KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ 1+X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalwcet1bb6in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalwcet1bb4in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalwcet1bb5in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalwcet1bb1in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalwcet1bbin

Problem after Preprocessing

Start: evalwcet1start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: evalwcet1bb1in, evalwcet1bb4in, evalwcet1bb5in, evalwcet1bb6in, evalwcet1bbin, evalwcet1entryin, evalwcet1returnin, evalwcet1start, evalwcet1stop
Transitions:
t₆: evalwcet1bb1in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, 0) :|: X₀ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₇: evalwcet1bb1in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, 1+X₁) :|: 2+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₈: evalwcet1bb4in(X₀, X₁, X₂, X₃) → evalwcet1bb5in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₉: evalwcet1bb4in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, X₁-1) :|: 2 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₁₀: evalwcet1bb5in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, 0) :|: X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂
t₁₁: evalwcet1bb6in(X₀, X₁, X₂, X₃) → evalwcet1bbin(X₀, X₃, X₂-1, X₃) :|: 2 ≤ X₂ ∧ X₃ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃
t₁₂: evalwcet1bb6in(X₀, X₁, X₂, X₃) → evalwcet1returnin(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1 ∧ X₃ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃
t₃: evalwcet1bbin(X₀, X₁, X₂, X₃) → evalwcet1bb1in(X₀, X₁, X₂, X₃) :|: 1+E ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₄: evalwcet1bbin(X₀, X₁, X₂, X₃) → evalwcet1bb1in(X₀, X₁, X₂, X₃) :|: 1 ≤ E ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₅: evalwcet1bbin(X₀, X₁, X₂, X₃) → evalwcet1bb4in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₁: evalwcet1entryin(X₀, X₁, X₂, X₃) → evalwcet1bbin(X₀, 0, X₀, X₃) :|: 1 ≤ X₀
t₂: evalwcet1entryin(X₀, X₁, X₂, X₃) → evalwcet1returnin(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₃: evalwcet1returnin(X₀, X₁, X₂, X₃) → evalwcet1stop(X₀, X₁, X₂, X₃)
t₀: evalwcet1start(X₀, X₁, X₂, X₃) → evalwcet1entryin(X₀, X₁, X₂, X₃)

MPRF for transition t₃: evalwcet1bbin(X₀, X₁, X₂, X₃) → evalwcet1bb1in(X₀, X₁, X₂, X₃) :|: 1+E ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂-1]
• evalwcet1bb4in: [X₂]
• evalwcet1bb5in: [X₂]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₄: evalwcet1bbin(X₀, X₁, X₂, X₃) → evalwcet1bb1in(X₀, X₁, X₂, X₃) :|: 1 ≤ E ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂-1]
• evalwcet1bb4in: [X₂]
• evalwcet1bb5in: [X₂]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₅: evalwcet1bbin(X₀, X₁, X₂, X₃) → evalwcet1bb4in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂-1]
• evalwcet1bb4in: [X₂-1]
• evalwcet1bb5in: [X₂-1]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₆: evalwcet1bb1in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, 0) :|: X₀ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂]
• evalwcet1bb4in: [X₂]
• evalwcet1bb5in: [X₂]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₇: evalwcet1bb1in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, 1+X₁) :|: 2+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂]
• evalwcet1bb4in: [X₂-1]
• evalwcet1bb5in: [X₂-1]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₈: evalwcet1bb4in(X₀, X₁, X₂, X₃) → evalwcet1bb5in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂]
• evalwcet1bb4in: [X₂]
• evalwcet1bb5in: [X₂-1]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₉: evalwcet1bb4in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, X₁-1) :|: 2 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂-1]
• evalwcet1bb4in: [X₂]
• evalwcet1bb5in: [X₂]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₁₀: evalwcet1bb5in(X₀, X₁, X₂, X₃) → evalwcet1bb6in(X₀, X₁, X₂, 0) :|: X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalwcet1bb1in: [X₂]
• evalwcet1bb4in: [X₂]
• evalwcet1bb5in: [X₂]
• evalwcet1bb6in: [X₂-1]
• evalwcet1bbin: [X₂]

MPRF for transition t₁₁: evalwcet1bb6in(X₀, X₁, X₂, X₃) → evalwcet1bbin(X₀, X₃, X₂-1, X₃) :|: 2 ≤ X₂ ∧ X₃ ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• evalwcet1bb1in: [1+X₂]
• evalwcet1bb4in: [1+X₂]
• evalwcet1bb5in: [1+X₂]
• evalwcet1bb6in: [1+X₂]
• evalwcet1bbin: [1+X₂]

All Bounds

Timebounds

Overall timebound:9⋅X₀+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀ {O(n)}
t₄: X₀ {O(n)}
t₅: X₀ {O(n)}
t₆: X₀ {O(n)}
t₇: X₀ {O(n)}
t₈: X₀ {O(n)}
t₉: X₀ {O(n)}
t₁₀: X₀ {O(n)}
t₁₁: X₀+1 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 9⋅X₀+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀ {O(n)}
t₄: X₀ {O(n)}
t₅: X₀ {O(n)}
t₆: X₀ {O(n)}
t₇: X₀ {O(n)}
t₈: X₀ {O(n)}
t₉: X₀ {O(n)}
t₁₀: X₀ {O(n)}
t₁₁: X₀+1 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 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: 0 {O(1)}
t₁, X₂: X₀ {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+2 {O(n)}
t₃, X₂: X₀ {O(n)}
t₃, X₃: 3⋅X₀+X₃+8 {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀+2 {O(n)}
t₄, X₂: X₀ {O(n)}
t₄, X₃: 3⋅X₀+X₃+8 {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀+2 {O(n)}
t₅, X₂: X₀ {O(n)}
t₅, X₃: 3⋅X₀+X₃+8 {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀+2 {O(n)}
t₆, X₂: X₀ {O(n)}
t₆, X₃: 0 {O(1)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀+2 {O(n)}
t₇, X₂: X₀ {O(n)}
t₇, X₃: 2⋅X₀+6 {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: 1 {O(1)}
t₈, X₂: X₀ {O(n)}
t₈, X₃: 3⋅X₀+X₃+8 {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀+2 {O(n)}
t₉, X₂: X₀ {O(n)}
t₉, X₃: X₀+2 {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: 1 {O(1)}
t₁₀, X₂: X₀ {O(n)}
t₁₀, X₃: 0 {O(1)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₀+2 {O(n)}
t₁₁, X₂: X₀ {O(n)}
t₁₁, X₃: 3⋅X₀+8 {O(n)}
t₁₂, X₀: 4⋅X₀ {O(n)}
t₁₂, X₁: 3⋅X₀+7 {O(n)}
t₁₂, X₂: 1 {O(1)}
t₁₂, X₃: 3⋅X₀+8 {O(n)}
t₁₃, X₀: 5⋅X₀ {O(n)}
t₁₃, X₁: 3⋅X₀+X₁+7 {O(n)}
t₁₃, X₂: X₂+1 {O(n)}
t₁₃, X₃: 3⋅X₀+X₃+8 {O(n)}