KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃, X₄)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Eliminate variables [X₂] that do not contribute to the problem

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

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

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_bb3_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:
t₁₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₁₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₉: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

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

new bound:

X₃ {O(n)}

MPRF:

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

MPRF for transition t₁₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]

All Bounds

Timebounds

Overall timebound:2⋅X₃+5 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃ {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: X₃ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃+5 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃ {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: X₃ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: X₂ {O(n)}
t₁₃, X₁: X₃ {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: X₂ {O(n)}
t₁₄, X₁: X₃ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: 2⋅X₂ {O(n)}
t₁₅, X₁: 2⋅X₃ {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₆, X₀: 2⋅X₂ {O(n)}
t₁₆, X₁: 2⋅X₃ {O(n)}
t₁₆, X₂: 2⋅X₂ {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₃ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: 4⋅X₂ {O(n)}
t₁₈, X₁: 4⋅X₃ {O(n)}
t₁₈, X₂: 4⋅X₂ {O(n)}
t₁₈, X₃: 4⋅X₃ {O(n)}
t₁₉, X₀: X₀ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}