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(X₂, X₃, X₂, X₃) :|: 2 ≤ X₂
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀+X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀+X₁ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-X₀, 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 X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 2 ≤ X₀ 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(X₂, X₃, X₂, X₃) :|: 2 ≤ X₂
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀+X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀+X₁ ≤ 0 ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₃
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-X₀, X₂, X₃) :|: 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₁ ≤ 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)

MPRF for transition t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀+X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

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

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-X₀, X₂, X₃) :|: 2 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 0 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

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

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₂+X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₂+X₃+1 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₂+X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₂+X₃+1 {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: X₂ {O(n)}
t₁, X₁: X₃ {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: 2⋅X₂+X₃+2 {O(n)}
t₃, X₁: 2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+6⋅X₃+7⋅X₂+6 {O(n^2)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: 2⋅X₂+X₃+3 {O(n)}
t₄, X₁: 2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+7⋅X₂+7⋅X₃+6 {O(n^2)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: X₂+X₃+3 {O(n)}
t₅, X₁: 2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+6⋅X₃+7⋅X₂+6 {O(n^2)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 2⋅X₂+X₀+X₃+3 {O(n)}
t₆, X₁: 2⋅X₂⋅X₂+3⋅X₂⋅X₃+X₃⋅X₃+7⋅X₂+7⋅X₃+X₁+6 {O(n^2)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}