KoAT2 Proof Output

Initial Problem

Start: evalaaron2start
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalaaron2bb3in, evalaaron2bb4in, evalaaron2bb5in, evalaaron2bb6in, evalaaron2entryin, evalaaron2returnin, evalaaron2start, evalaaron2stop
Transitions:
t₆: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb4in(X₀, X₁, X₂) :|: 1+D ≤ 0
t₇: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb4in(X₀, X₁, X₂) :|: 1 ≤ D
t₈: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb5in(X₀, X₁, X₂)
t₉: evalaaron2bb4in(X₀, X₁, X₂) → evalaaron2bb6in(X₀, X₁, X₂-1-X₀)
t₁₀: evalaaron2bb5in(X₀, X₁, X₂) → evalaaron2bb6in(X₀, 1+X₀+X₁, X₂)
t₅: evalaaron2bb6in(X₀, X₁, X₂) → evalaaron2bb3in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂
t₃: evalaaron2bb6in(X₀, X₁, X₂) → evalaaron2returnin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁
t₄: evalaaron2bb6in(X₀, X₁, X₂) → evalaaron2returnin(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₁: evalaaron2entryin(X₀, X₁, X₂) → evalaaron2bb6in(X₀, X₂, X₁) :|: 0 ≤ X₀
t₂: evalaaron2entryin(X₀, X₁, X₂) → evalaaron2returnin(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₁₁: evalaaron2returnin(X₀, X₁, X₂) → evalaaron2stop(X₀, X₁, X₂)
t₀: evalaaron2start(X₀, X₁, X₂) → evalaaron2entryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₁ ≤ X₂ ∧ 0 ≤ X₀ for location evalaaron2bb5in

Found invariant X₁ ≤ X₂ ∧ 0 ≤ X₀ for location evalaaron2bb3in

Found invariant X₁ ≤ X₂ ∧ 0 ≤ X₀ for location evalaaron2bb4in

Found invariant 0 ≤ X₀ for location evalaaron2bb6in

Cut unsatisfiable transition [t₄: evalaaron2bb6in→evalaaron2returnin]

Problem after Preprocessing

Start: evalaaron2start
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalaaron2bb3in, evalaaron2bb4in, evalaaron2bb5in, evalaaron2bb6in, evalaaron2entryin, evalaaron2returnin, evalaaron2start, evalaaron2stop
Transitions:
t₆: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb4in(X₀, X₁, X₂) :|: 1+D ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₇: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb4in(X₀, X₁, X₂) :|: 1 ≤ D ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₈: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb5in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂
t₉: evalaaron2bb4in(X₀, X₁, X₂) → evalaaron2bb6in(X₀, X₁, X₂-1-X₀) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂
t₁₀: evalaaron2bb5in(X₀, X₁, X₂) → evalaaron2bb6in(X₀, 1+X₀+X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂
t₅: evalaaron2bb6in(X₀, X₁, X₂) → evalaaron2bb3in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂
t₃: evalaaron2bb6in(X₀, X₁, X₂) → evalaaron2returnin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 0 ≤ X₀
t₁: evalaaron2entryin(X₀, X₁, X₂) → evalaaron2bb6in(X₀, X₂, X₁) :|: 0 ≤ X₀
t₂: evalaaron2entryin(X₀, X₁, X₂) → evalaaron2returnin(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₁₁: evalaaron2returnin(X₀, X₁, X₂) → evalaaron2stop(X₀, X₁, X₂)
t₀: evalaaron2start(X₀, X₁, X₂) → evalaaron2entryin(X₀, X₁, X₂)

MPRF for transition t₅: evalaaron2bb6in(X₀, X₁, X₂) → evalaaron2bb3in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

• evalaaron2bb3in: [X₂-X₁]
• evalaaron2bb4in: [X₂-X₁]
• evalaaron2bb5in: [X₂-X₁]
• evalaaron2bb6in: [1+X₂-X₁]

MPRF for transition t₆: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb4in(X₀, X₁, X₂) :|: 1+D ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

• evalaaron2bb3in: [1+X₂-X₁]
• evalaaron2bb4in: [X₂-X₁]
• evalaaron2bb5in: [X₂-X₁]
• evalaaron2bb6in: [1+X₂-X₁]

