KoAT2 Proof Output

Initial Problem

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_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_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_4(X₀, X₁, X₂, X₃, X₄)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₀: eval_start_5(X₀, X₁, X₂, X₃, X₄) → eval_start_6(X₀, X₁, nondef_0, X₃, X₄)
t₁₁: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂
t₁₂: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 0
t₁₃: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 0
t₁₄: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_start_0(X₀, X₁, X₂, X₃, X₄)
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀
t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_5(X₀, X₁, X₂, X₃, X₄)
t₁₅: eval_start_bb3_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

Cut unsatisfiable transition [t₁₂: eval_start_6→eval_start_bb1_in; t₁₃: eval_start_6→eval_start_bb1_in]

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ X₃ for location eval_start_bb1_in

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ for location eval_start_5

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location eval_start_stop

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ for location eval_start_6

Found invariant 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ 1+X₁ ≤ X₀ for location eval_start_bb2_in

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location eval_start_bb3_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_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_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_4(X₀, X₁, X₂, X₃, X₄)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₀: eval_start_5(X₀, X₁, X₂, X₃, X₄) → eval_start_6(X₀, X₁, nondef_0, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁₁: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁₄: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_start_0(X₀, X₁, X₂, X₃, X₄)
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_5(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₁₅: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_start_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

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

MPRF for transition t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_start_5(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

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

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

new bound:

X₃+X₄ {O(n)}

MPRF:

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

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

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_start_5: [X₃-X₁]
• eval_start_6: [X₃-X₁]
• eval_start_bb1_in: [X₃-X₁]
• eval_start_bb2_in: [X₃-X₁]

MPRF for transition t₁₄: eval_start_6(X₀, X₁, X₂, X₃, X₄) → eval_start_bb1_in(X₀-1, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₄ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

• eval_start_5: [X₀-X₄]
• eval_start_6: [X₀-X₄]
• eval_start_bb1_in: [X₀-X₄]
• eval_start_bb2_in: [X₀-X₄]

All Bounds

Timebounds

Overall timebound:5⋅X₃+5⋅X₄+9 {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: 1 {O(1)}
t₇: X₃+X₄ {O(n)}
t₈: 1 {O(1)}
t₉: X₃+X₄ {O(n)}
t₁₀: X₃+X₄ {O(n)}
t₁₁: X₃+X₄ {O(n)}
t₁₄: X₃+X₄ {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₃+5⋅X₄+9 {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: 1 {O(1)}
t₇: X₃+X₄ {O(n)}
t₈: 1 {O(1)}
t₉: X₃+X₄ {O(n)}
t₁₀: X₃+X₄ {O(n)}
t₁₁: X₃+X₄ {O(n)}
t₁₄: X₃+X₄ {O(n)}
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: 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: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: X₃ {O(n)}
t₆, X₁: X₄ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: 2⋅X₃+X₄ {O(n)}
t₇, X₁: 2⋅X₄+X₃ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₀: 2⋅X₄+5⋅X₃ {O(n)}
t₈, X₁: 2⋅X₃+5⋅X₄ {O(n)}
t₈, X₃: 3⋅X₃ {O(n)}
t₈, X₄: 3⋅X₄ {O(n)}
t₉, X₀: 2⋅X₃+X₄ {O(n)}
t₉, X₁: 2⋅X₄+X₃ {O(n)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₀: 2⋅X₃+X₄ {O(n)}
t₁₀, X₁: 2⋅X₄+X₃ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₀: 2⋅X₃+X₄ {O(n)}
t₁₁, X₁: 2⋅X₄+X₃ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₄, X₀: 2⋅X₃+X₄ {O(n)}
t₁₄, X₁: 2⋅X₄+X₃ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₅, X₀: 2⋅X₄+5⋅X₃ {O(n)}
t₁₅, X₁: 2⋅X₃+5⋅X₄ {O(n)}
t₁₅, X₃: 3⋅X₃ {O(n)}
t₁₅, X₄: 3⋅X₄ {O(n)}