KoAT2 Proof Output

Initial Problem

Start: f0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars: H
Locations: f0, f10, f18, f21, f32, f41
Transitions:
t₀: f0(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f10(H, 0, X₂, X₃, X₄, X₅, X₆)
t₁: f10(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f10(X₀, 1+X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₂
t₉: f10(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f18(X₀, X₁, X₂, X₂, 0, X₅, X₆) :|: X₂ ≤ X₁
t₂: f18(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f21(X₀, X₁, X₂, X₃, X₄, 0, X₆) :|: 2+X₄ ≤ X₃
t₈: f18(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f32(X₀, X₁, X₂, X₃, 0, X₅, X₆) :|: X₃ ≤ 1+X₄
t₇: f21(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f18(X₀, X₁, X₂, X₃, 1+X₄, X₅, X₆) :|: X₃ ≤ 1+X₄+X₅
t₃: f21(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f21(X₀, X₁, X₂, X₃, X₄, 1+X₅, X₆) :|: 2+X₄+X₅ ≤ X₃
t₄: f21(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f21(X₀, X₁, X₂, X₃, X₄, 1+X₅, H) :|: 2+X₄+X₅ ≤ X₃
t₅: f32(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f32(X₀, X₁, X₂, X₃, 1+X₄, X₅, X₆) :|: 2+X₄ ≤ X₃
t₆: f32(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → f41(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ 1+X₄

Preprocessing

Eliminate variables [H; X₀; X₆] that do not contribute to the problem

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f18

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f32

Found invariant 0 ≤ X₀ for location f10

Found invariant 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 2+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location f21

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ X₂ ≤ 1+X₃ ∧ X₁ ≤ 1+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f41

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: f0, f10, f18, f21, f32, f41
Transitions:
t₂₂: f0(X₀, X₁, X₂, X₃, X₄) → f10(0, X₁, X₂, X₃, X₄)
t₂₃: f10(X₀, X₁, X₂, X₃, X₄) → f10(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂₄: f10(X₀, X₁, X₂, X₃, X₄) → f18(X₀, X₁, X₁, 0, X₄) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₂₅: f18(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, 0) :|: 2+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₆: f18(X₀, X₁, X₂, X₃, X₄) → f32(X₀, X₁, X₂, 0, X₄) :|: X₂ ≤ 1+X₃ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₂₇: f21(X₀, X₁, X₂, X₃, X₄) → f18(X₀, X₁, X₂, 1+X₃, X₄) :|: X₂ ≤ 1+X₃+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄
t₂₈: f21(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, 1+X₄) :|: 2+X₃+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄
t₂₉: f21(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, 1+X₄) :|: 2+X₃+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄
t₃₀: f32(X₀, X₁, X₂, X₃, X₄) → f32(X₀, X₁, X₂, 1+X₃, X₄) :|: 2+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃
t₃₁: f32(X₀, X₁, X₂, X₃, X₄) → f41(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 1+X₃ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃

MPRF for transition t₂₃: f10(X₀, X₁, X₂, X₃, X₄) → f10(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• f10: [X₁-X₀]

MPRF for transition t₂₅: f18(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, 0) :|: 2+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• f18: [1+X₀-X₃]
• f21: [X₀-X₃]

MPRF for transition t₂₇: f21(X₀, X₁, X₂, X₃, X₄) → f18(X₀, X₁, X₂, 1+X₃, X₄) :|: X₂ ≤ 1+X₃+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• f18: [X₀-1-X₃]
• f21: [X₀-1-X₃]

MPRF for transition t₂₈: f21(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, 1+X₄) :|: 2+X₃+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}

MPRF:

• f18: [X₂]
• f21: [X₂-X₄]

MPRF for transition t₂₉: f21(X₀, X₁, X₂, X₃, X₄) → f21(X₀, X₁, X₂, X₃, 1+X₄) :|: 2+X₃+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}

MPRF:

• f18: [X₂]
• f21: [X₂-X₄]

Cut unsatisfiable transition [t₂₇: f21→f18; t₆₃: f21→f18]

