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:
t₃: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂)
t₄: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂) :|: 1+X₁ ≤ X₂
t₅: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₁: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂)
t₆: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₇: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀
t₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₉: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-1, 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 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location eval_foo_bb2_in

Found invariant 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location eval_foo_bb1_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:
t₃: eval_foo_0(X₀, X₁, X₂) → eval_foo_1(X₀, nondef.0, X₂)
t₄: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂) :|: 1+X₁ ≤ X₂
t₅: eval_foo_1(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₁: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_0(X₀, X₁, X₂)
t₆: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂
t₇: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂
t₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂
t₉: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-1, X₁, X₂) :|: 1+X₁ ≤ X₂ ∧ X₀ ≤ 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)

MPRF for transition t₇: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

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

Found invariant 1+X₂ ≤ 0 ∧ 3+X₁+X₂ ≤ 0 ∧ 3+X₀+X₂ ≤ 0 ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 4+X₀+X₁ ≤ 0 ∧ 2+X₀ ≤ 0 for location eval_foo_bb1_in_v1

Found invariant 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant 1+X₂ ≤ 0 ∧ 3+X₁+X₂ ≤ 0 ∧ 2+X₀+X₂ ≤ 0 ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0 for location eval_foo_bb2_in_v2

Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb1_in

Found invariant 1 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: inf {Infinity}
t₇: X₂ {O(n)}
t₈: 1 {O(1)}
t₉: inf {Infinity}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: inf {Infinity}
t₇: X₂ {O(n)}
t₈: 1 {O(1)}
t₉: inf {Infinity}
t₁₀: 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: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: 0 {O(1)}
t₈, X₂: 2⋅X₂ {O(n)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₂: 3⋅X₂ {O(n)}