KoAT2 Proof Output

Initial Problem

Start: evalDis2start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalDis2bb1in, evalDis2bb2in, evalDis2bb3in, evalDis2bbin, evalDis2entryin, evalDis2returnin, evalDis2start, evalDis2stop
Transitions:
t₆: evalDis2bb1in(X₀, X₁, X₂) → evalDis2bb3in(X₀, X₁, 1+X₂)
t₇: evalDis2bb2in(X₀, X₁, X₂) → evalDis2bb3in(X₀, 1+X₁, X₂)
t₂: evalDis2bb3in(X₀, X₁, X₂) → evalDis2bbin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀
t₃: evalDis2bb3in(X₀, X₁, X₂) → evalDis2returnin(X₀, X₁, X₂) :|: X₀ ≤ X₂
t₄: evalDis2bbin(X₀, X₁, X₂) → evalDis2bb1in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁
t₅: evalDis2bbin(X₀, X₁, X₂) → evalDis2bb2in(X₀, X₁, X₂) :|: X₁ ≤ X₂
t₁: evalDis2entryin(X₀, X₁, X₂) → evalDis2bb3in(X₁, X₂, X₀)
t₈: evalDis2returnin(X₀, X₁, X₂) → evalDis2stop(X₀, X₁, X₂)
t₀: evalDis2start(X₀, X₁, X₂) → evalDis2entryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₀ ≤ X₂ for location evalDis2stop

Found invariant 1+X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location evalDis2bb2in

Found invariant 1+X₂ ≤ X₀ for location evalDis2bbin

Found invariant X₀ ≤ X₂ for location evalDis2returnin

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ for location evalDis2bb1in

Problem after Preprocessing

Start: evalDis2start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalDis2bb1in, evalDis2bb2in, evalDis2bb3in, evalDis2bbin, evalDis2entryin, evalDis2returnin, evalDis2start, evalDis2stop
Transitions:
t₆: evalDis2bb1in(X₀, X₁, X₂) → evalDis2bb3in(X₀, X₁, 1+X₂) :|: 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁
t₇: evalDis2bb2in(X₀, X₁, X₂) → evalDis2bb3in(X₀, 1+X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₁ ≤ X₂
t₂: evalDis2bb3in(X₀, X₁, X₂) → evalDis2bbin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀
t₃: evalDis2bb3in(X₀, X₁, X₂) → evalDis2returnin(X₀, X₁, X₂) :|: X₀ ≤ X₂
t₄: evalDis2bbin(X₀, X₁, X₂) → evalDis2bb1in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀
t₅: evalDis2bbin(X₀, X₁, X₂) → evalDis2bb2in(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1+X₂ ≤ X₀
t₁: evalDis2entryin(X₀, X₁, X₂) → evalDis2bb3in(X₁, X₂, X₀)
t₈: evalDis2returnin(X₀, X₁, X₂) → evalDis2stop(X₀, X₁, X₂) :|: X₀ ≤ X₂
t₀: evalDis2start(X₀, X₁, X₂) → evalDis2entryin(X₀, X₁, X₂)

MPRF for transition t₄: evalDis2bbin(X₀, X₁, X₂) → evalDis2bb1in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• evalDis2bb1in: [X₀-1-X₂]
• evalDis2bb2in: [X₀-X₂]
• evalDis2bb3in: [X₀-X₂]
• evalDis2bbin: [X₀-X₂]

MPRF for transition t₅: evalDis2bbin(X₀, X₁, X₂) → evalDis2bb2in(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1+X₂ ≤ X₀ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

• evalDis2bb1in: [X₀-X₁]
• evalDis2bb2in: [X₀-1-X₁]
• evalDis2bb3in: [X₀-X₁]
• evalDis2bbin: [X₀-X₁]

MPRF for transition t₆: evalDis2bb1in(X₀, X₁, X₂) → evalDis2bb3in(X₀, X₁, 1+X₂) :|: 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• evalDis2bb1in: [X₀-X₂]
• evalDis2bb2in: [X₀-X₂]
• evalDis2bb3in: [X₀-X₂]
• evalDis2bbin: [X₀-X₂]

MPRF for transition t₇: evalDis2bb2in(X₀, X₁, X₂) → evalDis2bb3in(X₀, 1+X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

• evalDis2bb1in: [X₀-X₁]
• evalDis2bb2in: [X₀-X₁]
• evalDis2bb3in: [X₀-X₁]
• evalDis2bbin: [X₀-X₁]

knowledge_propagation leads to new time bound 2⋅X₁+X₀+X₂+1 {O(n)} for transition t₂: evalDis2bb3in(X₀, X₁, X₂) → evalDis2bbin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀

All Bounds

Timebounds

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

Costbounds

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