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f18

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f32

Found invariant 0 ≤ X₀ for location f10

Found invariant X₄ ≤ 0 ∧ X₄ ≤ X₃ ∧ 2+X₄ ≤ X₂ ∧ 2+X₄ ≤ X₁ ∧ 2+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 2+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location f21

Found invariant 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 3 ≤ X₂+X₄ ∧ 3 ≤ X₁+X₄ ∧ 3 ≤ X₀+X₄ ∧ 2+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location f21_v1

MPRF for transition t₃₀: f32(X₀, X₁, X₂, X₃, X₄) → f32(X₀, X₁, X₂, 1+X₃, X₄) :|: 2+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• f32: [1+X₀-X₃]

All Bounds

Timebounds

Overall timebound:4⋅X₁⋅X₁+13⋅X₁+7 {O(n^2)}
t₂₂: 1 {O(1)}
t₂₃: X₁ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₁+1 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₁+1 {O(n)}
t₂₈: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₂₉: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₃₀: 2⋅X₁+1 {O(n)}
t₃₁: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁⋅X₁+13⋅X₁+7 {O(n^2)}
t₂₂: 1 {O(1)}
t₂₃: X₁ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₁+1 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₁+1 {O(n)}
t₂₈: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₂₉: 2⋅X₁⋅X₁+4⋅X₁ {O(n^2)}
t₃₀: 2⋅X₁+1 {O(n)}
t₃₁: 1 {O(1)}

Sizebounds

t₂₂, X₀: 0 {O(1)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₃, X₀: X₁ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₄, X₀: X₁ {O(n)}
t₂₄, X₁: 2⋅X₁ {O(n)}
t₂₄, X₂: 2⋅X₁ {O(n)}
t₂₄, X₃: 0 {O(1)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₅, X₀: X₁ {O(n)}
t₂₅, X₁: 2⋅X₁ {O(n)}
t₂₅, X₂: 2⋅X₁ {O(n)}
t₂₅, X₃: X₁+1 {O(n)}
t₂₅, X₄: 0 {O(1)}
t₂₆, X₀: 2⋅X₁ {O(n)}
t₂₆, X₁: 4⋅X₁ {O(n)}
t₂₆, X₂: 4⋅X₁ {O(n)}
t₂₆, X₃: 0 {O(1)}
t₂₆, X₄: 8⋅X₁⋅X₁+16⋅X₁+2⋅X₄ {O(n^2)}
t₂₇, X₀: X₁ {O(n)}
t₂₇, X₁: 2⋅X₁ {O(n)}
t₂₇, X₂: 2⋅X₁ {O(n)}
t₂₇, X₃: X₁+1 {O(n)}
t₂₇, X₄: 8⋅X₁⋅X₁+16⋅X₁ {O(n^2)}
t₂₈, X₀: X₁ {O(n)}
t₂₈, X₁: 2⋅X₁ {O(n)}
t₂₈, X₂: 2⋅X₁ {O(n)}
t₂₈, X₃: X₁+1 {O(n)}
t₂₈, X₄: 4⋅X₁⋅X₁+8⋅X₁ {O(n^2)}
t₂₉, X₀: X₁ {O(n)}
t₂₉, X₁: 2⋅X₁ {O(n)}
t₂₉, X₂: 2⋅X₁ {O(n)}
t₂₉, X₃: X₁+1 {O(n)}
t₂₉, X₄: 4⋅X₁⋅X₁+8⋅X₁ {O(n^2)}
t₃₀, X₀: 2⋅X₁ {O(n)}
t₃₀, X₁: 4⋅X₁ {O(n)}
t₃₀, X₂: 4⋅X₁ {O(n)}
t₃₀, X₃: 2⋅X₁+1 {O(n)}
t₃₀, X₄: 8⋅X₁⋅X₁+16⋅X₁+2⋅X₄ {O(n^2)}
t₃₁, X₀: 4⋅X₁ {O(n)}
t₃₁, X₁: 8⋅X₁ {O(n)}
t₃₁, X₂: 8⋅X₁ {O(n)}
t₃₁, X₃: 2⋅X₁+1 {O(n)}
t₃₁, X₄: 16⋅X₁⋅X₁+32⋅X₁+4⋅X₄ {O(n^2)}