KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_5(X₀, X₁, X₂, X₃, X₄, X₅)
t₇: eval_start_5(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_6(X₀, X₁, X₂, X₃, X₄, X₅)
t₈: eval_start_6(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅)
t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₃
t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀
t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂
t₁₂: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂ ∧ X₂ ≤ X₁
t₁₃: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂ ∧ X₂ ≤ X₁
t₁₄: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₁
t₁₅: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Cut unsatisfiable transition [t₁₂: eval_start_bb2_in→eval_start_bb1_in; t₁₃: eval_start_bb2_in→eval_start_bb1_in]

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location eval_start_bb1_in

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₀ for location eval_start_stop

Found invariant X₅ ≤ X₁ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1+X₀ ≤ X₃ for location eval_start_bb2_in

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₀ for location eval_start_bb3_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_5(X₀, X₁, X₂, X₃, X₄, X₅)
t₇: eval_start_5(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_6(X₀, X₁, X₂, X₃, X₄, X₅)
t₈: eval_start_6(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅)
t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₄: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₅: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₂+X₅ {O(n)}

MPRF:

• eval_start_bb1_in: [X₂-X₁]
• eval_start_bb2_in: [X₂-X₁]

MPRF for transition t₁₄: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_start_bb1_in: [X₃-X₀]
• eval_start_bb2_in: [X₃-X₀]

knowledge_propagation leads to new time bound X₂+X₃+X₄+X₅+1 {O(n)} for transition t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+2⋅X₄+2⋅X₅+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: X₂+X₃+X₄+X₅+1 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: X₂+X₅ {O(n)}
t₁₄: X₃+X₄ {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+2⋅X₄+2⋅X₅+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: X₂+X₃+X₄+X₅+1 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: X₂+X₅ {O(n)}
t₁₄: X₃+X₄ {O(n)}
t₁₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: X₄ {O(n)}
t₈, X₁: X₅ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₉, X₀: 2⋅X₄+X₃ {O(n)}
t₉, X₁: 2⋅X₅+X₂ {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₀: 3⋅X₄+X₃ {O(n)}
t₁₀, X₁: 3⋅X₅+X₂ {O(n)}
t₁₀, X₂: 2⋅X₂ {O(n)}
t₁₀, X₃: 2⋅X₃ {O(n)}
t₁₀, X₄: 2⋅X₄ {O(n)}
t₁₀, X₅: 2⋅X₅ {O(n)}
t₁₁, X₀: 2⋅X₄+X₃ {O(n)}
t₁₁, X₁: 2⋅X₅+X₂ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₁, X₅: X₅ {O(n)}
t₁₄, X₀: 2⋅X₄+X₃ {O(n)}
t₁₄, X₁: 2⋅X₅+X₂ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₄, X₅: X₅ {O(n)}
t₁₅, X₀: 3⋅X₄+X₃ {O(n)}
t₁₅, X₁: 3⋅X₅+X₂ {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₅, X₄: 2⋅X₄ {O(n)}
t₁₅, X₅: 2⋅X₅ {O(n)}