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, X₁, X₂, X₃) :|: 1 ≤ X₃
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₁, X₂, X₃) :|: 1+X₃ ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: X₀ ≤ X₃
t₇: 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

Eliminate variables [X₁] that do not contribute to the problem

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

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

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
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:
t₁₅: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₁, X₁, X₂) :|: 1 ≤ X₂
t₁₆: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ 0
t₁₇: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₉: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-1, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁
t₂₁: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂)
t₂₂: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₁₇: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

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

MPRF for transition t₁₉: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-1, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]

All Bounds

Timebounds

Overall timebound:3⋅X₁+5 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: X₁ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₁+5 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: X₁ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}

Sizebounds

t₁₅, X₀: X₁ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₇, X₀: X₁ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₀: 2⋅X₁ {O(n)}
t₁₈, X₁: 2⋅X₁ {O(n)}
t₁₈, X₂: 2⋅X₂ {O(n)}
t₁₉, X₀: X₁ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: X₁ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: 2⋅X₁+X₀ {O(n)}
t₂₁, X₁: 3⋅X₁ {O(n)}
t₂₁, X₂: 3⋅X₂ {O(n)}
t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}