KoAT2 Proof Output

Initial Problem

Start: eval_size07_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉
Temp_Vars:
Locations: eval_size07_bb0_in, eval_size07_bb1_in, eval_size07_bb2_in, eval_size07_bb3_in, eval_size07_bb4_in, eval_size07_bb5_in, eval_size07_start, eval_size07_stop
Transitions:
t₁: eval_size07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb1_in(X₆, X₇, X₈, X₉, X₄, X₅, X₆, X₇, X₈, X₉)
t₂: eval_size07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₀
t₃: eval_size07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₁, X₃, X₆, X₇, X₈, X₉) :|: X₀ ≤ 0
t₄: eval_size07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb1_in(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, 2+2⋅(X₀)²+X₃-4⋅X₀, X₄, X₅, X₆, X₇, X₈, X₉)
t₅: eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₄+X₅
t₆: eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: X₄+X₅ ≤ 0
t₇: eval_size07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄-1, X₅-1, X₆, X₇, X₈, X₉)
t₈: eval_size07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉)
t₀: eval_size07_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉)

Preprocessing

Found invariant X₀ ≤ X₆ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₄+X₅ ∧ 1 ≤ X₁+X₅ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₁+X₃ ∧ X₀ ≤ 0 for location eval_size07_bb4_in

Found invariant X₀ ≤ X₆ ∧ X₄+X₅ ≤ 0 ∧ X₅ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size07_bb5_in

Found invariant 1 ≤ X₆ ∧ 2 ≤ X₀+X₆ ∧ X₀ ≤ X₆ ∧ 1 ≤ X₀ for location eval_size07_bb2_in

Found invariant X₀ ≤ X₆ ∧ X₅ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size07_bb3_in

Found invariant X₀ ≤ X₆ for location eval_size07_bb1_in

Found invariant X₀ ≤ X₆ ∧ X₄+X₅ ≤ 0 ∧ X₅ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size07_stop

Problem after Preprocessing

Start: eval_size07_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉
Temp_Vars:
Locations: eval_size07_bb0_in, eval_size07_bb1_in, eval_size07_bb2_in, eval_size07_bb3_in, eval_size07_bb4_in, eval_size07_bb5_in, eval_size07_start, eval_size07_stop
Transitions:
t₁: eval_size07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb1_in(X₆, X₇, X₈, X₉, X₄, X₅, X₆, X₇, X₈, X₉)
t₂: eval_size07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₀ ∧ X₀ ≤ X₆
t₃: eval_size07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₁, X₃, X₆, X₇, X₈, X₉) :|: X₀ ≤ 0 ∧ X₀ ≤ X₆
t₄: eval_size07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb1_in(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, 2+2⋅(X₀)²+X₃-4⋅X₀, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₀ ∧ 1 ≤ X₆ ∧ 2 ≤ X₀+X₆ ∧ X₀ ≤ X₆
t₅: eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₄+X₅ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₆ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₃
t₆: eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: X₄+X₅ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₆ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₃
t₇: eval_size07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄-1, X₅-1, X₆, X₇, X₈, X₉) :|: 1 ≤ X₁+X₃ ∧ 1 ≤ X₁+X₅ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₆ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₃
t₈: eval_size07_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: X₀ ≤ 0 ∧ X₀ ≤ X₆ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₃ ∧ X₄+X₅ ≤ 0
t₀: eval_size07_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉)

MPRF for transition t₂: eval_size07_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₀ ∧ X₀ ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• eval_size07_bb1_in: [X₀]
• eval_size07_bb2_in: [X₀-1]

MPRF for transition t₄: eval_size07_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb1_in(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, 2⋅Temp_Int₃₁₄+2⋅Temp_Int₃₁₅+X₃-4⋅X₀, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ Temp_Int₃₁₅ ∧ X₀ ≤ Temp_Int₃₁₅ ∧ 1 ≤ X₀ ∧ 1 ≤ X₆ ∧ 2 ≤ X₀+X₆ ∧ X₀ ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• eval_size07_bb1_in: [X₀]
• eval_size07_bb2_in: [X₀]

MPRF for transition t₅: eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) :|: 1 ≤ X₄+X₅ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₆ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₃ of depth 1:

new bound:

20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₈+2⋅X₉+6⋅X₇+8⋅X₆ {O(n^3)}

MPRF:

• eval_size07_bb3_in: [X₃+X₄]
• eval_size07_bb4_in: [X₃+X₄-1]

MPRF for transition t₇: eval_size07_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉) → eval_size07_bb3_in(X₀, X₁, X₂, X₃, X₄-1, X₅-1, X₆, X₇, X₈, X₉) :|: 1 ≤ X₁+X₃ ∧ 1 ≤ X₁+X₅ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₄+X₅ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₆ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₃ of depth 1:

new bound:

