KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 2+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ 1 ≤ X₀+X₄ ∧ X₀ ≤ 1+X₄ ∧ 2+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_start_7

Found invariant 0 ≤ X₃ for location eval_start_bb1_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_start_stop

Found invariant 1+X₄ ≤ X₂ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₂ for location eval_start_bb2_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ 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_3, eval_start_6, eval_start_7, 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_3(X₀, X₁, X₂, X₃, X₄)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, X₁, X₂, 0, X₄)
t₁₁: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_7(X₀, nondef_0, X₂, X₃, X₄) :|: X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄
t₁₂: eval_start_7(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, X₁, X₂, X₀, X₄) :|: 1 ≤ X₁ ∧ X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄
t₁₃: eval_start_7(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₀) :|: X₁ ≤ 0 ∧ X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ 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, X₃) :|: 1+X₃ ≤ X₂ ∧ 0 ≤ X₃
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₃ ∧ 0 ≤ X₃
t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, X₁, X₂, 1+X₄, X₄) :|: X₂ ≤ 1+X₄ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(1+X₄, X₁, X₂, X₃, X₄) :|: 2+X₄ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄
t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_6(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄
t₁₄: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₃ ∧ 0 ≤ X₃
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, X₃) :|: 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_start_6: [X₂-X₀]
• eval_start_7: [X₂-X₀]
• eval_start_bb1_in: [X₂-X₃]
• eval_start_bb2_in: [X₂-1-X₄]
• eval_start_bb3_in: [X₂-1-X₄]

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

new bound:

2⋅X₂+3 {O(n)}

MPRF:

• eval_start_6: [2⋅X₂-4-X₃-X₄]
• eval_start_7: [2⋅X₂-4-X₃-X₄]
• eval_start_bb1_in: [2⋅X₂-3-2⋅X₃]
• eval_start_bb2_in: [2⋅X₂-3-X₃-X₄]
• eval_start_bb3_in: [2⋅X₂-4-X₃-X₄]

MPRF for transition t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, X₁, X₂, 1+X₄, X₄) :|: X₂ ≤ 1+X₄ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 1+X₄ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_start_6: [X₂-X₄]
• eval_start_7: [X₂-X₀]
• eval_start_bb1_in: [X₂-X₃]
• eval_start_bb2_in: [X₂-X₄]
• eval_start_bb3_in: [X₂-X₄]

MPRF for transition t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_6(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_start_6: [X₂-2-X₄]
• eval_start_7: [X₂+X₄-2⋅X₀]
• eval_start_bb1_in: [X₂-1-X₃]
• eval_start_bb2_in: [X₂-1-X₄]
• eval_start_bb3_in: [X₂-1-X₄]

MPRF for transition t₁₁: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_7(X₀, nondef_0, X₂, X₃, X₄) :|: X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_start_6: [X₂-1-X₄]
• eval_start_7: [X₂-2-X₄]
• eval_start_bb1_in: [X₂-1-X₃]
• eval_start_bb2_in: [X₂-1-X₄]
• eval_start_bb3_in: [X₂-1-X₄]

MPRF for transition t₁₂: eval_start_7(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, X₁, X₂, X₀, X₄) :|: 1 ≤ X₁ ∧ X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_start_6: [X₂-X₀]
• eval_start_7: [X₂-X₀]
• eval_start_bb1_in: [X₂-1-X₃]
• eval_start_bb2_in: [X₂-1-X₄]
• eval_start_bb3_in: [X₂-X₀]

MPRF for transition t₁₃: eval_start_7(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₀) :|: X₁ ≤ 0 ∧ X₀ ≤ 1+X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1+X₄ ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₂+X₃ ∧ 2+X₃ ≤ X₂ ∧ 2 ≤ X₂+X₄ ∧ 2+X₄ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_start_6: [X₂-X₀]
• eval_start_7: [X₂-1-X₄]
• eval_start_bb1_in: [X₂-1-X₃]
• eval_start_bb2_in: [X₂-1-X₄]
• eval_start_bb3_in: [X₂-1-X₄]

All Bounds

Timebounds

Overall timebound:8⋅X₂+15 {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 {O(n)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+3 {O(n)}
t₉: X₂ {O(n)}
t₁₀: X₂+1 {O(n)}
t₁₁: X₂+1 {O(n)}
t₁₂: X₂+1 {O(n)}
t₁₃: X₂+1 {O(n)}
t₁₄: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₂+15 {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 {O(n)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+3 {O(n)}
t₉: X₂ {O(n)}
t₁₀: X₂+1 {O(n)}
t₁₁: X₂+1 {O(n)}
t₁₂: X₂+1 {O(n)}
t₁₃: X₂+1 {O(n)}
t₁₄: 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: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: 0 {O(1)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 2⋅X₂+X₀+3 {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: 2⋅X₂+3 {O(n)}
t₆, X₄: 2⋅X₂+3 {O(n)}
t₇, X₀: 2⋅X₀+4⋅X₂+6 {O(n)}
t₇, X₂: 3⋅X₂ {O(n)}
t₇, X₃: 4⋅X₂+8 {O(n)}
t₇, X₄: 4⋅X₂+X₄+6 {O(n)}
t₈, X₀: 2⋅X₂+3 {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: 2⋅X₂+3 {O(n)}
t₈, X₄: 4⋅X₂+6 {O(n)}
t₉, X₀: 4⋅X₂+X₀+6 {O(n)}
t₉, X₂: 2⋅X₂ {O(n)}
t₉, X₃: 4⋅X₂+8 {O(n)}
t₉, X₄: 4⋅X₂+6 {O(n)}
t₁₀, X₀: 2⋅X₂+3 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: 2⋅X₂+3 {O(n)}
t₁₀, X₄: 4⋅X₂+6 {O(n)}
t₁₁, X₀: 2⋅X₂+3 {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: 2⋅X₂+3 {O(n)}
t₁₁, X₄: 4⋅X₂+6 {O(n)}
t₁₂, X₀: 2⋅X₂+3 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: 2⋅X₂+3 {O(n)}
t₁₂, X₄: 4⋅X₂+6 {O(n)}
t₁₃, X₀: 2⋅X₂+3 {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: 2⋅X₂+3 {O(n)}
t₁₃, X₄: 2⋅X₂+3 {O(n)}
t₁₄, X₀: 2⋅X₀+4⋅X₂+6 {O(n)}
t₁₄, X₂: 3⋅X₂ {O(n)}
t₁₄, X₃: 4⋅X₂+8 {O(n)}
t₁₄, X₄: 4⋅X₂+X₄+6 {O(n)}