KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 0 ≤ X₃ ∧ X₀ ≤ X₂ for location eval_start_bb1_in

Found invariant 1+X₃ ≤ 0 for location eval_start_9

Found invariant 1+X₃ ≤ 0 for location eval_start_bb6_in

Found invariant 0 ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_start_bb5_in

Found invariant 1+X₃ ≤ 0 for location eval_start_8

Found invariant 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_start_bb2_in

Found invariant X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_bb3_in

Found invariant X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 3 ≤ X₂+X₄ ∧ 3 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 2 ≤ X₀ 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_2, 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)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₂, X₁, X₂, X₃, X₄) :|: 0 ≤ X₃
t₅: eval_start_2(X₀, X₁, X₂, X₃, X₄) → eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ 0
t₁₄: eval_start_8(X₀, X₁, X₂, X₃, X₄) → eval_start_9(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ 0
t₁₅: eval_start_9(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ 0
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) :|: 1+X₃ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₃ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₀-1-X₃, X₂, X₃, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃
t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₁, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₄ ≤ X₃ ∧ 0 ≤ X₄
t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₄ ≤ X₃ ∧ 0 ≤ X₄
t₁₁: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄-1) :|: 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 2 ≤ X₃+X₄ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₄ ∧ 3 ≤ X₂+X₃ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁+X₃ ∧ X₄ ≤ X₃
t₁₂: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₁₃: eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄) → eval_start_8(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ 0
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) :|: 1+X₃ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀-X₃]
• eval_start_bb2_in: [X₀-1-X₃]
• eval_start_bb3_in: [X₁]
• eval_start_bb4_in: [X₁]

MPRF for transition t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₀-1-X₃, X₂, X₃, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₀]
• eval_start_bb3_in: [X₁]
• eval_start_bb4_in: [X₁]

MPRF for transition t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₄ ≤ X₃ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀+X₃]
• eval_start_bb2_in: [X₀+X₃]
• eval_start_bb3_in: [1+X₁+X₃+X₄]
• eval_start_bb4_in: [X₁+X₃+X₄]

MPRF for transition t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₁, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₄ ≤ X₃ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₀]
• eval_start_bb3_in: [X₀]
• eval_start_bb4_in: [X₀]

MPRF for transition t₁₁: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄-1) :|: 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 2 ≤ X₃+X₄ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₄ ∧ 3 ≤ X₂+X₃ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁+X₃ ∧ X₄ ≤ X₃ of depth 1:

new bound:

2⋅X₂+2 {O(n)}

MPRF:

• eval_start_bb1_in: [2⋅X₀-2]
• eval_start_bb2_in: [2⋅X₀-2]
• eval_start_bb3_in: [X₀+X₁+X₄-1]
• eval_start_bb4_in: [X₀+X₁+X₄-1]

All Bounds

Timebounds

Overall timebound:2⋅X₃+6⋅X₂+13 {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: X₂+X₃ {O(n)}
t₇: 1 {O(1)}
t₈: X₂ {O(n)}
t₉: X₂+X₃ {O(n)}
t₁₀: X₂ {O(n)}
t₁₁: 2⋅X₂+2 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃+6⋅X₂+13 {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: X₂+X₃ {O(n)}
t₇: 1 {O(1)}
t₈: X₂ {O(n)}
t₉: X₂+X₃ {O(n)}
t₁₀: X₂ {O(n)}
t₁₁: 2⋅X₂+2 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 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: X₂ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: X₂ {O(n)}
t₆, X₁: 2⋅X₂+X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: 2⋅X₂ {O(n)}
t₇, X₁: 2⋅X₂+X₁ {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: X₂ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₃ {O(n)}
t₉, X₀: X₂ {O(n)}
t₉, X₁: X₂ {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₃ {O(n)}
t₁₀, X₀: X₂ {O(n)}
t₁₀, X₁: 2⋅X₂ {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₀, X₄: 0 {O(1)}
t₁₁, X₀: X₂ {O(n)}
t₁₁, X₁: X₂ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₃ {O(n)}
t₁₂, X₀: 2⋅X₂ {O(n)}
t₁₂, X₁: 2⋅X₂+X₁ {O(n)}
t₁₂, X₂: 2⋅X₂ {O(n)}
t₁₂, X₃: 2⋅X₃ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: X₀ {O(n)}
t₁₃, X₁: X₁ {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₃, X₄: X₄ {O(n)}
t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₅, X₀: X₀ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}