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_speed_popl10_nested_single_bb3_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_speed_popl10_nested_single_stop

Found invariant 0 ≤ X₃ for location eval_speed_popl10_nested_single_bb1_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_speed_popl10_nested_single_bb4_in

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_speed_popl10_nested_single_bb2_in

Problem after Preprocessing

Start: eval_speed_popl10_nested_single_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_speed_popl10_nested_single_2, eval_speed_popl10_nested_single_3, eval_speed_popl10_nested_single_bb0_in, eval_speed_popl10_nested_single_bb1_in, eval_speed_popl10_nested_single_bb2_in, eval_speed_popl10_nested_single_bb3_in, eval_speed_popl10_nested_single_bb4_in, eval_speed_popl10_nested_single_start, eval_speed_popl10_nested_single_stop
Transitions:
t₈: eval_speed_popl10_nested_single_2(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_3(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
t₉: eval_speed_popl10_nested_single_3(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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
t₁₀: eval_speed_popl10_nested_single_3(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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_speed_popl10_nested_single_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_bb1_in(X₀, X₁, X₂, 0, X₄)
t₂: eval_speed_popl10_nested_single_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_bb2_in(X₀, X₁, X₂, X₃, X₃) :|: 1+X₃ ≤ X₂ ∧ 0 ≤ X₃
t₃: eval_speed_popl10_nested_single_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₃ ∧ 0 ≤ X₃
t₅: eval_speed_popl10_nested_single_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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
t₄: eval_speed_popl10_nested_single_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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
t₆: eval_speed_popl10_nested_single_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_2(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
t₁₁: eval_speed_popl10_nested_single_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_stop(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₃ ∧ 0 ≤ X₃
t₀: eval_speed_popl10_nested_single_start(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: eval_speed_popl10_nested_single_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_bb2_in(X₀, X₁, X₂, X₃, X₃) :|: 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_speed_popl10_nested_single_2: [X₂-X₀]
• eval_speed_popl10_nested_single_3: [X₂-X₀]
• eval_speed_popl10_nested_single_bb1_in: [X₂-X₃]
• eval_speed_popl10_nested_single_bb2_in: [X₂-1-X₄]
• eval_speed_popl10_nested_single_bb3_in: [X₂-1-X₄]

MPRF for transition t₄: eval_speed_popl10_nested_single_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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:

X₂+1 {O(n)}

MPRF:

• eval_speed_popl10_nested_single_2: [X₂-1-X₀]
• eval_speed_popl10_nested_single_3: [X₂+X₄-2⋅X₀]
• eval_speed_popl10_nested_single_bb1_in: [X₂-1-X₃]
• eval_speed_popl10_nested_single_bb2_in: [X₂-1-X₄]
• eval_speed_popl10_nested_single_bb3_in: [X₂-1-X₀]

MPRF for transition t₅: eval_speed_popl10_nested_single_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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_speed_popl10_nested_single_2: [X₂-X₃]
• eval_speed_popl10_nested_single_3: [X₂-X₃]
• eval_speed_popl10_nested_single_bb1_in: [X₂-X₃]
• eval_speed_popl10_nested_single_bb2_in: [X₂-X₃]
• eval_speed_popl10_nested_single_bb3_in: [X₂-X₃]

MPRF for transition t₆: eval_speed_popl10_nested_single_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_2(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_speed_popl10_nested_single_2: [X₂-2-X₄]
• eval_speed_popl10_nested_single_3: [X₂-1-X₀]
• eval_speed_popl10_nested_single_bb1_in: [X₂-1-X₃]
• eval_speed_popl10_nested_single_bb2_in: [X₂-1-X₄]
• eval_speed_popl10_nested_single_bb3_in: [X₂-1-X₄]

MPRF for transition t₈: eval_speed_popl10_nested_single_2(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_3(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_speed_popl10_nested_single_2: [1+X₂-X₄]
• eval_speed_popl10_nested_single_3: [X₂-X₄]
• eval_speed_popl10_nested_single_bb1_in: [1+X₂-X₃]
• eval_speed_popl10_nested_single_bb2_in: [1+X₂-X₄]
• eval_speed_popl10_nested_single_bb3_in: [1+X₂-X₄]

MPRF for transition t₉: eval_speed_popl10_nested_single_3(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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_speed_popl10_nested_single_2: [X₂-X₀]
• eval_speed_popl10_nested_single_3: [X₂-X₀]
• eval_speed_popl10_nested_single_bb1_in: [X₂-1-X₃]
• eval_speed_popl10_nested_single_bb2_in: [X₂-1-X₄]
• eval_speed_popl10_nested_single_bb3_in: [X₂-X₀]

MPRF for transition t₁₀: eval_speed_popl10_nested_single_3(X₀, X₁, X₂, X₃, X₄) → eval_speed_popl10_nested_single_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:

All Bounds

Timebounds

Overall timebound:7⋅X₂+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₄: X₂+1 {O(n)}
t₅: X₂ {O(n)}
t₆: X₂+1 {O(n)}
t₈: X₂+1 {O(n)}
t₉: X₂+1 {O(n)}
t₁₀: X₂+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 7⋅X₂+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₄: X₂+1 {O(n)}
t₅: X₂ {O(n)}
t₆: X₂+1 {O(n)}
t₈: X₂+1 {O(n)}
t₉: X₂+1 {O(n)}
t₁₀: X₂+1 {O(n)}
t₁₁: 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: 0 {O(1)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₀+X₂+1 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₂+1 {O(n)}
t₂, X₄: X₂+1 {O(n)}
t₃, X₀: 2⋅X₀+2⋅X₂+2 {O(n)}
t₃, X₂: 3⋅X₂ {O(n)}
t₃, X₃: 2⋅X₂+4 {O(n)}
t₃, X₄: 2⋅X₂+X₄+2 {O(n)}
t₄, X₀: X₂+1 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₂+1 {O(n)}
t₄, X₄: 2⋅X₂+2 {O(n)}
t₅, X₀: 2⋅X₂+X₀+2 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₂+4 {O(n)}
t₅, X₄: 2⋅X₂+2 {O(n)}
t₆, X₀: X₂+1 {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₂+1 {O(n)}
t₆, X₄: 2⋅X₂+2 {O(n)}
t₈, X₀: X₂+1 {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₂+1 {O(n)}
t₈, X₄: 2⋅X₂+2 {O(n)}
t₉, X₀: X₂+1 {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₂+1 {O(n)}
t₉, X₄: 2⋅X₂+2 {O(n)}
t₁₀, X₀: X₂+1 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₂+1 {O(n)}
t₁₀, X₄: X₂+1 {O(n)}
t₁₁, X₀: 2⋅X₀+2⋅X₂+2 {O(n)}
t₁₁, X₂: 3⋅X₂ {O(n)}
t₁₁, X₃: 2⋅X₂+4 {O(n)}
t₁₁, X₄: 2⋅X₂+X₄+2 {O(n)}