KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ for location eval_start_bb1_in

Found invariant X₀ ≤ 0 for location eval_start_9

Found invariant X₀ ≤ 0 for location eval_start_bb5_in

Found invariant X₀ ≤ 0 for location eval_start_8

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_start_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_start_bb3_in

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location eval_start_bb4_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_8, eval_start_9, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_bb4_in, eval_start_bb5_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_bb1_in(X₀, X₁, X₁, 0) :|: 1 ≤ X₀
t₇: eval_start_4(X₀, X₁, X₂, X₃) → eval_start_bb5_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₅: eval_start_8(X₀, X₁, X₂, X₃) → eval_start_9(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₆: eval_start_9(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₉: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb4_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃
t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_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
t₁₀: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_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
t₁₂: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_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
t₁₃: eval_start_bb4_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ 0 ≤ X₃
t₁₄: eval_start_bb5_in(X₀, X₁, X₂, X₃) → eval_start_8(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• eval_start_bb1_in: [2⋅X₂+X₃-1]
• eval_start_bb2_in: [2⋅X₂+X₃-2]
• eval_start_bb3_in: [2⋅X₂+X₃-2]

MPRF for transition t₁₀: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_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:

2⋅X₁+1 {O(n)}

MPRF:

• eval_start_bb1_in: [X₁+X₂-1]
• eval_start_bb2_in: [X₁+X₂-1]
• eval_start_bb3_in: [X₁+X₂-2]

MPRF for transition t₁₁: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_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_start_bb1_in: [X₀+X₁+X₂+X₃-1]
• eval_start_bb2_in: [X₀+X₁+X₂+X₃-1]
• eval_start_bb3_in: [X₀+X₁+X₂+X₃-1]

MPRF for transition t₁₂: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_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:

X₁ {O(n)}

MPRF:

• eval_start_bb1_in: [X₂]
• eval_start_bb2_in: [X₂]
• eval_start_bb3_in: [X₂]

All Bounds

Timebounds

Overall timebound:7⋅X₁+X₀+16 {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: 2⋅X₁+1 {O(n)}
t₉: 1 {O(1)}
t₁₀: 2⋅X₁+1 {O(n)}
t₁₁: 2⋅X₁+X₀+1 {O(n)}
t₁₂: X₁ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 7⋅X₁+X₀+16 {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: 2⋅X₁+1 {O(n)}
t₉: 1 {O(1)}
t₁₀: 2⋅X₁+1 {O(n)}
t₁₁: 2⋅X₁+X₀+1 {O(n)}
t₁₂: X₁ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 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: X₁ {O(n)}
t₆, X₃: 0 {O(1)}
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: X₁ {O(n)}
t₈, X₃: X₁ {O(n)}
t₉, X₀: 2⋅X₀ {O(n)}
t₉, X₁: 2⋅X₁ {O(n)}
t₉, X₂: 2⋅X₁ {O(n)}
t₉, X₃: X₁ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₀, X₂: X₁ {O(n)}
t₁₀, X₃: X₁ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: X₁ {O(n)}
t₁₁, X₃: 0 {O(1)}
t₁₂, X₀: X₀ {O(n)}
t₁₂, X₁: X₁ {O(n)}
t₁₂, X₂: X₁ {O(n)}
t₁₂, X₃: X₁ {O(n)}
t₁₃, X₀: 2⋅X₀ {O(n)}
t₁₃, X₁: 2⋅X₁ {O(n)}
t₁₃, X₂: 2⋅X₁ {O(n)}
t₁₃, X₃: X₁ {O(n)}
t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: X₀ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}