KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 0 ≤ X₀ for location evalrealselectbb6in

Found invariant X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ for location evalrealselectstop

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb1in

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb4in

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb5in

Found invariant 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbbin

Found invariant X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ for location evalrealselectreturnin

Problem after Preprocessing

Start: evalrealselectstart
Program_Vars: X₀, X₁, X₂
Temp_Vars: D, E
Locations: evalrealselectbb1in, evalrealselectbb4in, evalrealselectbb5in, evalrealselectbb6in, evalrealselectbbin, evalrealselectentryin, evalrealselectreturnin, evalrealselectstart, evalrealselectstop
Transitions:
t₇: evalrealselectbb1in(X₀, X₁, X₂) → evalrealselectbb4in(X₀, X₁, 1+X₂) :|: 1+E ≤ D ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀
t₈: evalrealselectbb1in(X₀, X₁, X₂) → evalrealselectbb4in(X₀, X₁, 1+X₂) :|: D ≤ E ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀
t₅: evalrealselectbb4in(X₀, X₁, X₂) → evalrealselectbb1in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₆: evalrealselectbb4in(X₀, X₁, X₂) → evalrealselectbb5in(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₉: evalrealselectbb5in(X₀, X₁, X₂) → evalrealselectbb6in(1+X₀, X₁, X₂) :|: 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₁ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₂: evalrealselectbb6in(X₀, X₁, X₂) → evalrealselectbbin(X₀, X₁, X₂) :|: 2+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₃: evalrealselectbb6in(X₀, X₁, X₂) → evalrealselectreturnin(X₀, X₁, X₂) :|: X₁ ≤ 1+X₀ ∧ 0 ≤ X₀
t₄: evalrealselectbbin(X₀, X₁, X₂) → evalrealselectbb4in(X₀, X₁, 1+X₀) :|: 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 0 ≤ X₀
t₁: evalrealselectentryin(X₀, X₁, X₂) → evalrealselectbb6in(0, X₁, X₂)
t₁₀: evalrealselectreturnin(X₀, X₁, X₂) → evalrealselectstop(X₀, X₁, X₂) :|: X₁ ≤ 1+X₀ ∧ 0 ≤ X₀
t₀: evalrealselectstart(X₀, X₁, X₂) → evalrealselectentryin(X₀, X₁, X₂)

MPRF for transition t₂: evalrealselectbb6in(X₀, X₁, X₂) → evalrealselectbbin(X₀, X₁, X₂) :|: 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalrealselectbb1in: [X₁-2-X₀]
• evalrealselectbb4in: [X₁-2-X₀]
• evalrealselectbb5in: [X₂-2-X₀]
• evalrealselectbb6in: [X₁-1-X₀]
• evalrealselectbbin: [X₁-2-X₀]

MPRF for transition t₄: evalrealselectbbin(X₀, X₁, X₂) → evalrealselectbb4in(X₀, X₁, 1+X₀) :|: 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalrealselectbb1in: [X₁-2-X₀]
• evalrealselectbb4in: [X₁-2-X₀]
• evalrealselectbb5in: [X₁-2-X₀]
• evalrealselectbb6in: [X₁-1-X₀]
• evalrealselectbbin: [X₁-1-X₀]

MPRF for transition t₆: evalrealselectbb4in(X₀, X₁, X₂) → evalrealselectbb5in(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalrealselectbb1in: [X₁-X₀]
• evalrealselectbb4in: [X₁-X₀]
• evalrealselectbb5in: [X₂-1-X₀]
• evalrealselectbb6in: [X₁-X₀]
• evalrealselectbbin: [X₁-X₀]

MPRF for transition t₉: evalrealselectbb5in(X₀, X₁, X₂) → evalrealselectbb6in(1+X₀, X₁, X₂) :|: 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₁ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalrealselectbb1in: [X₁-X₀]
• evalrealselectbb4in: [X₁-X₀]
• evalrealselectbb5in: [X₁-X₀]
• evalrealselectbb6in: [X₁-X₀]
• evalrealselectbbin: [X₁-X₀]

MPRF for transition t₅: evalrealselectbb4in(X₀, X₁, X₂) → evalrealselectbb1in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁⋅X₁+X₁ {O(n^2)}

MPRF:

• evalrealselectbb1in: [X₁-X₂]
• evalrealselectbb4in: [1+X₁-X₂]
• evalrealselectbb5in: [X₁-X₂]
• evalrealselectbb6in: [X₁]
• evalrealselectbbin: [X₁]

MPRF for transition t₇: evalrealselectbb1in(X₀, X₁, X₂) → evalrealselectbb4in(X₀, X₁, 1+X₂) :|: 1+E ≤ D ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+X₁ {O(n^2)}

MPRF:

• evalrealselectbb1in: [X₁-X₂]
• evalrealselectbb4in: [X₁-X₂]
• evalrealselectbb5in: [X₁-X₂]
• evalrealselectbb6in: [X₁]
• evalrealselectbbin: [X₁]

MPRF for transition t₈: evalrealselectbb1in(X₀, X₁, X₂) → evalrealselectbb4in(X₀, X₁, 1+X₂) :|: D ≤ E ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+X₁ {O(n^2)}

MPRF:

• evalrealselectbb1in: [X₁-X₂]
• evalrealselectbb4in: [X₁-X₂]
• evalrealselectbb5in: [X₁-X₂]
• evalrealselectbb6in: [X₁]
• evalrealselectbbin: [X₁]

Cut unsatisfiable transition [t₆: evalrealselectbb4in→evalrealselectbb5in; t₅₈: evalrealselectbb4in→evalrealselectbb5in]

Found invariant 0 ≤ X₀ for location evalrealselectbb6in

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb1in_v1

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb4in_v1

Found invariant X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ for location evalrealselectstop

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb4in

Found invariant 1+X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbb1in_v2

Found invariant 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrealselectbbin

Found invariant X₁ ≤ 1+X₀ ∧ 0 ≤ X₀ for location evalrealselectreturnin

All Bounds

Timebounds

Overall timebound:3⋅X₁⋅X₁+7⋅X₁+6 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₁+1 {O(n)}
t₅: X₁⋅X₁+X₁ {O(n^2)}
t₆: X₁ {O(n)}
t₇: X₁⋅X₁+X₁ {O(n^2)}
t₈: X₁⋅X₁+X₁ {O(n^2)}
t₉: X₁ {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₁⋅X₁+7⋅X₁+6 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₁+1 {O(n)}
t₅: X₁⋅X₁+X₁ {O(n^2)}
t₆: X₁ {O(n)}
t₇: X₁⋅X₁+X₁ {O(n^2)}
t₈: X₁⋅X₁+X₁ {O(n^2)}
t₉: X₁ {O(n)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 4⋅X₁⋅X₁+6⋅X₁+X₂+2 {O(n^2)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 4⋅X₁⋅X₁+6⋅X₁+X₂+2 {O(n^2)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁+1 {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁⋅X₁+3⋅X₁+1 {O(n^2)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 4⋅X₁⋅X₁+6⋅X₁+2 {O(n^2)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 2⋅X₁⋅X₁+3⋅X₁+1 {O(n^2)}
t₈, X₀: X₁ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 2⋅X₁⋅X₁+3⋅X₁+1 {O(n^2)}
t₉, X₀: X₁ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: 4⋅X₁⋅X₁+6⋅X₁+2 {O(n^2)}
t₁₀, X₀: X₁ {O(n)}
t₁₀, X₁: 2⋅X₁ {O(n)}
t₁₀, X₂: 4⋅X₁⋅X₁+6⋅X₁+X₂+2 {O(n^2)}