KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₆: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂)
t₇: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₁ ≤ 0
t₈: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1 ≤ X₁
t₉: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(2+X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ 0
t₁: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂)
t₂: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: X₀ ≤ 254
t₃: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 255 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂)
t₁₀: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂)
t₀: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

Preprocessing

Found invariant X₂ ≤ 254 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 for location eval_foo_bb2_in

Found invariant X₂ ≤ 254 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 for location eval_foo_0

Found invariant X₂ ≤ 254 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 for location eval_foo_1

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₂ ≤ X₀ ∧ 255 ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ X₀ ∧ 255 ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
Temp_Vars: nondef.0
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₆: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂) :|: X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀
t₇: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₁ ≤ 0 ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀
t₈: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀
t₉: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(2+X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀
t₁: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂)
t₂: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: X₀ ≤ 254 ∧ X₂ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 255 ≤ X₀ ∧ X₂ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂) :|: X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀
t₁₀: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: 255 ≤ X₀ ∧ X₂ ≤ X₀
t₀: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: X₀ ≤ 254 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+255 {O(n)}

MPRF:

• eval_foo_0: [254-X₀]
• eval_foo_1: [254-X₀]
• eval_foo_bb1_in: [255-X₀]
• eval_foo_bb2_in: [254-X₀]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂) :|: X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+255 {O(n)}

MPRF:

• eval_foo_0: [254-X₀]
• eval_foo_1: [254-X₀]
• eval_foo_bb1_in: [255-X₀]
• eval_foo_bb2_in: [255-X₀]

MPRF for transition t₆: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂) :|: X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+255 {O(n)}

MPRF:

• eval_foo_0: [255-X₀]
• eval_foo_1: [254-X₀]
• eval_foo_bb1_in: [255-X₀]
• eval_foo_bb2_in: [255-X₀]

MPRF for transition t₇: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1+X₁ ≤ 0 ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+255 {O(n)}

MPRF:

• eval_foo_0: [255-X₀]
• eval_foo_1: [255-X₀]
• eval_foo_bb1_in: [255-X₀]
• eval_foo_bb2_in: [255-X₀]

MPRF for transition t₈: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+255 {O(n)}

MPRF:

• eval_foo_0: [255-X₀]
• eval_foo_1: [255-X₀]
• eval_foo_bb1_in: [255-X₀]
• eval_foo_bb2_in: [255-X₀]

MPRF for transition t₉: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(2+X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₀+X₂ ≤ 508 ∧ X₀ ≤ 254 ∧ X₂ ≤ 254 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+256 {O(n)}

MPRF:

• eval_foo_0: [255-X₀]
• eval_foo_1: [255-X₀]
• eval_foo_bb1_in: [256-X₀]
• eval_foo_bb2_in: [256-X₀]

All Bounds

Timebounds

Overall timebound:6⋅X₂+1535 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+255 {O(n)}
t₃: 1 {O(1)}
t₄: X₂+255 {O(n)}
t₆: X₂+255 {O(n)}
t₇: X₂+255 {O(n)}
t₈: X₂+255 {O(n)}
t₉: X₂+256 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₂+1535 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+255 {O(n)}
t₃: 1 {O(1)}
t₄: X₂+255 {O(n)}
t₆: X₂+255 {O(n)}
t₇: X₂+255 {O(n)}
t₈: X₂+255 {O(n)}
t₉: X₂+256 {O(n)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 5⋅X₂+1022 {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 16⋅X₂+3066 {O(n)}
t₃, X₂: 4⋅X₂ {O(n)}
t₄, X₀: 5⋅X₂+1022 {O(n)}
t₄, X₂: X₂ {O(n)}
t₆, X₀: 5⋅X₂+1022 {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: 5⋅X₂+1022 {O(n)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: 5⋅X₂+1022 {O(n)}
t₈, X₂: X₂ {O(n)}
t₉, X₀: 5⋅X₂+1022 {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: 16⋅X₂+3066 {O(n)}
t₁₀, X₂: 4⋅X₂ {O(n)}