KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_10, eval_start_2, eval_start_3, 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_bb6_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄) → eval_start_1(X₀, X₁, X₂, X₃, X₄)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄) → eval_start_2(X₀, X₁, X₂, X₃, X₄)
t₁₂: eval_start_10(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₂, X₂, X₃, X₄)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄) → eval_start_3(X₀, X₁, X₂, X₃, X₄)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₃, X₁, X₂, X₃, X₄)
t₁₀: eval_start_8(X₀, X₁, X₂, X₃, X₄) → eval_start_9(X₀, X₁, X₂, X₃, X₄)
t₁₁: eval_start_9(X₀, X₁, X₂, X₃, X₄) → eval_start_10(X₀, X₁, X₂, X₃, X₄)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_start_0(X₀, X₁, X₂, X₃, X₄)
t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 101 ≤ X₀
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 100
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, X₁, X₂, X₃, X₄)
t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_8(X₀, X₁, 50+X₀+X₄, X₃, X₄)
t₁₃: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₁
t₁₄: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ 0
t₁₅: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁-1, X₂, X₃, X₄)
t₁₆: eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant X₂ ≤ 150+X₄ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 100 for location eval_start_10

Found invariant X₀ ≤ X₃ for location eval_start_bb1_in

Found invariant X₂ ≤ 150+X₄ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 100 for location eval_start_9

Found invariant X₂ ≤ 150+X₄ ∧ X₁ ≤ 150+X₄ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ 0 ∧ X₀+X₁ ≤ 99 ∧ X₀ ≤ 100 for location eval_start_stop

Found invariant X₂ ≤ 150+X₄ ∧ X₁ ≤ 150+X₄ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ 0 ∧ X₀+X₁ ≤ 99 ∧ X₀ ≤ 100 for location eval_start_bb6_in

Found invariant 0 ≤ 150+X₄ ∧ 0 ≤ 150+X₂+X₄ ∧ X₂ ≤ 150+X₄ ∧ 0 ≤ 150+X₁+X₄ ∧ X₁ ≤ 150+X₄ ∧ X₀ ≤ 250+X₄ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 100+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ 100+X₁ ∧ X₀ ≤ 100 for location eval_start_bb5_in

Found invariant X₂ ≤ 150+X₄ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 100 for location eval_start_8

Found invariant 101 ≤ X₃ ∧ 202 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 101 ≤ X₀ for location eval_start_bb2_in

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 100 for location eval_start_bb3_in

Found invariant X₂ ≤ 150+X₄ ∧ X₁ ≤ 150+X₄ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 100 for location eval_start_bb4_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_10, eval_start_2, eval_start_3, 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_bb6_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄) → eval_start_1(X₀, X₁, X₂, X₃, X₄)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄) → eval_start_2(X₀, X₁, X₂, X₃, X₄)
t₁₂: eval_start_10(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₂, X₂, X₃, X₄) :|: X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₃
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄) → eval_start_3(X₀, X₁, X₂, X₃, X₄)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₃, X₁, X₂, X₃, X₄)
t₁₀: eval_start_8(X₀, X₁, X₂, X₃, X₄) → eval_start_9(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₃
t₁₁: eval_start_9(X₀, X₁, X₂, X₃, X₄) → eval_start_10(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₃
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_start_0(X₀, X₁, X₂, X₃, X₄)
t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 101 ≤ X₀ ∧ X₀ ≤ X₃
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 100 ∧ X₀ ≤ X₃
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: 101 ≤ X₀ ∧ 101 ≤ X₃ ∧ 202 ≤ X₀+X₃ ∧ X₀ ≤ X₃
t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_8(X₀, X₁, 50+X₀+X₄, X₃, X₄) :|: X₀ ≤ 100 ∧ X₀ ≤ X₃
t₁₃: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₁ ∧ X₁ ≤ 150+X₄ ∧ X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂
t₁₄: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ 0 ∧ X₁ ≤ 150+X₄ ∧ X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂
t₁₅: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁-1, X₂, X₃, X₄) :|: X₀ ≤ 250+X₄ ∧ 0 ≤ 150+X₁+X₄ ∧ X₁ ≤ 150+X₄ ∧ 0 ≤ 150+X₂+X₄ ∧ X₂ ≤ 150+X₄ ∧ 0 ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ 100+X₁ ∧ X₀ ≤ 100+X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₁₆: eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 150+X₄ ∧ X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀+X₁ ≤ 99 ∧ 1+X₁ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 101 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃+100 {O(n)}

MPRF:

• eval_start_bb1_in: [X₀-100]
• eval_start_bb2_in: [X₀-101]

MPRF for transition t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: 101 ≤ X₀ ∧ 101 ≤ X₃ ∧ 202 ≤ X₀+X₃ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃+100 {O(n)}

