KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₄ for location eval_start_bb1_in

Found invariant 2 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 4 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_start_5

Found invariant X₀ ≤ X₄ ∧ X₀ ≤ 0 for location eval_start_stop

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_bb2_in

Found invariant X₀ ≤ X₄ ∧ X₀ ≤ 0 for location eval_start_bb4_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef_0
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_bb4_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)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₄, X₁, X₂, X₃, X₄)
t₁₀: eval_start_5(X₀, X₁, X₂, X₃, X₄) → eval_start_6(X₀, X₁, X₂, nondef_0, X₄) :|: X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄
t₁₁: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₂, X₁, X₂, X₃, X₄) :|: 1 ≤ X₃ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄
t₁₂: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₂, X₂, X₃, X₄) :|: X₃ ≤ 0 ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_start_0(X₀, X₁, X₂, X₃, X₄)
t₅: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₀, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄
t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₁-1, X₁, X₂, X₃, X₄) :|: X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄
t₇: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₁-1, X₃, X₄) :|: 2 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄
t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_5(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄
t₁₃: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₅: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₀, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_start_5: [X₂]
• eval_start_6: [X₂]
• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₁-1]
• eval_start_bb3_in: [X₂]

MPRF for transition t₇: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₁-1, X₃, X₄) :|: 2 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_start_5: [X₂]
• eval_start_6: [X₂]
• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₁]
• eval_start_bb3_in: [X₁-1]

MPRF for transition t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₁-1, X₁, X₂, X₃, X₄) :|: X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_start_5: [X₂]
• eval_start_6: [X₂]
• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₁]
• eval_start_bb3_in: [X₁]

MPRF for transition t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_5(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_start_5: [X₁-2]
• eval_start_6: [X₁-2]
• eval_start_bb1_in: [X₀-1]
• eval_start_bb2_in: [X₁-1]
• eval_start_bb3_in: [X₂]

MPRF for transition t₁₀: eval_start_5(X₀, X₁, X₂, X₃, X₄) → eval_start_6(X₀, X₁, X₂, nondef_0, X₄) :|: X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_start_5: [X₂]
• eval_start_6: [X₁-2]
• eval_start_bb1_in: [X₀-1]
• eval_start_bb2_in: [X₁-1]
• eval_start_bb3_in: [X₁-1]

MPRF for transition t₁₁: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₂, X₁, X₂, X₃, X₄) :|: 1 ≤ X₃ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_start_5: [X₂]
• eval_start_6: [X₂]
• eval_start_bb1_in: [X₀-1]
• eval_start_bb2_in: [X₁-1]
• eval_start_bb3_in: [X₂]

MPRF for transition t₁₂: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₂, X₂, X₃, X₄) :|: X₃ ≤ 0 ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 2 ≤ X₄ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_start_5: [X₁-1]
• eval_start_6: [X₁-1]
• eval_start_bb1_in: [X₀-1]
• eval_start_bb2_in: [X₁-1]
• eval_start_bb3_in: [X₂]

All Bounds

Timebounds

Overall timebound:7⋅X₄+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₄ {O(n)}
t₆: 1 {O(1)}
t₇: X₄ {O(n)}
t₈: X₄ {O(n)}
t₉: X₄+1 {O(n)}
t₁₀: X₄+1 {O(n)}
t₁₁: X₄+1 {O(n)}
t₁₂: X₄+1 {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 7⋅X₄+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₄ {O(n)}
t₆: 1 {O(1)}
t₇: X₄ {O(n)}
t₈: X₄ {O(n)}
t₉: X₄+1 {O(n)}
t₁₀: X₄+1 {O(n)}
t₁₁: X₄+1 {O(n)}
t₁₂: X₄+1 {O(n)}
t₁₃: 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: X₄ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₄ {O(n)}
t₅, X₁: 2⋅X₄ {O(n)}
t₅, X₂: 4⋅X₄+X₂ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: X₄ {O(n)}
t₆, X₁: X₁+1 {O(n)}
t₆, X₂: 2⋅X₂+8⋅X₄ {O(n)}
t₆, X₄: 3⋅X₄ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: 2⋅X₄ {O(n)}
t₇, X₂: 4⋅X₄ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₀: 0 {O(1)}
t₈, X₁: 1 {O(1)}
t₈, X₂: 8⋅X₄+X₂ {O(n)}
t₈, X₄: 2⋅X₄ {O(n)}
t₉, X₀: X₄ {O(n)}
t₉, X₁: 2⋅X₄ {O(n)}
t₉, X₂: 4⋅X₄ {O(n)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₀: X₄ {O(n)}
t₁₀, X₁: 2⋅X₄ {O(n)}
t₁₀, X₂: 4⋅X₄ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₀: X₄ {O(n)}
t₁₁, X₁: 2⋅X₄ {O(n)}
t₁₁, X₂: 4⋅X₄ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₂, X₀: X₄ {O(n)}
t₁₂, X₁: 2⋅X₄ {O(n)}
t₁₂, X₂: 4⋅X₄ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: X₄ {O(n)}
t₁₃, X₁: X₁+1 {O(n)}
t₁₃, X₂: 2⋅X₂+8⋅X₄ {O(n)}
t₁₃, X₄: 3⋅X₄ {O(n)}