KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location evalperfectbb9in

Found invariant X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb3in

Found invariant X₃ ≤ X₀ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb4in

Found invariant 1+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₀ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb5in

Found invariant 2 ≤ X₀ for location evalperfectbb1in

Found invariant 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb8in

Problem after Preprocessing

Start: evalperfectstart
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: evalperfectbb1in, evalperfectbb3in, evalperfectbb4in, evalperfectbb5in, evalperfectbb8in, evalperfectbb9in, evalperfectentryin, evalperfectreturnin, evalperfectstart, evalperfectstop
Transitions:
t₃: evalperfectbb1in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₀, X₀-1, X₃) :|: 2 ≤ X₀
t₈: evalperfectbb3in(X₀, X₁, X₂, X₃) → evalperfectbb4in(X₀, X₁, X₂, X₃-X₂) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₃
t₆: evalperfectbb4in(X₀, X₁, X₂, X₃) → evalperfectbb3in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀
t₇: evalperfectbb4in(X₀, X₁, X₂, X₃) → evalperfectbb5in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀
t₉: evalperfectbb5in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₁-X₂, X₂-1, X₃) :|: 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀
t₁₀: evalperfectbb5in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₁, X₂-1, X₃) :|: 1+X₃ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀
t₁₁: evalperfectbb5in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₁, X₂-1, X₃) :|: 1 ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀
t₄: evalperfectbb8in(X₀, X₁, X₂, X₃) → evalperfectbb4in(X₀, X₁, X₂, X₀) :|: 1 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₅: evalperfectbb8in(X₀, X₁, X₂, X₃) → evalperfectbb9in(X₁, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₁₂: evalperfectbb9in(X₀, X₁, X₂, X₃) → evalperfectreturnin(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₁₃: evalperfectbb9in(X₀, X₁, X₂, X₃) → evalperfectreturnin(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₁₄: evalperfectbb9in(X₀, X₁, X₂, X₃) → evalperfectreturnin(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₂: evalperfectentryin(X₀, X₁, X₂, X₃) → evalperfectbb1in(X₀, X₁, X₂, X₃) :|: 2 ≤ X₀
t₁: evalperfectentryin(X₀, X₁, X₂, X₃) → evalperfectreturnin(X₀, X₁, X₂, X₃) :|: X₀ ≤ 1
t₁₅: evalperfectreturnin(X₀, X₁, X₂, X₃) → evalperfectstop(X₀, X₁, X₂, X₃)
t₀: evalperfectstart(X₀, X₁, X₂, X₃) → evalperfectentryin(X₀, X₁, X₂, X₃)

MPRF for transition t₄: evalperfectbb8in(X₀, X₁, X₂, X₃) → evalperfectbb4in(X₀, X₁, X₂, X₀) :|: 1 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalperfectbb3in: [X₂]
• evalperfectbb4in: [X₂]
• evalperfectbb5in: [X₂]
• evalperfectbb8in: [1+X₂]

MPRF for transition t₇: evalperfectbb4in(X₀, X₁, X₂, X₃) → evalperfectbb5in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalperfectbb3in: [X₂]
• evalperfectbb4in: [X₂]
• evalperfectbb5in: [X₂-1]
• evalperfectbb8in: [X₂]

MPRF for transition t₉: evalperfectbb5in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₁-X₂, X₂-1, X₃) :|: 0 ≤ X₃ ∧ X₃ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalperfectbb3in: [X₂]
• evalperfectbb4in: [X₂]
• evalperfectbb5in: [X₂]
• evalperfectbb8in: [X₂]

MPRF for transition t₁₀: evalperfectbb5in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₁, X₂-1, X₃) :|: 1+X₃ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ of depth 1:

new bound:

2⋅X₀+2 {O(n)}

MPRF:

• evalperfectbb3in: [X₀+X₂-2]
• evalperfectbb4in: [X₀+X₂-2]
• evalperfectbb5in: [X₀+X₂-2]
• evalperfectbb8in: [X₀+X₂-2]

MPRF for transition t₁₁: evalperfectbb5in(X₀, X₁, X₂, X₃) → evalperfectbb8in(X₀, X₁, X₂-1, X₃) :|: 1 ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 2+X₃ ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalperfectbb3in: [X₂]
• evalperfectbb4in: [X₂]
• evalperfectbb5in: [X₂]
• evalperfectbb8in: [X₂]

