KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₁ ≤ 1 for location evalfreturnin

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₁ for location evalfbb1in

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₁ for location evalfbb2in

Found invariant 2 ≤ X₁ for location evalfbbin

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₁ for location evalfbb3in

Found invariant X₁ ≤ 1 for location evalfstop

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₁ for location evalfbb4in

Problem after Preprocessing

Start: evalfstart
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: evalfbb1in, evalfbb2in, evalfbb3in, evalfbb4in, evalfbb5in, evalfbbin, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₁₀: evalfbb1in(X₀, X₁, X₂, X₃) → evalfbb2in(X₀, X₁, X₂, X₃-1) :|: X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃
t₆: evalfbb2in(X₀, X₁, X₂, X₃) → evalfbb3in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₅: evalfbb2in(X₀, X₁, X₂, X₃) → evalfbb4in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₇: evalfbb3in(X₀, X₁, X₂, X₃) → evalfbb1in(X₀, X₁, X₂, X₃) :|: 1+E ≤ 0 ∧ X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃
t₈: evalfbb3in(X₀, X₁, X₂, X₃) → evalfbb1in(X₀, X₁, X₂, X₃) :|: 1 ≤ E ∧ X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃
t₉: evalfbb3in(X₀, X₁, X₂, X₃) → evalfbb4in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃
t₁₁: evalfbb4in(X₀, X₁, X₂, X₃) → evalfbb5in(1+X₃-X₂, X₂-1, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂
t₂: evalfbb5in(X₀, X₁, X₂, X₃) → evalfbbin(X₀, X₁, X₂, X₃) :|: 2 ≤ X₁
t₃: evalfbb5in(X₀, X₁, X₂, X₃) → evalfreturnin(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1
t₄: evalfbbin(X₀, X₁, X₂, X₃) → evalfbb2in(X₀, X₁, X₁-1, X₀+X₁-1) :|: 2 ≤ X₁
t₁: evalfentryin(X₀, X₁, X₂, X₃) → evalfbb5in(X₁, X₁, X₂, X₃)
t₁₂: evalfreturnin(X₀, X₁, X₂, X₃) → evalfstop(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1
t₀: evalfstart(X₀, X₁, X₂, X₃) → evalfentryin(X₀, X₁, X₂, X₃)

MPRF for transition t₂: evalfbb5in(X₀, X₁, X₂, X₃) → evalfbbin(X₀, X₁, X₂, X₃) :|: 2 ≤ X₁ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁]
• evalfbb4in: [X₁]
• evalfbb5in: [1+X₁]
• evalfbbin: [X₁]

MPRF for transition t₄: evalfbbin(X₀, X₁, X₂, X₃) → evalfbb2in(X₀, X₁, X₁-1, X₀+X₁-1) :|: 2 ≤ X₁ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁]
• evalfbb4in: [X₁]
• evalfbb5in: [2+X₁]
• evalfbbin: [1+X₁]

MPRF for transition t₅: evalfbb2in(X₀, X₁, X₂, X₃) → evalfbb4in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalfbb1in: [X₂]
• evalfbb2in: [X₂]
• evalfbb3in: [X₂]
• evalfbb4in: [X₂-2]
• evalfbb5in: [X₁-1]
• evalfbbin: [X₁-1]

MPRF for transition t₆: evalfbb2in(X₀, X₁, X₂, X₃) → evalfbb3in(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ of depth 1:

new bound:

3⋅X₁+3 {O(n)}

MPRF:

• evalfbb1in: [X₂+X₃-2]
• evalfbb2in: [X₂+X₃-1]
• evalfbb3in: [X₂+X₃-2]
• evalfbb4in: [X₂+X₃-4]
• evalfbb5in: [X₀+2⋅X₁-3]
• evalfbbin: [X₀+2⋅X₁-3]

MPRF for transition t₇: evalfbb3in(X₀, X₁, X₂, X₃) → evalfbb1in(X₀, X₁, X₂, X₃) :|: 1+E ≤ 0 ∧ X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃ of depth 1:

new bound:

3⋅X₁ {O(n)}

MPRF:

• evalfbb1in: [X₁+X₃-3]
• evalfbb2in: [X₂+X₃-1]
• evalfbb3in: [X₁+X₃-2]
• evalfbb4in: [X₁+X₃-2]
• evalfbb5in: [X₀+2⋅X₁]
• evalfbbin: [X₀+2⋅X₁]

