KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedpldi2_bb1_in

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedpldi2_bb2_in

Found invariant 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedpldi2_bb3_in

Problem after Preprocessing

Start: eval_speedpldi2_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_speedpldi2_0, eval_speedpldi2_1, eval_speedpldi2_2, eval_speedpldi2_bb0_in, eval_speedpldi2_bb1_in, eval_speedpldi2_bb2_in, eval_speedpldi2_bb3_in, eval_speedpldi2_bb4_in, eval_speedpldi2_start, eval_speedpldi2_stop
Transitions:
t₂: eval_speedpldi2_0(X₀, X₁, X₂, X₃) → eval_speedpldi2_1(X₀, X₁, X₂, X₃)
t₃: eval_speedpldi2_1(X₀, X₁, X₂, X₃) → eval_speedpldi2_2(X₀, X₁, X₂, X₃)
t₄: eval_speedpldi2_2(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb1_in(X₀, X₁, X₁, 0) :|: 1 ≤ X₀ ∧ 0 ≤ X₁
t₅: eval_speedpldi2_2(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0
t₆: eval_speedpldi2_2(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁: eval_speedpldi2_bb0_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_0(X₀, X₁, X₂, X₃)
t₇: eval_speedpldi2_bb1_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₈: eval_speedpldi2_bb1_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb4_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₀: eval_speedpldi2_bb2_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb1_in(X₀, X₁, X₂, 0) :|: X₀ ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₉: eval_speedpldi2_bb2_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₁₁: eval_speedpldi2_bb3_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb1_in(X₀, X₁, X₂-1, 1+X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₁₂: eval_speedpldi2_bb4_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_stop(X₀, X₁, X₂, X₃)
t₀: eval_speedpldi2_start(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₇: eval_speedpldi2_bb1_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• eval_speedpldi2_bb1_in: [1+2⋅X₂+X₃]
• eval_speedpldi2_bb2_in: [2⋅X₂+X₃]
• eval_speedpldi2_bb3_in: [2⋅X₂+X₃]

MPRF for transition t₉: eval_speedpldi2_bb2_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_speedpldi2_bb1_in: [X₂]
• eval_speedpldi2_bb2_in: [X₂]
• eval_speedpldi2_bb3_in: [X₂-1]

MPRF for transition t₁₀: eval_speedpldi2_bb2_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb1_in(X₀, X₁, X₂, 0) :|: X₀ ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₁+X₀+1 {O(n)}

MPRF:

• eval_speedpldi2_bb1_in: [X₀+X₁+X₂+X₃-1]
• eval_speedpldi2_bb2_in: [X₀+X₁+X₂+X₃-1]
• eval_speedpldi2_bb3_in: [X₀+X₁+X₂+X₃-1]

MPRF for transition t₁₁: eval_speedpldi2_bb3_in(X₀, X₁, X₂, X₃) → eval_speedpldi2_bb1_in(X₀, X₁, X₂-1, 1+X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

• eval_speedpldi2_bb1_in: [X₀+X₂-1]
• eval_speedpldi2_bb2_in: [X₀+X₂-1]
• eval_speedpldi2_bb3_in: [X₀+X₂-1]

All Bounds

Timebounds

Overall timebound:2⋅X₀+6⋅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: 2⋅X₁+1 {O(n)}
t₈: 1 {O(1)}
t₉: X₁ {O(n)}
t₁₀: 2⋅X₁+X₀+1 {O(n)}
t₁₁: X₀+X₁+1 {O(n)}
t₁₂: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+6⋅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: 2⋅X₁+1 {O(n)}
t₈: 1 {O(1)}
t₉: X₁ {O(n)}
t₁₀: 2⋅X₁+X₀+1 {O(n)}
t₁₁: X₀+X₁+1 {O(n)}
t₁₂: 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: X₁ {O(n)}
t₄, X₃: 0 {O(1)}
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: X₁ {O(n)}
t₇, X₃: X₀+X₁+1 {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 0 {O(1)}
t₈, X₃: X₀+X₁+1 {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: X₁ {O(n)}
t₉, X₃: X₀+X₁+1 {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₀, X₂: X₁ {O(n)}
t₁₀, X₃: 0 {O(1)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: X₁ {O(n)}
t₁₁, X₃: X₀+X₁+1 {O(n)}
t₁₂, X₀: 4⋅X₀ {O(n)}
t₁₂, X₁: 4⋅X₁ {O(n)}
t₁₂, X₂: 2⋅X₂ {O(n)}
t₁₂, X₃: 2⋅X₃+X₀+X₁+1 {O(n)}