MPRF for transition t₆: evalperfectbb4in(X₀, X₁, X₂, X₃) → evalperfectbb3in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ of depth 1:

new bound:

4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}

MPRF:

• evalperfectbb3in: [1+X₃-2⋅X₂]
• evalperfectbb4in: [1+X₃-X₂]
• evalperfectbb5in: [X₃-X₂]
• evalperfectbb8in: [1+X₀]

MPRF for transition t₈: evalperfectbb3in(X₀, X₁, X₂, X₃) → evalperfectbb4in(X₀, X₁, X₂, X₃-X₂) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₃ of depth 1:

new bound:

4⋅X₀⋅X₀+3⋅X₀ {O(n^2)}

MPRF:

• evalperfectbb3in: [X₃]
• evalperfectbb4in: [X₃]
• evalperfectbb5in: [X₃]
• evalperfectbb8in: [X₀]

Cut unsatisfiable transition [t₇: evalperfectbb4in→evalperfectbb5in; t₈₄: evalperfectbb4in→evalperfectbb5in]

Found invariant X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location evalperfectbb9in

Found invariant X₃ ≤ X₀ ∧ 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb4in

Found invariant 1+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb5in

Found invariant 2 ≤ X₀ for location evalperfectbb1in

Found invariant X₃ ≤ X₀ ∧ 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 4 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb3in_v1

Found invariant 1+X₃ ≤ X₀ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb4in_v1

Found invariant 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb8in

Found invariant 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₀ for location evalperfectbb3in_v2

Cut unsatisfiable transition [t₁₀: evalperfectbb5in→evalperfectbb8in]

All Bounds

Timebounds

Overall timebound:8⋅X₀⋅X₀+16⋅X₀+15 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: X₀+1 {O(n)}
t₅: 1 {O(1)}
t₆: 4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 4⋅X₀⋅X₀+3⋅X₀ {O(n^2)}
t₉: X₀ {O(n)}
t₁₀: 2⋅X₀+2 {O(n)}
t₁₁: X₀ {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₀⋅X₀+16⋅X₀+15 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: X₀+1 {O(n)}
t₅: 1 {O(1)}
t₆: 4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
t₇: X₀ {O(n)}
t₈: 4⋅X₀⋅X₀+3⋅X₀ {O(n^2)}
t₉: X₀ {O(n)}
t₁₀: 2⋅X₀+2 {O(n)}
t₁₁: X₀ {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)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₀ {O(n)}
t₃, X₂: X₀ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₄, X₂: X₀ {O(n)}
t₄, X₃: 4⋅X₀ {O(n)}
t₅, X₀: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₅, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₅, X₂: 0 {O(1)}
t₅, X₃: 0 {O(1)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₆, X₂: X₀ {O(n)}
t₆, X₃: 4⋅X₀ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₇, X₂: X₀ {O(n)}
t₇, X₃: 4⋅X₀ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₈, X₂: X₀ {O(n)}
t₈, X₃: 4⋅X₀ {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₉, X₂: X₀ {O(n)}
t₉, X₃: 0 {O(1)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₀, X₂: X₀ {O(n)}
t₁₀, X₃: 4⋅X₀ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₁, X₂: X₀ {O(n)}
t₁₁, X₃: 4⋅X₀ {O(n)}
t₁₂, X₀: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₂, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₂, X₂: 0 {O(1)}
t₁₂, X₃: 0 {O(1)}
t₁₃, X₀: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₃, X₁: X₀⋅X₀+2⋅X₀ {O(n^2)}
t₁₃, X₂: 0 {O(1)}
t₁₃, X₃: 0 {O(1)}
t₁₄, X₀: 0 {O(1)}
t₁₄, X₁: 0 {O(1)}
t₁₄, X₂: 0 {O(1)}
t₁₄, X₃: 0 {O(1)}
t₁₅, X₀: 2⋅X₀⋅X₀+5⋅X₀ {O(n^2)}
t₁₅, X₁: 2⋅X₀⋅X₀+4⋅X₀+X₁ {O(n^2)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}