MPRF:

• eval_start_bb1_in: [X₀-100]
• eval_start_bb2_in: [X₀-100]

MPRF for transition t₁₃: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₁ ∧ X₁ ≤ 150+X₄ ∧ X₂ ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₂ of depth 1:

new bound:

2⋅X₃+2⋅X₄+51 {O(n)}

MPRF:

• eval_start_bb4_in: [1+X₁]
• eval_start_bb5_in: [X₁]

MPRF for transition t₁₅: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁-1, X₂, X₃, X₄) :|: X₀ ≤ 250+X₄ ∧ 0 ≤ 150+X₁+X₄ ∧ X₁ ≤ 150+X₄ ∧ 0 ≤ 150+X₂+X₄ ∧ X₂ ≤ 150+X₄ ∧ 0 ≤ 150+X₄ ∧ X₀ ≤ 100 ∧ X₀ ≤ 100+X₁ ∧ X₀ ≤ 100+X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₃+2⋅X₄+51 {O(n)}

MPRF:

• eval_start_bb4_in: [1+X₁]
• eval_start_bb5_in: [1+X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₄+6⋅X₃+315 {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: X₃+100 {O(n)}
t₇: 1 {O(1)}
t₈: X₃+100 {O(n)}
t₉: 1 {O(1)}
t₁₀: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₃+2⋅X₄+51 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₃+2⋅X₄+51 {O(n)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₄+6⋅X₃+315 {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: X₃+100 {O(n)}
t₇: 1 {O(1)}
t₈: X₃+100 {O(n)}
t₉: 1 {O(1)}
t₁₀: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₃+2⋅X₄+51 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₃+2⋅X₄+51 {O(n)}
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)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₃ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: X₃ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: 2⋅X₃ {O(n)}
t₇, X₁: 2⋅X₁ {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₇, X₄: 2⋅X₄ {O(n)}
t₈, X₀: X₃ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₉, X₀: 2⋅X₃ {O(n)}
t₉, X₁: 2⋅X₁ {O(n)}
t₉, X₂: 2⋅X₃+2⋅X₄+50 {O(n)}
t₉, X₃: 2⋅X₃ {O(n)}
t₉, X₄: 2⋅X₄ {O(n)}
t₁₀, X₀: 2⋅X₃ {O(n)}
t₁₀, X₁: 2⋅X₁ {O(n)}
t₁₀, X₂: 2⋅X₃+2⋅X₄+50 {O(n)}
t₁₀, X₃: 2⋅X₃ {O(n)}
t₁₀, X₄: 2⋅X₄ {O(n)}
t₁₁, X₀: 2⋅X₃ {O(n)}
t₁₁, X₁: 2⋅X₁ {O(n)}
t₁₁, X₂: 2⋅X₃+2⋅X₄+50 {O(n)}
t₁₁, X₃: 2⋅X₃ {O(n)}
t₁₁, X₄: 2⋅X₄ {O(n)}
t₁₂, X₀: 2⋅X₃ {O(n)}
t₁₂, X₁: 2⋅X₃+2⋅X₄+50 {O(n)}
t₁₂, X₂: 2⋅X₃+2⋅X₄+50 {O(n)}
t₁₂, X₃: 2⋅X₃ {O(n)}
t₁₂, X₄: 2⋅X₄ {O(n)}
t₁₃, X₀: 2⋅X₃ {O(n)}
t₁₃, X₁: 2⋅X₃+2⋅X₄+51 {O(n)}
t₁₃, X₂: 2⋅X₃+2⋅X₄+50 {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₃, X₄: 2⋅X₄ {O(n)}
t₁₄, X₀: 4⋅X₃ {O(n)}
t₁₄, X₁: 4⋅X₃+4⋅X₄+101 {O(n)}
t₁₄, X₂: 4⋅X₃+4⋅X₄+100 {O(n)}
t₁₄, X₃: 4⋅X₃ {O(n)}
t₁₄, X₄: 4⋅X₄ {O(n)}
t₁₅, X₀: 2⋅X₃ {O(n)}
t₁₅, X₁: 2⋅X₃+2⋅X₄+51 {O(n)}
t₁₅, X₂: 2⋅X₃+2⋅X₄+50 {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₅, X₄: 2⋅X₄ {O(n)}
t₁₆, X₀: 4⋅X₃ {O(n)}
t₁₆, X₁: 4⋅X₃+4⋅X₄+101 {O(n)}
t₁₆, X₂: 4⋅X₃+4⋅X₄+100 {O(n)}
t₁₆, X₃: 4⋅X₃ {O(n)}
t₁₆, X₄: 4⋅X₄ {O(n)}