MPRF for transition t₈: evalfbb3in(X₀, X₁, X₂, X₃) → evalfbb1in(X₀, X₁, X₂, X₃) :|: 1 ≤ E ∧ X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃ of depth 1:

new bound:

2⋅X₁ {O(n)}

MPRF:

• evalfbb1in: [X₃-1]
• evalfbb2in: [X₃]
• evalfbb3in: [X₃]
• evalfbb4in: [X₃]
• evalfbb5in: [X₀+X₁]
• evalfbbin: [X₀+X₁]

MPRF for transition t₉: evalfbb3in(X₀, X₁, X₂, X₃) → evalfbb4in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃ of depth 1:

new bound:

2⋅X₁ {O(n)}

MPRF:

• evalfbb1in: [X₁+X₃-X₂]
• evalfbb2in: [X₁+X₃-X₂]
• evalfbb3in: [1+X₃]
• evalfbb4in: [X₃]
• evalfbb5in: [X₀+X₁]
• evalfbbin: [X₀+X₁]

MPRF for transition t₁₀: evalfbb1in(X₀, X₁, X₂, X₃) → evalfbb2in(X₀, X₁, X₂, X₃-1) :|: X₁ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₃ ∧ X₂ ≤ X₃ of depth 1:

new bound:

2⋅X₁ {O(n)}

MPRF:

• evalfbb1in: [X₃]
• evalfbb2in: [X₃]
• evalfbb3in: [X₃]
• evalfbb4in: [X₃]
• evalfbb5in: [X₀+X₁]
• evalfbbin: [X₀+X₁]

MPRF for transition t₁₁: evalfbb4in(X₀, X₁, X₂, X₃) → evalfbb5in(1+X₃-X₂, X₂-1, X₂, X₃) :|: X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₁+X₂ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalfbb1in: [X₂]
• evalfbb2in: [X₂]
• evalfbb3in: [X₂]
• evalfbb4in: [X₂]
• evalfbb5in: [X₁]
• evalfbbin: [X₁-1]

All Bounds

Timebounds

Overall timebound:16⋅X₁+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₁+2 {O(n)}
t₅: X₁+1 {O(n)}
t₆: 3⋅X₁+3 {O(n)}
t₇: 3⋅X₁ {O(n)}
t₈: 2⋅X₁ {O(n)}
t₉: 2⋅X₁ {O(n)}
t₁₀: 2⋅X₁ {O(n)}
t₁₁: X₁ {O(n)}
t₁₂: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₁+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₁+2 {O(n)}
t₅: X₁+1 {O(n)}
t₆: 3⋅X₁+3 {O(n)}
t₇: 3⋅X₁ {O(n)}
t₈: 2⋅X₁ {O(n)}
t₉: 2⋅X₁ {O(n)}
t₁₀: 2⋅X₁ {O(n)}
t₁₁: X₁ {O(n)}
t₁₂: 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: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 3⋅X₁+X₂ {O(n)}
t₂, X₃: 8⋅X₁⋅X₁+14⋅X₁+X₃ {O(n^2)}
t₃, X₀: 4⋅X₁⋅X₁+8⋅X₁ {O(n^2)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 3⋅X₁+X₂ {O(n)}
t₃, X₃: 8⋅X₁⋅X₁+14⋅X₁+X₃ {O(n^2)}
t₄, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁ {O(n)}
t₄, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₅, X₀: 8⋅X₁⋅X₁+14⋅X₁ {O(n^2)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁ {O(n)}
t₅, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₆, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₁ {O(n)}
t₆, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₇, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁ {O(n)}
t₇, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₈, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₁ {O(n)}
t₈, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₉, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: X₁ {O(n)}
t₉, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₁₀, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₁₀, X₁: X₁ {O(n)}
t₁₀, X₂: X₁ {O(n)}
t₁₀, X₃: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₁₁, X₀: 4⋅X₁⋅X₁+7⋅X₁ {O(n^2)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: 3⋅X₁ {O(n)}
t₁₁, X₃: 8⋅X₁⋅X₁+14⋅X₁ {O(n^2)}
t₁₂, X₀: 4⋅X₁⋅X₁+8⋅X₁ {O(n^2)}
t₁₂, X₁: 2⋅X₁ {O(n)}
t₁₂, X₂: 3⋅X₁+X₂ {O(n)}
t₁₂, X₃: 8⋅X₁⋅X₁+14⋅X₁+X₃ {O(n^2)}