KoAT2 Proof Output

Initial Problem

Start: eval_size04_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀
Temp_Vars:
Locations: eval_size04_bb0_in, eval_size04_bb1_in, eval_size04_bb2_in, eval_size04_bb3_in, eval_size04_bb4_in, eval_size04_bb5_in, eval_size04_start, eval_size04_stop
Transitions:
t₁: eval_size04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb1_in(X₇, X₈, X₉, X₁₀, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀)
t₂: eval_size04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₀
t₃: eval_size04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₁, X₂, X₃, X₇, X₈, X₉, X₁₀) :|: X₀ ≤ 0
t₄: eval_size04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb1_in(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, 1+(X₀)²+X₃-2⋅X₀, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀)
t₅: eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₄+X₅+X₆
t₆: eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: X₄+X₅+X₆ ≤ 0
t₇: eval_size04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄-1, X₅-1, X₆-1, X₇, X₈, X₉, X₁₀)
t₈: eval_size04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀)
t₀: eval_size04_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀)

Preprocessing

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_stop

Found invariant X₀ ≤ X₇ for location eval_size04_bb1_in

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_bb5_in

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₀+X₇ ∧ X₀ ≤ X₇ ∧ 1 ≤ X₀ for location eval_size04_bb2_in

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_bb3_in

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_bb4_in

Problem after Preprocessing

Start: eval_size04_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀
Temp_Vars:
Locations: eval_size04_bb0_in, eval_size04_bb1_in, eval_size04_bb2_in, eval_size04_bb3_in, eval_size04_bb4_in, eval_size04_bb5_in, eval_size04_start, eval_size04_stop
Transitions:
t₁: eval_size04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb1_in(X₇, X₈, X₉, X₁₀, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀)
t₂: eval_size04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₀ ∧ X₀ ≤ X₇
t₃: eval_size04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₁, X₂, X₃, X₇, X₈, X₉, X₁₀) :|: X₀ ≤ 0 ∧ X₀ ≤ X₇
t₄: eval_size04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb1_in(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, 1+(X₀)²+X₃-2⋅X₀, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₀ ∧ 1 ≤ X₇ ∧ 2 ≤ X₀+X₇ ∧ X₀ ≤ X₇
t₅: eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₄+X₅+X₆ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₇ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₂ ∧ X₆ ≤ X₃
t₆: eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: X₄+X₅+X₆ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₇ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₂ ∧ X₆ ≤ X₃
t₇: eval_size04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄-1, X₅-1, X₆-1, X₇, X₈, X₉, X₁₀) :|: X₀ ≤ 0 ∧ X₀ ≤ X₇ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₂ ∧ X₆ ≤ X₃
t₈: eval_size04_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: X₀ ≤ 0 ∧ X₀ ≤ X₇ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₂ ∧ X₆ ≤ X₃
t₀: eval_size04_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀)

MPRF for transition t₂: eval_size04_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₀ ∧ X₀ ≤ X₇ of depth 1:

new bound:

X₇ {O(n)}

MPRF:

• eval_size04_bb1_in: [X₀]
• eval_size04_bb2_in: [X₀-1]

MPRF for transition t₄: eval_size04_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb1_in(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, Temp_Int₃₃₆+Temp_Int₃₃₇+X₃-2⋅X₀, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ Temp_Int₃₃₇ ∧ X₀ ≤ Temp_Int₃₃₇ ∧ 1 ≤ X₀ ∧ 1 ≤ X₇ ∧ 2 ≤ X₀+X₇ ∧ X₀ ≤ X₇ of depth 1:

new bound:

X₇ {O(n)}

MPRF:

• eval_size04_bb1_in: [X₀]
• eval_size04_bb2_in: [X₀]

MPRF for transition t₅: eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) :|: 1 ≤ X₄+X₅+X₆ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₇ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₂ ∧ X₆ ≤ X₃ of depth 1:

new bound:

14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₈+2⋅X₁₀+4⋅X₇+7⋅X₉ {O(n^3)}

MPRF:

