KoAT2 Proof Output

Initial Problem

Start: evalNestedMultiplestart
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: F
Locations: evalNestedMultiplebb1in, evalNestedMultiplebb2in, evalNestedMultiplebb3in, evalNestedMultiplebb4in, evalNestedMultiplebb5in, evalNestedMultipleentryin, evalNestedMultiplereturnin, evalNestedMultiplestart, evalNestedMultiplestop
Transitions:
t₉: evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, 1+X₄)
t₅: evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₂
t₄: evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₄
t₆: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) :|: 1+F ≤ 0
t₇: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ F
t₈: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄)
t₁₀: evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb5in(X₀, 1+X₁, X₂, X₄, X₄)
t₂: evalNestedMultiplebb5in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₃) :|: 1+X₁ ≤ X₀
t₃: evalNestedMultiplebb5in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplereturnin(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₁: evalNestedMultipleentryin(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb5in(X₁, X₀, X₃, X₂, X₄)
t₁₁: evalNestedMultiplereturnin(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplestop(X₀, X₁, X₂, X₃, X₄)
t₀: evalNestedMultiplestart(X₀, X₁, X₂, X₃, X₄) → evalNestedMultipleentryin(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location evalNestedMultiplebb1in

Found invariant 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location evalNestedMultiplebb3in

Found invariant X₃ ≤ X₄ ∧ 1+X₁ ≤ X₀ for location evalNestedMultiplebb4in

Found invariant X₃ ≤ X₄ ∧ 1+X₁ ≤ X₀ for location evalNestedMultiplebb2in

Found invariant X₀ ≤ X₁ for location evalNestedMultiplereturnin

Found invariant X₀ ≤ X₁ for location evalNestedMultiplestop

Problem after Preprocessing

Start: evalNestedMultiplestart
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: F
Locations: evalNestedMultiplebb1in, evalNestedMultiplebb2in, evalNestedMultiplebb3in, evalNestedMultiplebb4in, evalNestedMultiplebb5in, evalNestedMultipleentryin, evalNestedMultiplereturnin, evalNestedMultiplestart, evalNestedMultiplestop
Transitions:
t₉: evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, 1+X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄
t₅: evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ X₃ ≤ X₄
t₄: evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₄ ∧ 1+X₁ ≤ X₀ ∧ X₃ ≤ X₄
t₆: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) :|: 1+F ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄
t₇: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ F ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄
t₈: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄
t₁₀: evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb5in(X₀, 1+X₁, X₂, X₄, X₄) :|: 1+X₁ ≤ X₀ ∧ X₃ ≤ X₄
t₂: evalNestedMultiplebb5in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₃) :|: 1+X₁ ≤ X₀
t₃: evalNestedMultiplebb5in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplereturnin(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₁: evalNestedMultipleentryin(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb5in(X₁, X₀, X₃, X₂, X₄)
t₁₁: evalNestedMultiplereturnin(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplestop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₀: evalNestedMultiplestart(X₀, X₁, X₂, X₃, X₄) → evalNestedMultipleentryin(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: evalNestedMultiplebb5in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₃) :|: 1+X₁ ≤ X₀ of depth 1:

new bound:

X₀+X₁+1 {O(n)}

MPRF:

• evalNestedMultiplebb1in: [X₀-X₁]
• evalNestedMultiplebb2in: [X₀-X₁]
• evalNestedMultiplebb3in: [X₀-X₁]
• evalNestedMultiplebb4in: [X₀-X₁]
• evalNestedMultiplebb5in: [1+X₀-X₁]

MPRF for transition t₄: evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₄ ∧ 1+X₁ ≤ X₀ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• evalNestedMultiplebb1in: [X₀-X₁]
• evalNestedMultiplebb2in: [X₀-X₁]
• evalNestedMultiplebb3in: [X₀-X₁]
• evalNestedMultiplebb4in: [X₀-1-X₁]
• evalNestedMultiplebb5in: [X₀-X₁]

MPRF for transition t₅: evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₀+X₁+X₂+X₃ {O(n)}

MPRF:

• evalNestedMultiplebb1in: [X₀+X₂-1-X₁-X₄]
• evalNestedMultiplebb2in: [X₀+X₂-X₁-X₄]
• evalNestedMultiplebb3in: [X₀+X₂-1-X₁-X₄]
• evalNestedMultiplebb4in: [X₀+X₂-1-X₁-X₄]
• evalNestedMultiplebb5in: [X₀+X₂-X₁-X₃]

MPRF for transition t₆: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) :|: 1+F ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• evalNestedMultiplebb1in: [X₂-1-X₄]
• evalNestedMultiplebb2in: [X₂-X₄]
• evalNestedMultiplebb3in: [X₂-X₄]
• evalNestedMultiplebb4in: [X₂-X₄]
• evalNestedMultiplebb5in: [X₂-X₃]

MPRF for transition t₇: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ F ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄ of depth 1:

new bound:

2⋅X₂+2⋅X₃ {O(n)}

MPRF:

• evalNestedMultiplebb1in: [2⋅X₂-1-X₃-X₄]
• evalNestedMultiplebb2in: [2⋅X₂-X₃-X₄]
• evalNestedMultiplebb3in: [2⋅X₂-X₃-X₄]
• evalNestedMultiplebb4in: [2⋅X₂-X₃-X₄]
• evalNestedMultiplebb5in: [2⋅X₂-2⋅X₃]

MPRF for transition t₈: evalNestedMultiplebb3in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

MPRF for transition t₉: evalNestedMultiplebb1in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb2in(X₀, X₁, X₂, X₃, 1+X₄) :|: 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ 1+X₄ ≤ X₂ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• evalNestedMultiplebb1in: [X₂-X₄]
• evalNestedMultiplebb2in: [X₂-X₄]
• evalNestedMultiplebb3in: [X₂-X₄]
• evalNestedMultiplebb4in: [X₂-X₄]
• evalNestedMultiplebb5in: [X₂-X₃]

MPRF for transition t₁₀: evalNestedMultiplebb4in(X₀, X₁, X₂, X₃, X₄) → evalNestedMultiplebb5in(X₀, 1+X₁, X₂, X₄, X₄) :|: 1+X₁ ≤ X₀ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

All Bounds

Timebounds

Overall timebound:5⋅X₀+5⋅X₁+5⋅X₂+5⋅X₃+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₀+X₁ {O(n)}
t₅: X₀+X₁+X₂+X₃ {O(n)}
t₆: X₂+X₃ {O(n)}
t₇: 2⋅X₂+2⋅X₃ {O(n)}
t₈: X₀+X₁ {O(n)}
t₉: X₂+X₃ {O(n)}
t₁₀: X₀+X₁ {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₀+5⋅X₁+5⋅X₂+5⋅X₃+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₀+X₁ {O(n)}
t₅: X₀+X₁+X₂+X₃ {O(n)}
t₆: X₂+X₃ {O(n)}
t₇: 2⋅X₂+2⋅X₃ {O(n)}
t₈: X₀+X₁ {O(n)}
t₉: X₂+X₃ {O(n)}
t₁₀: X₀+X₁ {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₃ {O(n)}
t₁, X₃: X₂ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: 2⋅X₀+X₁ {O(n)}
t₂, X₂: X₃ {O(n)}
t₂, X₃: 3⋅X₂+X₃ {O(n)}
t₂, X₄: 2⋅X₂+X₃ {O(n)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 3⋅X₀+X₁ {O(n)}
t₃, X₂: 2⋅X₃ {O(n)}
t₃, X₃: 3⋅X₂+X₃ {O(n)}
t₃, X₄: 2⋅X₃+4⋅X₂+X₄ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: 2⋅X₀+X₁ {O(n)}
t₄, X₂: X₃ {O(n)}
t₄, X₃: 2⋅X₃+6⋅X₂ {O(n)}
t₄, X₄: 2⋅X₂+X₃ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: 2⋅X₀+X₁ {O(n)}
t₅, X₂: X₃ {O(n)}
t₅, X₃: 3⋅X₂+X₃ {O(n)}
t₅, X₄: 2⋅X₂+X₃ {O(n)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: 2⋅X₀+X₁ {O(n)}
t₆, X₂: X₃ {O(n)}
t₆, X₃: 3⋅X₂+X₃ {O(n)}
t₆, X₄: 2⋅X₂+X₃ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: 2⋅X₀+X₁ {O(n)}
t₇, X₂: X₃ {O(n)}
t₇, X₃: 3⋅X₂+X₃ {O(n)}
t₇, X₄: 2⋅X₂+X₃ {O(n)}
t₈, X₀: X₁ {O(n)}
t₈, X₁: 2⋅X₀+X₁ {O(n)}
t₈, X₂: X₃ {O(n)}
t₈, X₃: 3⋅X₂+X₃ {O(n)}
t₈, X₄: 2⋅X₂+X₃ {O(n)}
t₉, X₀: X₁ {O(n)}
t₉, X₁: 2⋅X₀+X₁ {O(n)}
t₉, X₂: X₃ {O(n)}
t₉, X₃: 3⋅X₂+X₃ {O(n)}
t₉, X₄: 2⋅X₂+X₃ {O(n)}
t₁₀, X₀: X₁ {O(n)}
t₁₀, X₁: 2⋅X₀+X₁ {O(n)}
t₁₀, X₂: X₃ {O(n)}
t₁₀, X₃: 2⋅X₂+X₃ {O(n)}
t₁₀, X₄: 2⋅X₃+4⋅X₂ {O(n)}
t₁₁, X₀: 2⋅X₁ {O(n)}
t₁₁, X₁: 3⋅X₀+X₁ {O(n)}
t₁₁, X₂: 2⋅X₃ {O(n)}
t₁₁, X₃: 3⋅X₂+X₃ {O(n)}
t₁₁, X₄: 2⋅X₃+4⋅X₂+X₄ {O(n)}