KoAT2 Proof Output

Initial Problem

Start: evalaxstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalaxbb1in, evalaxbb2in, evalaxbb3in, evalaxbbin, evalaxentryin, evalaxreturnin, evalaxstart, evalaxstop
Transitions:
t₅: evalaxbb1in(X₀, X₁, X₂) → evalaxbb2in(X₀, 1+X₁, X₂)
t₃: evalaxbb2in(X₀, X₁, X₂) → evalaxbb1in(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂
t₄: evalaxbb2in(X₀, X₁, X₂) → evalaxbb3in(X₀, X₁, X₂) :|: X₂ ≤ 1+X₁
t₆: evalaxbb3in(X₀, X₁, X₂) → evalaxbbin(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 3+X₀ ≤ X₂
t₇: evalaxbb3in(X₀, X₁, X₂) → evalaxreturnin(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂
t₈: evalaxbb3in(X₀, X₁, X₂) → evalaxreturnin(X₀, X₁, X₂) :|: X₂ ≤ 2+X₀
t₂: evalaxbbin(X₀, X₁, X₂) → evalaxbb2in(X₀, 0, X₂)
t₁: evalaxentryin(X₀, X₁, X₂) → evalaxbbin(0, X₁, X₂)
t₉: evalaxreturnin(X₀, X₁, X₂) → evalaxstop(X₀, X₁, X₂)
t₀: evalaxstart(X₀, X₁, X₂) → evalaxentryin(X₀, X₁, X₂)

Preprocessing

Cut unsatisfiable transition [t₇: evalaxbb3in→evalaxreturnin]

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

Found invariant 0 ≤ X₀ for location evalaxbbin

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb1in

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb2in

Found invariant X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb3in

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

Problem after Preprocessing

Start: evalaxstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalaxbb1in, evalaxbb2in, evalaxbb3in, evalaxbbin, evalaxentryin, evalaxreturnin, evalaxstart, evalaxstop
Transitions:
t₅: evalaxbb1in(X₀, X₁, X₂) → evalaxbb2in(X₀, 1+X₁, X₂) :|: 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₃: evalaxbb2in(X₀, X₁, X₂) → evalaxbb1in(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₄: evalaxbb2in(X₀, X₁, X₂) → evalaxbb3in(X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₆: evalaxbb3in(X₀, X₁, X₂) → evalaxbbin(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 3+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₈: evalaxbb3in(X₀, X₁, X₂) → evalaxreturnin(X₀, X₁, X₂) :|: X₂ ≤ 2+X₀ ∧ X₂ ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₂: evalaxbbin(X₀, X₁, X₂) → evalaxbb2in(X₀, 0, X₂) :|: 0 ≤ X₀
t₁: evalaxentryin(X₀, X₁, X₂) → evalaxbbin(0, X₁, X₂)
t₉: evalaxreturnin(X₀, X₁, X₂) → evalaxstop(X₀, X₁, X₂) :|: X₂ ≤ 2+X₀ ∧ X₂ ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₀: evalaxstart(X₀, X₁, X₂) → evalaxentryin(X₀, X₁, X₂)

MPRF for transition t₆: evalaxbb3in(X₀, X₁, X₂) → evalaxbbin(1+X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 3+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+2 {O(n)}

MPRF:

• evalaxbb1in: [X₂-2-X₀]
• evalaxbb2in: [X₂-2-X₀]
• evalaxbb3in: [X₂-2-X₀]
• evalaxbbin: [X₂-2-X₀]

knowledge_propagation leads to new time bound X₂+3 {O(n)} for transition t₂: evalaxbbin(X₀, X₁, X₂) → evalaxbb2in(X₀, 0, X₂) :|: 0 ≤ X₀

MPRF for transition t₃: evalaxbb2in(X₀, X₁, X₂) → evalaxbb1in(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+3⋅X₂ {O(n^2)}

MPRF:

• evalaxbb1in: [X₂-2-X₁]
• evalaxbb2in: [X₂-1-X₁]
• evalaxbb3in: [X₂-1-X₁]
• evalaxbbin: [X₂]

MPRF for transition t₄: evalaxbb2in(X₀, X₁, X₂) → evalaxbb3in(X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+3 {O(n)}

MPRF:

• evalaxbb1in: [1]
• evalaxbb2in: [1]
• evalaxbb3in: [0]
• evalaxbbin: [1]

MPRF for transition t₅: evalaxbb1in(X₀, X₁, X₂) → evalaxbb2in(X₀, 1+X₁, X₂) :|: 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+4⋅X₂+3 {O(n^2)}

MPRF:

• evalaxbb1in: [1+X₂-X₁]
• evalaxbb2in: [1+X₂-X₁]
• evalaxbb3in: [1+X₂-X₁]
• evalaxbbin: [1+X₂]

Found invariant 3 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb1in_v2

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

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

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb1in_v1

Found invariant 0 ≤ X₀ for location evalaxbbin

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb2in

Found invariant X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb3in

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location evalaxbb2in_v1

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+10⋅X₂+15 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+3 {O(n)}
t₃: X₂⋅X₂+3⋅X₂ {O(n^2)}
t₄: X₂+3 {O(n)}
t₅: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₆: X₂+2 {O(n)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+10⋅X₂+15 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+3 {O(n)}
t₃: X₂⋅X₂+3⋅X₂ {O(n^2)}
t₄: X₂+3 {O(n)}
t₅: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₆: X₂+2 {O(n)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₂+2 {O(n)}
t₂, X₁: 0 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂+2 {O(n)}
t₃, X₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₂+2 {O(n)}
t₄, X₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₂+2 {O(n)}
t₅, X₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₂+2 {O(n)}
t₆, X₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₆, X₂: X₂ {O(n)}
t₈, X₀: X₂+2 {O(n)}
t₈, X₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₈, X₂: X₂ {O(n)}
t₉, X₀: X₂+2 {O(n)}
t₉, X₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₉, X₂: X₂ {O(n)}