20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₈+2⋅X₉+6⋅X₇+8⋅X₆ {O(n^3)}

MPRF:

• eval_size07_bb3_in: [X₃+X₄]
• eval_size07_bb4_in: [X₃+X₄]

Found invariant X₀ ≤ X₆ ∧ 1+X₅ ≤ X₃ ∧ 0 ≤ 1+X₄+X₅ ∧ 0 ≤ X₁+X₅ ∧ 1+X₄ ≤ X₁ ∧ 0 ≤ X₃+X₄ ∧ 1 ≤ X₁+X₃ ∧ X₀ ≤ 0 for location eval_size07_bb3_in_v1

Found invariant X₀ ≤ X₆ ∧ X₄+X₅ ≤ 0 ∧ X₅ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size07_bb5_in

Found invariant X₀ ≤ X₆ ∧ 1+X₅ ≤ X₃ ∧ 1 ≤ X₄+X₅ ∧ 2 ≤ X₁+X₅ ∧ 1+X₄ ≤ X₁ ∧ 2 ≤ X₃+X₄ ∧ 3 ≤ X₁+X₃ ∧ X₀ ≤ 0 for location eval_size07_bb4_in_v2

Found invariant X₀ ≤ X₆ for location eval_size07_bb1_in

Found invariant X₀ ≤ X₆ ∧ X₅ ≤ X₃ ∧ 1 ≤ X₄+X₅ ∧ X₃ ≤ X₅ ∧ 1 ≤ X₁+X₅ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₃+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₁+X₃ ∧ X₀ ≤ 0 for location eval_size07_bb4_in_v1

Found invariant 1 ≤ X₆ ∧ 2 ≤ X₀+X₆ ∧ X₀ ≤ X₆ ∧ 1 ≤ X₀ for location eval_size07_bb2_in

Found invariant X₀ ≤ X₆ ∧ X₅ ≤ X₃ ∧ X₃ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_size07_bb3_in

Found invariant X₀ ≤ X₆ ∧ X₄+X₅ ≤ 0 ∧ X₅ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size07_stop

