KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
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, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀+X₁ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀+X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

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

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_stop

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

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:
t₁₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀+X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₁₆: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀+X₁ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₂+X₃+X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₂-X₁-X₃]
• eval_foo_bb2_in: [X₂-1-X₁-X₃]

knowledge_propagation leads to new time bound X₂+X₃+X₄ {O(n)} for transition t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+X₃+X₄ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₂+X₃+X₄ {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+X₃+X₄ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₂+X₃+X₄ {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₃ {O(n)}
t₁₁, X₁: X₄ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₂, X₀: 2⋅X₃+X₂+X₄ {O(n)}
t₁₂, X₁: 2⋅X₄+X₂+X₃ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: 3⋅X₃+X₂+X₄ {O(n)}
t₁₃, X₁: 3⋅X₄+X₂+X₃ {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₃, X₄: 2⋅X₄ {O(n)}
t₁₄, X₀: 2⋅X₃+X₂+X₄ {O(n)}
t₁₄, X₁: 2⋅X₄+X₂+X₃ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₅, X₀: 3⋅X₃+X₂+X₄ {O(n)}
t₁₅, X₁: 3⋅X₄+X₂+X₃ {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₅, X₄: 2⋅X₄ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₆, X₄: X₄ {O(n)}