KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalSimpleMultiplebb2in

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

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalSimpleMultiplestop

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalSimpleMultiplebb3in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalSimpleMultiplereturnin

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalSimpleMultiplebb1in

Problem after Preprocessing

Start: evalSimpleMultiplestart
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: evalSimpleMultiplebb1in, evalSimpleMultiplebb2in, evalSimpleMultiplebb3in, evalSimpleMultiplebbin, evalSimpleMultipleentryin, evalSimpleMultiplereturnin, evalSimpleMultiplestart, evalSimpleMultiplestop
Transitions:
t₆: evalSimpleMultiplebb1in(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb3in(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₇: evalSimpleMultiplebb2in(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb3in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁
t₂: evalSimpleMultiplebb3in(X₀, X₁, X₂, X₃) → evalSimpleMultiplebbin(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₃: evalSimpleMultiplebb3in(X₀, X₁, X₂, X₃) → evalSimpleMultiplereturnin(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₄: evalSimpleMultiplebbin(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb1in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₅: evalSimpleMultiplebbin(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb2in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₁: evalSimpleMultipleentryin(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb3in(0, 0, X₂, X₃)
t₈: evalSimpleMultiplereturnin(X₀, X₁, X₂, X₃) → evalSimpleMultiplestop(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ X₂ ≤ X₁
t₀: evalSimpleMultiplestart(X₀, X₁, X₂, X₃) → evalSimpleMultipleentryin(X₀, X₁, X₂, X₃)

MPRF for transition t₄: evalSimpleMultiplebbin(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb1in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• evalSimpleMultiplebb1in: [X₃-1-X₀]
• evalSimpleMultiplebb2in: [X₃-X₀]
• evalSimpleMultiplebb3in: [X₃-X₀]
• evalSimpleMultiplebbin: [X₃-X₀]

MPRF for transition t₅: evalSimpleMultiplebbin(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb2in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• evalSimpleMultiplebb1in: [X₂-X₁]
• evalSimpleMultiplebb2in: [X₂-1-X₁]
• evalSimpleMultiplebb3in: [X₂-X₁]
• evalSimpleMultiplebbin: [X₂-X₁]

MPRF for transition t₆: evalSimpleMultiplebb1in(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb3in(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• evalSimpleMultiplebb1in: [X₃-X₀]
• evalSimpleMultiplebb2in: [X₃-X₀]
• evalSimpleMultiplebb3in: [X₃-X₀]
• evalSimpleMultiplebbin: [X₃-X₀]

MPRF for transition t₇: evalSimpleMultiplebb2in(X₀, X₁, X₂, X₃) → evalSimpleMultiplebb3in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• evalSimpleMultiplebb1in: [X₂-X₁]
• evalSimpleMultiplebb2in: [X₂-X₁]
• evalSimpleMultiplebb3in: [X₂-X₁]
• evalSimpleMultiplebbin: [X₂-X₁]

knowledge_propagation leads to new time bound X₂+X₃+1 {O(n)} for transition t₂: evalSimpleMultiplebb3in(X₀, X₁, X₂, X₃) → evalSimpleMultiplebbin(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁

All Bounds

Timebounds

Overall timebound:3⋅X₂+3⋅X₃+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₃ {O(n)}
t₅: X₂ {O(n)}
t₆: X₃ {O(n)}
t₇: X₂ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+3⋅X₃+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₃ {O(n)}
t₅: X₂ {O(n)}
t₆: X₃ {O(n)}
t₇: X₂ {O(n)}
t₈: 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: 0 {O(1)}
t₁, X₁: 0 {O(1)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₃ {O(n)}
t₂, X₁: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₃ {O(n)}
t₃, X₁: X₂ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₃ {O(n)}
t₄, X₁: X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₃ {O(n)}
t₅, X₁: X₂ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₃ {O(n)}
t₆, X₁: X₂ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₃ {O(n)}
t₇, X₁: X₂ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₃ {O(n)}
t₈, X₁: X₂ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}