KoAT2 Proof Output

Initial Problem

Start: evalSequentialSinglestart
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: evalSequentialSinglebb1in, evalSequentialSinglebb2in, evalSequentialSinglebb4in, evalSequentialSinglebb5in, evalSequentialSinglebbin, evalSequentialSingleentryin, evalSequentialSinglereturnin, evalSequentialSinglestart, evalSequentialSinglestop
Transitions:
t₃: evalSequentialSinglebb1in(X₀, X₁) → evalSequentialSinglebb2in(X₀, X₁) :|: 1+X₀ ≤ X₁
t₂: evalSequentialSinglebb1in(X₀, X₁) → evalSequentialSinglebb5in(X₀, X₁) :|: X₁ ≤ X₀
t₆: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebb5in(X₀, X₁)
t₄: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebbin(X₀, X₁) :|: 1+C ≤ 0
t₅: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebbin(X₀, X₁) :|: 1 ≤ C
t₁₀: evalSequentialSinglebb4in(X₀, X₁) → evalSequentialSinglebb5in(1+X₀, X₁)
t₈: evalSequentialSinglebb5in(X₀, X₁) → evalSequentialSinglebb4in(X₀, X₁) :|: 1+X₀ ≤ X₁
t₉: evalSequentialSinglebb5in(X₀, X₁) → evalSequentialSinglereturnin(X₀, X₁) :|: X₁ ≤ X₀
t₇: evalSequentialSinglebbin(X₀, X₁) → evalSequentialSinglebb1in(1+X₀, X₁)
t₁: evalSequentialSingleentryin(X₀, X₁) → evalSequentialSinglebb1in(0, X₁)
t₁₁: evalSequentialSinglereturnin(X₀, X₁) → evalSequentialSinglestop(X₀, X₁)
t₀: evalSequentialSinglestart(X₀, X₁) → evalSequentialSingleentryin(X₀, X₁)

Preprocessing

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

Found invariant 0 ≤ X₀ for location evalSequentialSinglebb1in

Found invariant 0 ≤ X₀ for location evalSequentialSinglebb5in

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalSequentialSinglebb4in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalSequentialSinglestop

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalSequentialSinglebb2in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location evalSequentialSinglereturnin

Problem after Preprocessing

Start: evalSequentialSinglestart
Program_Vars: X₀, X₁
Temp_Vars: C
Locations: evalSequentialSinglebb1in, evalSequentialSinglebb2in, evalSequentialSinglebb4in, evalSequentialSinglebb5in, evalSequentialSinglebbin, evalSequentialSingleentryin, evalSequentialSinglereturnin, evalSequentialSinglestart, evalSequentialSinglestop
Transitions:
t₃: evalSequentialSinglebb1in(X₀, X₁) → evalSequentialSinglebb2in(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂: evalSequentialSinglebb1in(X₀, X₁) → evalSequentialSinglebb5in(X₀, X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₆: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebb5in(X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₄: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebbin(X₀, X₁) :|: 1+C ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₅: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebbin(X₀, X₁) :|: 1 ≤ C ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₁₀: evalSequentialSinglebb4in(X₀, X₁) → evalSequentialSinglebb5in(1+X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₈: evalSequentialSinglebb5in(X₀, X₁) → evalSequentialSinglebb4in(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₉: evalSequentialSinglebb5in(X₀, X₁) → evalSequentialSinglereturnin(X₀, X₁) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₇: evalSequentialSinglebbin(X₀, X₁) → evalSequentialSinglebb1in(1+X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₁: evalSequentialSingleentryin(X₀, X₁) → evalSequentialSinglebb1in(0, X₁)
t₁₁: evalSequentialSinglereturnin(X₀, X₁) → evalSequentialSinglestop(X₀, X₁) :|: 0 ≤ X₀ ∧ X₁ ≤ X₀
t₀: evalSequentialSinglestart(X₀, X₁) → evalSequentialSingleentryin(X₀, X₁)

MPRF for transition t₃: evalSequentialSinglebb1in(X₀, X₁) → evalSequentialSinglebb2in(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalSequentialSinglebb1in: [X₁-X₀]
• evalSequentialSinglebb2in: [X₁-1-X₀]
• evalSequentialSinglebbin: [X₁-1-X₀]

MPRF for transition t₄: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebbin(X₀, X₁) :|: 1+C ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalSequentialSinglebb1in: [X₁-X₀]
• evalSequentialSinglebb2in: [X₁-X₀]
• evalSequentialSinglebbin: [X₁-1-X₀]

MPRF for transition t₅: evalSequentialSinglebb2in(X₀, X₁) → evalSequentialSinglebbin(X₀, X₁) :|: 1 ≤ C ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalSequentialSinglebb1in: [X₁-X₀]
• evalSequentialSinglebb2in: [X₁-X₀]
• evalSequentialSinglebbin: [X₁-1-X₀]

MPRF for transition t₇: evalSequentialSinglebbin(X₀, X₁) → evalSequentialSinglebb1in(1+X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• evalSequentialSinglebb1in: [X₁-X₀]
• evalSequentialSinglebb2in: [X₁-X₀]
• evalSequentialSinglebbin: [X₁-X₀]

MPRF for transition t₈: evalSequentialSinglebb5in(X₀, X₁) → evalSequentialSinglebb4in(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₁ {O(n)}

MPRF:

• evalSequentialSinglebb4in: [X₁-1-X₀]
• evalSequentialSinglebb5in: [X₁-X₀]

MPRF for transition t₁₀: evalSequentialSinglebb4in(X₀, X₁) → evalSequentialSinglebb5in(1+X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₁ {O(n)}

MPRF:

• evalSequentialSinglebb4in: [X₁-X₀]
• evalSequentialSinglebb5in: [X₁-X₀]

All Bounds

Timebounds

Overall timebound:8⋅X₁+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₁ {O(n)}
t₄: X₁ {O(n)}
t₅: X₁ {O(n)}
t₆: 1 {O(1)}
t₇: X₁ {O(n)}
t₈: 2⋅X₁ {O(n)}
t₉: 1 {O(1)}
t₁₀: 2⋅X₁ {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₁+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₁ {O(n)}
t₄: X₁ {O(n)}
t₅: X₁ {O(n)}
t₆: 1 {O(1)}
t₇: X₁ {O(n)}
t₈: 2⋅X₁ {O(n)}
t₉: 1 {O(1)}
t₁₀: 2⋅X₁ {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: X₁ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₈, X₀: 3⋅X₁ {O(n)}
t₈, X₁: X₁ {O(n)}
t₉, X₀: 4⋅X₁ {O(n)}
t₉, X₁: 3⋅X₁ {O(n)}
t₁₀, X₀: 3⋅X₁ {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₁, X₀: 4⋅X₁ {O(n)}
t₁₁, X₁: 3⋅X₁ {O(n)}