10: [30-5⋅X₀]
• f13: [31-5⋅X₀-X₁]

MPRF for transition t₉: f13(X₀, X₁, X₂) → f10(1+X₀, X₁, X₂) :|: 6 ≤ X₁ ∧ X₀+X₁ ≤ 11 ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ X₀ ≤ 5 ∧ X₀ ≤ 4+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

7 {O(1)}

MPRF:

• f10: [6-X₀]
• f13: [6-X₀]

MPRF for transition t₃: f21(X₀, X₁, X₂) → f24(X₀, 1, X₂) :|: X₀ ≤ 5 ∧ 1 ≤ X₀ of depth 1:

new bound:

7 {O(1)}

MPRF:

• f21: [6-X₀]
• f24: [5-X₀]
• f27: [5-X₀]

MPRF for transition t₄: f24(X₀, X₁, X₂) → f27(X₀, X₁, 1) :|: X₁ ≤ 5 ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

49 {O(1)}

MPRF:

• f21: [-X₁]
• f24: [6-X₁]
• f27: [5-X₁]

MPRF for transition t₅: f27(X₀, X₁, X₂) → f27(X₀, X₁, 1+X₂) :|: X₂ ≤ 5 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ 6 ∧ X₂ ≤ 5+X₀ ∧ X₂ ≤ 5+X₁ ∧ X₁ ≤ 5 ∧ X₁ ≤ 4+X₀ ∧ X₁ ≤ 4+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ of depth 1:

new bound:

385 {O(1)}

MPRF:

• f21: [-5⋅X₁]
• f24: [50-5⋅X₁]
• f27: [51-5⋅X₁-X₂]

MPRF for transition t₆: f27(X₀, X₁, X₂) → f24(X₀, 1+X₁, X₂) :|: 6 ≤ X₂ ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ 6 ∧ X₂ ≤ 5+X₀ ∧ X₂ ≤ 5+X₁ ∧ X₁ ≤ 5 ∧ X₁ ≤ 4+X₀ ∧ X₁ ≤ 4+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ of depth 1:

new bound:

49 {O(1)}

MPRF:

• f21: [-X₁]
• f24: [6-X₁]
• f27: [6-X₁]

MPRF for transition t₇: f24(X₀, X₁, X₂) → f21(1+X₀, X₁, X₂) :|: 6 ≤ X₁ ∧ X₁ ≤ 6 ∧ X₁ ≤ 5+X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

7 {O(1)}

MPRF:

• f21: [4-2⋅X₀]
• f24: [1]
• f27: [1]

All Bounds

Timebounds

Overall timebound:549 {O(1)}
t₀: 1 {O(1)}
t₁: 7 {O(1)}
t₂: 35 {O(1)}
t₃: 7 {O(1)}
t₄: 49 {O(1)}
t₅: 385 {O(1)}
t₆: 49 {O(1)}
t₇: 7 {O(1)}
t₈: 1 {O(1)}
t₉: 7 {O(1)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 549 {O(1)}
t₀: 1 {O(1)}
t₁: 7 {O(1)}
t₂: 35 {O(1)}
t₃: 7 {O(1)}
t₄: 49 {O(1)}
t₅: 385 {O(1)}
t₆: 49 {O(1)}
t₇: 7 {O(1)}
t₈: 1 {O(1)}
t₉: 7 {O(1)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: 1 {O(1)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 5 {O(1)}
t₁, X₁: 1 {O(1)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 5 {O(1)}
t₂, X₁: 6 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 5 {O(1)}
t₃, X₁: 1 {O(1)}
t₃, X₂: X₂+6 {O(n)}
t₄, X₀: 5 {O(1)}
t₄, X₁: 5 {O(1)}
t₄, X₂: 1 {O(1)}
t₅, X₀: 5 {O(1)}
t₅, X₁: 5 {O(1)}
t₅, X₂: 6 {O(1)}
t₆, X₀: 5 {O(1)}
t₆, X₁: 6 {O(1)}
t₆, X₂: 6 {O(1)}
t₇, X₀: 6 {O(1)}
t₇, X₁: 6 {O(1)}
t₇, X₂: 6 {O(1)}
t₈, X₀: 6 {O(1)}
t₈, X₁: 6 {O(1)}
t₈, X₂: 6 {O(1)}
t₉, X₀: 6 {O(1)}
t₉, X₁: 6 {O(1)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: 1 {O(1)}
t₁₀, X₁: 6 {O(1)}
t₁₀, X₂: X₂ {O(n)}