KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃
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) → eval_start_1(X₀, X₁, X₂, X₃)
t₃: eval_start_1(X₀, X₁, X₂, X₃) → eval_start_2(X₀, X₁, X₂, X₃)
t₄: eval_start_2(X₀, X₁, X₂, X₃) → eval_start_3(X₀, X₁, X₂, X₃)
t₅: eval_start_3(X₀, X₁, X₂, X₃) → eval_start_4(X₀, X₁, X₂, X₃)
t₆: eval_start_4(X₀, X₁, X₂, X₃) → eval_start_5(X₀, X₁, X₂, X₃)
t₇: eval_start_5(X₀, X₁, X₂, X₃) → eval_start_6(X₀, X₁, X₂, X₃)
t₈: eval_start_6(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, 0, 0)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁
t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂
t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, X₂, 1+X₃) :|: 1+X₃ ≤ X₀
t₁₂: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: 1+X₃ ≤ X₀ ∧ X₀ ≤ X₃
t₁₃: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ X₀ ≤ X₃
t₁₄: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, 1+X₂, X₃) :|: X₀ ≤ X₃
t₁₅: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃)
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

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

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ for location eval_start_bb1_in

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ for location eval_start_stop

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_start_bb2_in

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ for location eval_start_bb3_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃
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) → eval_start_1(X₀, X₁, X₂, X₃)
t₃: eval_start_1(X₀, X₁, X₂, X₃) → eval_start_2(X₀, X₁, X₂, X₃)
t₄: eval_start_2(X₀, X₁, X₂, X₃) → eval_start_3(X₀, X₁, X₂, X₃)
t₅: eval_start_3(X₀, X₁, X₂, X₃) → eval_start_4(X₀, X₁, X₂, X₃)
t₆: eval_start_4(X₀, X₁, X₂, X₃) → eval_start_5(X₀, X₁, X₂, X₃)
t₇: eval_start_5(X₀, X₁, X₂, X₃) → eval_start_6(X₀, X₁, X₂, X₃)
t₈: eval_start_6(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, 0, 0)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₀: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, X₂, 1+X₃) :|: 1+X₃ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₄: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, 1+X₂, X₃) :|: X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₅: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, X₂, 1+X₃) :|: 1+X₃ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

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

MPRF for transition t₁₄: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₀, X₁, 1+X₂, X₃) :|: X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

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

knowledge_propagation leads to new time bound X₀+X₁+1 {O(n)} for transition t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃

All Bounds

Timebounds

Overall timebound:2⋅X₀+2⋅X₁+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: X₀+X₁+1 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: X₀ {O(n)}
t₁₄: X₁ {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+2⋅X₁+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: X₀+X₁+1 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: X₀ {O(n)}
t₁₄: X₁ {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)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 0 {O(1)}
t₈, X₃: 0 {O(1)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: X₁ {O(n)}
t₉, X₃: X₀ {O(n)}
t₁₀, X₀: 2⋅X₀ {O(n)}
t₁₀, X₁: 2⋅X₁ {O(n)}
t₁₀, X₂: X₁ {O(n)}
t₁₀, X₃: X₀ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: X₁ {O(n)}
t₁₁, X₃: X₀ {O(n)}
t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₁ {O(n)}
t₁₄, X₃: X₀ {O(n)}
t₁₅, X₀: 2⋅X₀ {O(n)}
t₁₅, X₁: 2⋅X₁ {O(n)}
t₁₅, X₂: X₁ {O(n)}
t₁₅, X₃: X₀ {O(n)}