• eval_size04_bb3_in: [X₁+X₃+X₅]
• eval_size04_bb4_in: [X₁+X₃+X₅-1]

knowledge_propagation leads to new time bound 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₈+2⋅X₁₀+4⋅X₇+7⋅X₉ {O(n^3)} for transition t₇: eval_size04_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀) → eval_size04_bb3_in(X₀, X₁, X₂, X₃, X₄-1, X₅-1, X₆-1, X₇, X₈, X₉, X₁₀) :|: X₀ ≤ 0 ∧ X₀ ≤ X₇ ∧ X₄ ≤ X₁ ∧ X₅ ≤ X₂ ∧ X₆ ≤ X₃

Found invariant X₀ ≤ X₇ for location eval_size04_bb1_in

Found invariant X₀ ≤ X₇ ∧ 1+X₆ ≤ X₃ ∧ 1+X₅ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_bb3_in_v1

Found invariant 1 ≤ X₇ ∧ 2 ≤ X₀+X₇ ∧ X₀ ≤ X₇ ∧ 1 ≤ X₀ for location eval_size04_bb2_in

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₃ ≤ X₆ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_size04_bb4_in_v1

Found invariant X₀ ≤ X₇ ∧ 1+X₆ ≤ X₃ ∧ 1+X₅ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_bb4_in_v2

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_stop

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₅ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 0 for location eval_size04_bb5_in

Found invariant X₀ ≤ X₇ ∧ X₆ ≤ X₃ ∧ X₃ ≤ X₆ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₀ ≤ 0 for location eval_size04_bb3_in