All Bounds

Timebounds

Overall timebound:40⋅X₆⋅X₆⋅X₆+32⋅X₆⋅X₆+12⋅X₇+18⋅X₆+4⋅X₈+4⋅X₉+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₆ {O(n)}
t₃: 1 {O(1)}
t₄: X₆ {O(n)}
t₅: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₈+2⋅X₉+6⋅X₇+8⋅X₆ {O(n^3)}
t₆: 1 {O(1)}
t₇: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₈+2⋅X₉+6⋅X₇+8⋅X₆ {O(n^3)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 40⋅X₆⋅X₆⋅X₆+32⋅X₆⋅X₆+12⋅X₇+18⋅X₆+4⋅X₈+4⋅X₉+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₆ {O(n)}
t₃: 1 {O(1)}
t₄: X₆ {O(n)}
t₅: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₈+2⋅X₉+6⋅X₇+8⋅X₆ {O(n^3)}
t₆: 1 {O(1)}
t₇: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₈+2⋅X₉+6⋅X₇+8⋅X₆ {O(n^3)}
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)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₀, X₆: X₆ {O(n)}
t₀, X₇: X₇ {O(n)}
t₀, X₈: X₈ {O(n)}
t₀, X₉: X₉ {O(n)}
t₁, X₀: X₆ {O(n)}
t₁, X₁: X₇ {O(n)}
t₁, X₂: X₈ {O(n)}
t₁, X₃: X₉ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₁, X₆: X₆ {O(n)}
t₁, X₇: X₇ {O(n)}
t₁, X₈: X₈ {O(n)}
t₁, X₉: X₉ {O(n)}
t₂, X₀: X₆ {O(n)}
t₂, X₁: 2⋅X₈+5⋅X₇ {O(n)}
t₂, X₂: 12⋅X₇+5⋅X₈ {O(n)}
t₂, X₃: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+8⋅X₆+X₉ {O(n^3)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₂, X₆: X₆ {O(n)}
t₂, X₇: X₇ {O(n)}
t₂, X₈: X₈ {O(n)}
t₂, X₉: X₉ {O(n)}
t₃, X₀: 2⋅X₆ {O(n)}
t₃, X₁: 2⋅X₈+6⋅X₇ {O(n)}
t₃, X₂: 12⋅X₇+6⋅X₈ {O(n)}
t₃, X₃: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₉+8⋅X₆ {O(n^3)}
t₃, X₄: 2⋅X₈+6⋅X₇ {O(n)}
t₃, X₅: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₉+8⋅X₆ {O(n^3)}
t₃, X₆: 2⋅X₆ {O(n)}
t₃, X₇: 2⋅X₇ {O(n)}
t₃, X₈: 2⋅X₈ {O(n)}
t₃, X₉: 2⋅X₉ {O(n)}
t₄, X₀: X₆ {O(n)}
t₄, X₁: 2⋅X₈+5⋅X₇ {O(n)}
t₄, X₂: 12⋅X₇+5⋅X₈ {O(n)}
t₄, X₃: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+8⋅X₆+X₉ {O(n^3)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₄, X₆: X₆ {O(n)}
t₄, X₇: X₇ {O(n)}
t₄, X₈: X₈ {O(n)}
t₄, X₉: X₉ {O(n)}
t₅, X₀: 2⋅X₆ {O(n)}
t₅, X₁: 2⋅X₈+6⋅X₇ {O(n)}
t₅, X₂: 12⋅X₇+6⋅X₈ {O(n)}
t₅, X₃: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₉+8⋅X₆ {O(n^3)}
t₅, X₄: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+12⋅X₇+2⋅X₉+4⋅X₈+8⋅X₆ {O(n^3)}
t₅, X₅: 40⋅X₆⋅X₆⋅X₆+32⋅X₆⋅X₆+16⋅X₆+2⋅X₈+4⋅X₉+6⋅X₇ {O(n^3)}
t₅, X₆: 2⋅X₆ {O(n)}
t₅, X₇: 2⋅X₇ {O(n)}
t₅, X₈: 2⋅X₈ {O(n)}
t₅, X₉: 2⋅X₉ {O(n)}
t₆, X₀: 4⋅X₆ {O(n)}
t₆, X₁: 12⋅X₇+4⋅X₈ {O(n)}
t₆, X₂: 12⋅X₈+24⋅X₇ {O(n)}
t₆, X₃: 40⋅X₆⋅X₆⋅X₆+32⋅X₆⋅X₆+16⋅X₆+4⋅X₉ {O(n^3)}
t₆, X₄: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+18⋅X₇+2⋅X₉+6⋅X₈+8⋅X₆ {O(n^3)}
t₆, X₅: 60⋅X₆⋅X₆⋅X₆+48⋅X₆⋅X₆+2⋅X₈+24⋅X₆+6⋅X₇+6⋅X₉ {O(n^3)}
t₆, X₆: 4⋅X₆ {O(n)}
t₆, X₇: 4⋅X₇ {O(n)}
t₆, X₈: 4⋅X₈ {O(n)}
t₆, X₉: 4⋅X₉ {O(n)}
t₇, X₀: 2⋅X₆ {O(n)}
t₇, X₁: 2⋅X₈+6⋅X₇ {O(n)}
t₇, X₂: 12⋅X₇+6⋅X₈ {O(n)}
t₇, X₃: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+2⋅X₉+8⋅X₆ {O(n^3)}
t₇, X₄: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+12⋅X₇+2⋅X₉+4⋅X₈+8⋅X₆ {O(n^3)}
t₇, X₅: 40⋅X₆⋅X₆⋅X₆+32⋅X₆⋅X₆+16⋅X₆+2⋅X₈+4⋅X₉+6⋅X₇ {O(n^3)}
t₇, X₆: 2⋅X₆ {O(n)}
t₇, X₇: 2⋅X₇ {O(n)}
t₇, X₈: 2⋅X₈ {O(n)}
t₇, X₉: 2⋅X₉ {O(n)}
t₈, X₀: 4⋅X₆ {O(n)}
t₈, X₁: 12⋅X₇+4⋅X₈ {O(n)}
t₈, X₂: 12⋅X₈+24⋅X₇ {O(n)}
t₈, X₃: 40⋅X₆⋅X₆⋅X₆+32⋅X₆⋅X₆+16⋅X₆+4⋅X₉ {O(n^3)}
t₈, X₄: 20⋅X₆⋅X₆⋅X₆+16⋅X₆⋅X₆+18⋅X₇+2⋅X₉+6⋅X₈+8⋅X₆ {O(n^3)}
t₈, X₅: 60⋅X₆⋅X₆⋅X₆+48⋅X₆⋅X₆+2⋅X₈+24⋅X₆+6⋅X₇+6⋅X₉ {O(n^3)}
t₈, X₆: 4⋅X₆ {O(n)}
t₈, X₇: 4⋅X₇ {O(n)}
t₈, X₈: 4⋅X₈ {O(n)}
t₈, X₉: 4⋅X₉ {O(n)}