KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ for location eval_speed_popl10_simple_single_2_bb3_in

Found invariant X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location eval_speed_popl10_simple_single_2_bb5_in

Found invariant X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₂ for location eval_speed_popl10_simple_single_2_bb1_in

Found invariant X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_speed_popl10_simple_single_2_bb2_in

Found invariant X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location eval_speed_popl10_simple_single_2_bb4_in

Found invariant X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location eval_speed_popl10_simple_single_2_stop

Problem after Preprocessing

Start: eval_speed_popl10_simple_single_2_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_speed_popl10_simple_single_2_bb0_in, eval_speed_popl10_simple_single_2_bb1_in, eval_speed_popl10_simple_single_2_bb2_in, eval_speed_popl10_simple_single_2_bb3_in, eval_speed_popl10_simple_single_2_bb4_in, eval_speed_popl10_simple_single_2_bb5_in, eval_speed_popl10_simple_single_2_start, eval_speed_popl10_simple_single_2_stop
Transitions:
t₁: eval_speed_popl10_simple_single_2_bb0_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, 0, 0)
t₂: eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₃: eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₄: eval_speed_popl10_simple_single_2_bb2_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₅: eval_speed_popl10_simple_single_2_bb3_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₆: eval_speed_popl10_simple_single_2_bb3_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb5_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₇: eval_speed_popl10_simple_single_2_bb4_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₈: eval_speed_popl10_simple_single_2_bb5_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₀: eval_speed_popl10_simple_single_2_start(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_speed_popl10_simple_single_2_bb1_in: [X₁-X₂]
• eval_speed_popl10_simple_single_2_bb2_in: [X₁-1-X₃]
• eval_speed_popl10_simple_single_2_bb3_in: [X₁-X₂]
• eval_speed_popl10_simple_single_2_bb4_in: [X₁-X₂]

MPRF for transition t₄: eval_speed_popl10_simple_single_2_bb2_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_speed_popl10_simple_single_2_bb1_in: [X₁-X₃]
• eval_speed_popl10_simple_single_2_bb2_in: [X₁-X₂]
• eval_speed_popl10_simple_single_2_bb3_in: [X₁-X₃]
• eval_speed_popl10_simple_single_2_bb4_in: [X₁-X₃]

MPRF for transition t₅: eval_speed_popl10_simple_single_2_bb3_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_speed_popl10_simple_single_2_bb1_in: [1+X₀-X₃]
• eval_speed_popl10_simple_single_2_bb2_in: [1+X₀-X₃]
• eval_speed_popl10_simple_single_2_bb3_in: [1+X₀-X₃]
• eval_speed_popl10_simple_single_2_bb4_in: [X₀-X₃]

MPRF for transition t₇: eval_speed_popl10_simple_single_2_bb4_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, 1+X₂, 1+X₃) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_speed_popl10_simple_single_2_bb1_in: [X₀-X₃]
• eval_speed_popl10_simple_single_2_bb2_in: [X₀-X₃]
• eval_speed_popl10_simple_single_2_bb3_in: [X₀-X₂]
• eval_speed_popl10_simple_single_2_bb4_in: [X₀-X₂]

knowledge_propagation leads to new time bound X₀+X₁+1 {O(n)} for transition t₃: eval_speed_popl10_simple_single_2_bb1_in(X₀, X₁, X₂, X₃) → eval_speed_popl10_simple_single_2_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃

All Bounds

Timebounds

Overall timebound:3⋅X₀+3⋅X₁+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: X₀+X₁+1 {O(n)}
t₄: X₁ {O(n)}
t₅: X₀+1 {O(n)}
t₆: 1 {O(1)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₀+3⋅X₁+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: X₀+X₁+1 {O(n)}
t₄: X₁ {O(n)}
t₅: X₀+1 {O(n)}
t₆: 1 {O(1)}
t₇: X₀ {O(n)}
t₈: 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: 0 {O(1)}
t₁, X₃: 0 {O(1)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁ {O(n)}
t₂, X₃: X₁ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₀+X₁ {O(n)}
t₃, X₃: X₀+X₁ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁ {O(n)}
t₄, X₃: X₁ {O(n)}
t₅, X₀: 2⋅X₀ {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: X₀+X₁ {O(n)}
t₅, X₃: X₀+X₁ {O(n)}
t₆, X₀: 2⋅X₀ {O(n)}
t₆, X₁: 2⋅X₁ {O(n)}
t₆, X₂: X₀+X₁ {O(n)}
t₆, X₃: X₀+X₁ {O(n)}
t₇, X₀: 2⋅X₀ {O(n)}
t₇, X₁: 2⋅X₁ {O(n)}
t₇, X₂: X₀+X₁ {O(n)}
t₇, X₃: X₀+X₁ {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: X₀+X₁ {O(n)}
t₈, X₃: X₀+X₁ {O(n)}