MPRF for transition t₇: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb4in(X₀, X₁, X₂) :|: 1 ≤ D ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

• evalaaron2bb3in: [1+X₂-X₁]
• evalaaron2bb4in: [X₂-X₁]
• evalaaron2bb5in: [X₂-X₁]
• evalaaron2bb6in: [1+X₂-X₁]

MPRF for transition t₈: evalaaron2bb3in(X₀, X₁, X₂) → evalaaron2bb5in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

• evalaaron2bb3in: [1+X₂-X₁]
• evalaaron2bb4in: [1+X₂-X₁]
• evalaaron2bb5in: [X₂-X₁]
• evalaaron2bb6in: [1+X₂-X₁]

MPRF for transition t₉: evalaaron2bb4in(X₀, X₁, X₂) → evalaaron2bb6in(X₀, X₁, X₂-1-X₀) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

• evalaaron2bb3in: [1+X₂-X₁]
• evalaaron2bb4in: [1+X₂-X₁]
• evalaaron2bb5in: [X₂-X₁]
• evalaaron2bb6in: [1+X₂-X₁]

MPRF for transition t₁₀: evalaaron2bb5in(X₀, X₁, X₂) → evalaaron2bb6in(X₀, 1+X₀+X₁, X₂) :|: 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

• evalaaron2bb3in: [1+X₂-X₁]
• evalaaron2bb4in: [X₂-X₁]
• evalaaron2bb5in: [1+X₂-X₁]
• evalaaron2bb6in: [1+X₂-X₁]

All Bounds

Timebounds

Overall timebound:6⋅X₁+6⋅X₂+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₅: X₁+X₂+1 {O(n)}
t₆: X₁+X₂+1 {O(n)}
t₇: X₁+X₂+1 {O(n)}
t₈: X₁+X₂+1 {O(n)}
t₉: X₁+X₂+1 {O(n)}
t₁₀: X₁+X₂+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₁+6⋅X₂+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₅: X₁+X₂+1 {O(n)}
t₆: X₁+X₂+1 {O(n)}
t₇: X₁+X₂+1 {O(n)}
t₈: X₁+X₂+1 {O(n)}
t₉: X₁+X₂+1 {O(n)}
t₁₀: X₁+X₂+1 {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₁ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 3⋅X₀ {O(n)}
t₃, X₁: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+5⋅X₂+2 {O(n^2)}
t₃, X₂: 4⋅X₀⋅X₁+4⋅X₀⋅X₂+2⋅X₂+5⋅X₁+8⋅X₀+2 {O(n^2)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀⋅X₁+X₀⋅X₂+2⋅X₀+2⋅X₂+X₁+1 {O(n^2)}
t₅, X₂: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+X₂+1 {O(n^2)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀⋅X₁+X₀⋅X₂+2⋅X₀+2⋅X₂+X₁+1 {O(n^2)}
t₆, X₂: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+X₂+1 {O(n^2)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀⋅X₁+X₀⋅X₂+2⋅X₀+2⋅X₂+X₁+1 {O(n^2)}
t₇, X₂: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+X₂+1 {O(n^2)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₀⋅X₁+X₀⋅X₂+2⋅X₀+2⋅X₂+X₁+1 {O(n^2)}
t₈, X₂: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+X₂+1 {O(n^2)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀⋅X₁+X₀⋅X₂+2⋅X₀+2⋅X₂+X₁+1 {O(n^2)}
t₉, X₂: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+X₂+1 {O(n^2)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₀⋅X₁+X₀⋅X₂+2⋅X₀+2⋅X₂+X₁+1 {O(n^2)}
t₁₀, X₂: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+2⋅X₁+4⋅X₀+X₂+1 {O(n^2)}
t₁₁, X₀: 4⋅X₀ {O(n)}
t₁₁, X₁: 2⋅X₀⋅X₁+2⋅X₀⋅X₂+3⋅X₁+4⋅X₀+5⋅X₂+2 {O(n^2)}
t₁₁, X₂: 4⋅X₀⋅X₁+4⋅X₀⋅X₂+3⋅X₂+5⋅X₁+8⋅X₀+2 {O(n^2)}