All Bounds

Timebounds

Overall timebound:28⋅X₇⋅X₇⋅X₇+20⋅X₇⋅X₇+10⋅X₇+14⋅X₉+22⋅X₈+4⋅X₁₀+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₇ {O(n)}
t₃: 1 {O(1)}
t₄: X₇ {O(n)}
t₅: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₈+2⋅X₁₀+4⋅X₇+7⋅X₉ {O(n^3)}
t₆: 1 {O(1)}
t₇: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₈+2⋅X₁₀+4⋅X₇+7⋅X₉ {O(n^3)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 28⋅X₇⋅X₇⋅X₇+20⋅X₇⋅X₇+10⋅X₇+14⋅X₉+22⋅X₈+4⋅X₁₀+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₇ {O(n)}
t₃: 1 {O(1)}
t₄: X₇ {O(n)}
t₅: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₈+2⋅X₁₀+4⋅X₇+7⋅X₉ {O(n^3)}
t₆: 1 {O(1)}
t₇: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₈+2⋅X₁₀+4⋅X₇+7⋅X₉ {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)}
t₀, X₁₀: X₁₀ {O(n)}
t₁, X₀: X₇ {O(n)}
t₁, X₁: X₈ {O(n)}
t₁, X₂: X₉ {O(n)}
t₁, X₃: X₁₀ {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)}
t₁, X₁₀: X₁₀ {O(n)}
t₂, X₀: X₇ {O(n)}
t₂, X₁: 2⋅X₉+4⋅X₈ {O(n)}
t₂, X₂: 4⋅X₉+6⋅X₈ {O(n)}
t₂, X₃: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+4⋅X₇+X₁₀ {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)}
t₂, X₁₀: X₁₀ {O(n)}
t₃, X₀: 2⋅X₇ {O(n)}
t₃, X₁: 2⋅X₉+5⋅X₈ {O(n)}
t₃, X₂: 5⋅X₉+6⋅X₈ {O(n)}
t₃, X₃: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+2⋅X₁₀+4⋅X₇ {O(n^3)}
t₃, X₄: 2⋅X₉+5⋅X₈ {O(n)}
t₃, X₅: 5⋅X₉+6⋅X₈ {O(n)}
t₃, X₆: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+2⋅X₁₀+4⋅X₇ {O(n^3)}
t₃, X₇: 2⋅X₇ {O(n)}
t₃, X₈: 2⋅X₈ {O(n)}
t₃, X₉: 2⋅X₉ {O(n)}
t₃, X₁₀: 2⋅X₁₀ {O(n)}
t₄, X₀: X₇ {O(n)}
t₄, X₁: 2⋅X₉+4⋅X₈ {O(n)}
t₄, X₂: 4⋅X₉+6⋅X₈ {O(n)}
t₄, X₃: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+4⋅X₇+X₁₀ {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)}
t₄, X₁₀: X₁₀ {O(n)}
t₅, X₀: 2⋅X₇ {O(n)}
t₅, X₁: 2⋅X₉+5⋅X₈ {O(n)}
t₅, X₂: 5⋅X₉+6⋅X₈ {O(n)}
t₅, X₃: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+2⋅X₁₀+4⋅X₇ {O(n^3)}
t₅, X₄: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+16⋅X₈+2⋅X₁₀+4⋅X₇+9⋅X₉ {O(n^3)}
t₅, X₅: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+12⋅X₉+17⋅X₈+2⋅X₁₀+4⋅X₇ {O(n^3)}
t₅, X₆: 28⋅X₇⋅X₇⋅X₇+20⋅X₇⋅X₇+11⋅X₈+4⋅X₁₀+7⋅X₉+8⋅X₇ {O(n^3)}
t₅, X₇: 2⋅X₇ {O(n)}
t₅, X₈: 2⋅X₈ {O(n)}
t₅, X₉: 2⋅X₉ {O(n)}
t₅, X₁₀: 2⋅X₁₀ {O(n)}
t₆, X₀: 4⋅X₇ {O(n)}
t₆, X₁: 10⋅X₈+4⋅X₉ {O(n)}
t₆, X₂: 10⋅X₉+12⋅X₈ {O(n)}
t₆, X₃: 28⋅X₇⋅X₇⋅X₇+20⋅X₇⋅X₇+4⋅X₁₀+8⋅X₇ {O(n^3)}
t₆, X₄: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₉+2⋅X₁₀+21⋅X₈+4⋅X₇ {O(n^3)}
t₆, X₅: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+17⋅X₉+2⋅X₁₀+23⋅X₈+4⋅X₇ {O(n^3)}
t₆, X₆: 42⋅X₇⋅X₇⋅X₇+30⋅X₇⋅X₇+11⋅X₈+12⋅X₇+6⋅X₁₀+7⋅X₉ {O(n^3)}
t₆, X₇: 4⋅X₇ {O(n)}
t₆, X₈: 4⋅X₈ {O(n)}
t₆, X₉: 4⋅X₉ {O(n)}
t₆, X₁₀: 4⋅X₁₀ {O(n)}
t₇, X₀: 2⋅X₇ {O(n)}
t₇, X₁: 2⋅X₉+5⋅X₈ {O(n)}
t₇, X₂: 5⋅X₉+6⋅X₈ {O(n)}
t₇, X₃: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+2⋅X₁₀+4⋅X₇ {O(n^3)}
t₇, X₄: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+16⋅X₈+2⋅X₁₀+4⋅X₇+9⋅X₉ {O(n^3)}
t₇, X₅: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+12⋅X₉+17⋅X₈+2⋅X₁₀+4⋅X₇ {O(n^3)}
t₇, X₆: 28⋅X₇⋅X₇⋅X₇+20⋅X₇⋅X₇+11⋅X₈+4⋅X₁₀+7⋅X₉+8⋅X₇ {O(n^3)}
t₇, X₇: 2⋅X₇ {O(n)}
t₇, X₈: 2⋅X₈ {O(n)}
t₇, X₉: 2⋅X₉ {O(n)}
t₇, X₁₀: 2⋅X₁₀ {O(n)}
t₈, X₀: 4⋅X₇ {O(n)}
t₈, X₁: 10⋅X₈+4⋅X₉ {O(n)}
t₈, X₂: 10⋅X₉+12⋅X₈ {O(n)}
t₈, X₃: 28⋅X₇⋅X₇⋅X₇+20⋅X₇⋅X₇+4⋅X₁₀+8⋅X₇ {O(n^3)}
t₈, X₄: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+11⋅X₉+2⋅X₁₀+21⋅X₈+4⋅X₇ {O(n^3)}
t₈, X₅: 14⋅X₇⋅X₇⋅X₇+10⋅X₇⋅X₇+17⋅X₉+2⋅X₁₀+23⋅X₈+4⋅X₇ {O(n^3)}
t₈, X₆: 42⋅X₇⋅X₇⋅X₇+30⋅X₇⋅X₇+11⋅X₈+12⋅X₇+6⋅X₁₀+7⋅X₉ {O(n^3)}
t₈, X₇: 4⋅X₇ {O(n)}
t₈, X₈: 4⋅X₈ {O(n)}
t₈, X₉: 4⋅X₉ {O(n)}
t₈, X₁₀: 4⋅X₁₀ {O(n)}