KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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) → eval_foo_bb1_in(X₃, X₁, X₂, X₃) :|: 1 ≤ X₂
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₂, X₁, X₂, X₃)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location eval_foo_bb2_in

Found invariant X₃ ≤ X₀ ∧ 1 ≤ X₂ for location eval_foo_bb1_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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) → eval_foo_bb1_in(X₃, X₁, X₂, X₃) :|: 1 ≤ X₂
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₂, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₁+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₁-X₀]
• eval_foo_bb2_in: [X₁-X₀]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₂, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₁ of depth 1:

new bound:

X₁+X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁+X₂-X₀]
• eval_foo_bb2_in: [1+X₁-X₀]

All Bounds

Timebounds

Overall timebound:2⋅X₁+2⋅X₃+X₂+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₁+X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₁+X₂+X₃ {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+2⋅X₃+X₂+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₁+X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₁+X₂+X₃ {O(n)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₁⋅X₂+X₂⋅X₃+2⋅X₂+X₃ {O(n^2)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₁⋅X₂+X₂⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: X₁⋅X₂+X₂⋅X₃+2⋅X₂+X₃ {O(n^2)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₁⋅X₂+X₂⋅X₃+2⋅X₂+2⋅X₃+X₀ {O(n^2)}
t₆, X₁: 3⋅X₁ {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}