KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb1_in(X₇, X₈, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 1+X₀ ≤ X₂
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 1+X₁ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)

Preprocessing

Eliminate variables [X₄; X₅; X₆] that do not contribute to the problem

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₂ ≤ X₀ for location eval_foo_stop

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₂ ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₁₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₂ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀ ∧ X₅ ≤ X₁
t₁₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₀ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₅ ≤ X₁
t₁₉: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₁₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₂ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₂+X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₂-X₀]
• eval_foo_bb2_in: [X₂-1-X₀]

MPRF for transition t₁₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₃ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₃+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₃-X₁]
• eval_foo_bb2_in: [X₃-1-X₁]

knowledge_propagation leads to new time bound X₂+X₃+X₄+X₅ {O(n)} for transition t₁₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀ ∧ X₅ ≤ X₁

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+2⋅X₄+2⋅X₅+4 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₂+X₄ {O(n)}
t₁₅: X₃+X₅ {O(n)}
t₁₆: 1 {O(1)}
t₁₇: X₂+X₃+X₄+X₅ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+2⋅X₄+2⋅X₅+4 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₂+X₄ {O(n)}
t₁₅: X₃+X₅ {O(n)}
t₁₆: 1 {O(1)}
t₁₇: X₂+X₃+X₄+X₅ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: X₄ {O(n)}
t₁₃, X₁: X₅ {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₃, X₄: X₄ {O(n)}
t₁₃, X₅: X₅ {O(n)}
t₁₄, X₀: 2⋅X₄+X₂+X₃+X₅ {O(n)}
t₁₄, X₁: 2⋅X₅+X₂+X₃+X₄ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₄, X₅: X₅ {O(n)}
t₁₅, X₀: 2⋅X₄+X₂+X₃+X₅ {O(n)}
t₁₅, X₁: 2⋅X₅+X₂+X₃+X₄ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₅, X₅: X₅ {O(n)}
t₁₆, X₀: 3⋅X₄+X₂+X₃+X₅ {O(n)}
t₁₆, X₁: 3⋅X₅+X₂+X₃+X₄ {O(n)}
t₁₆, X₂: 2⋅X₂ {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₆, X₄: 2⋅X₄ {O(n)}
t₁₆, X₅: 2⋅X₅ {O(n)}
t₁₇, X₀: 2⋅X₄+X₂+X₃+X₅ {O(n)}
t₁₇, X₁: 2⋅X₅+X₂+X₃+X₄ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₇, X₅: X₅ {O(n)}
t₁₈, X₀: 3⋅X₄+X₂+X₃+X₅ {O(n)}
t₁₈, X₁: 3⋅X₅+X₂+X₃+X₄ {O(n)}
t₁₈, X₂: 2⋅X₂ {O(n)}
t₁₈, X₃: 2⋅X₃ {O(n)}
t₁₈, X₄: 2⋅X₄ {O(n)}
t₁₈, X₅: 2⋅X₅ {O(n)}
t₁₉, X₀: X₀ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}