KoAT2 Proof Output

Initial Problem

Start: evalEx6start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalEx6bb1in, evalEx6bb2in, evalEx6bb3in, evalEx6bbin, evalEx6entryin, evalEx6returnin, evalEx6start, evalEx6stop
Transitions:
t₆: evalEx6bb1in(X₀, X₁, X₂) → evalEx6bb3in(X₀, 1+X₁, X₂)
t₇: evalEx6bb2in(X₀, X₁, X₂) → evalEx6bb3in(1+X₀, X₁, X₂)
t₂: evalEx6bb3in(X₀, X₁, X₂) → evalEx6bbin(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂
t₃: evalEx6bb3in(X₀, X₁, X₂) → evalEx6returnin(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₄: evalEx6bbin(X₀, X₁, X₂) → evalEx6bb1in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀
t₅: evalEx6bbin(X₀, X₁, X₂) → evalEx6bb2in(X₀, X₁, X₂) :|: X₀ ≤ X₁
t₁: evalEx6entryin(X₀, X₁, X₂) → evalEx6bb3in(X₁, X₀, X₂)
t₈: evalEx6returnin(X₀, X₁, X₂) → evalEx6stop(X₀, X₁, X₂)
t₀: evalEx6start(X₀, X₁, X₂) → evalEx6entryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₂ ≤ X₁ for location evalEx6returnin

Found invariant X₂ ≤ X₁ for location evalEx6stop

Found invariant 1+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location evalEx6bb1in

Found invariant 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₀ ≤ X₁ for location evalEx6bb2in

Found invariant 1+X₁ ≤ X₂ for location evalEx6bbin

Problem after Preprocessing

Start: evalEx6start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalEx6bb1in, evalEx6bb2in, evalEx6bb3in, evalEx6bbin, evalEx6entryin, evalEx6returnin, evalEx6start, evalEx6stop
Transitions:
t₆: evalEx6bb1in(X₀, X₁, X₂) → evalEx6bb3in(X₀, 1+X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂
t₇: evalEx6bb2in(X₀, X₁, X₂) → evalEx6bb3in(1+X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₁
t₂: evalEx6bb3in(X₀, X₁, X₂) → evalEx6bbin(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂
t₃: evalEx6bb3in(X₀, X₁, X₂) → evalEx6returnin(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₄: evalEx6bbin(X₀, X₁, X₂) → evalEx6bb1in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂
t₅: evalEx6bbin(X₀, X₁, X₂) → evalEx6bb2in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1+X₁ ≤ X₂
t₁: evalEx6entryin(X₀, X₁, X₂) → evalEx6bb3in(X₁, X₀, X₂)
t₈: evalEx6returnin(X₀, X₁, X₂) → evalEx6stop(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₀: evalEx6start(X₀, X₁, X₂) → evalEx6entryin(X₀, X₁, X₂)

MPRF for transition t₄: evalEx6bbin(X₀, X₁, X₂) → evalEx6bb1in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ of depth 1:

new bound:

X₀+X₂ {O(n)}

MPRF:

• evalEx6bb1in: [X₂-1-X₁]
• evalEx6bb2in: [X₂-X₁]
• evalEx6bb3in: [X₂-X₁]
• evalEx6bbin: [X₂-X₁]

MPRF for transition t₅: evalEx6bbin(X₀, X₁, X₂) → evalEx6bb2in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1+X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

• evalEx6bb1in: [X₂-X₀]
• evalEx6bb2in: [X₂-1-X₀]
• evalEx6bb3in: [X₂-X₀]
• evalEx6bbin: [X₂-X₀]

MPRF for transition t₆: evalEx6bb1in(X₀, X₁, X₂) → evalEx6bb3in(X₀, 1+X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ of depth 1:

new bound:

X₀+X₂ {O(n)}

MPRF:

• evalEx6bb1in: [X₂-X₁]
• evalEx6bb2in: [X₂-X₁]
• evalEx6bb3in: [X₂-X₁]
• evalEx6bbin: [X₂-X₁]

MPRF for transition t₇: evalEx6bb2in(X₀, X₁, X₂) → evalEx6bb3in(1+X₀, X₁, X₂) :|: 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

• evalEx6bb1in: [X₂-X₀]
• evalEx6bb2in: [X₂-X₀]
• evalEx6bb3in: [X₂-X₀]
• evalEx6bbin: [X₂-X₀]

knowledge_propagation leads to new time bound 2⋅X₂+X₀+X₁+1 {O(n)} for transition t₂: evalEx6bb3in(X₀, X₁, X₂) → evalEx6bbin(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂

All Bounds

Timebounds

Overall timebound:3⋅X₀+3⋅X₁+6⋅X₂+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₂+X₀+X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₀+X₂ {O(n)}
t₅: X₁+X₂ {O(n)}
t₆: X₀+X₂ {O(n)}
t₇: X₁+X₂ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₀+3⋅X₁+6⋅X₂+5 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₂+X₀+X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₀+X₂ {O(n)}
t₅: X₁+X₂ {O(n)}
t₆: X₀+X₂ {O(n)}
t₇: X₁+X₂ {O(n)}
t₈: 1 {O(1)}

Sizebounds

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