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(X₃, 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 ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀
t₈: 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

Cut unsatisfiable transition [t₅: eval_foo_bb2_in→eval_foo_bb1_in; t₆: eval_foo_bb2_in→eval_foo_bb1_in]

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

Found invariant X₄ ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₄ ≤ X₁ ∧ X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location eval_foo_stop

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 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:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₄, X₂, X₃, X₄)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: X₁ ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₂₃: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₃-X₀]
• eval_foo_bb2_in: [X₃-X₀]

MPRF for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, 1+X₁, X₂, X₃, X₄) :|: X₁ ≤ X₀ ∧ 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₃+X₄ {O(n)}

MPRF:

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

knowledge_propagation leads to new time bound 2⋅X₃+X₂+X₄+1 {O(n)} for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₄ ≤ X₁

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₄+4⋅X₃+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₃+X₂+X₄+1 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+X₃ {O(n)}
t₂₁: X₃+X₄ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₄+4⋅X₃+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₃+X₂+X₄+1 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+X₃ {O(n)}
t₂₁: X₃+X₄ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₄ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: 2⋅X₂+X₃ {O(n)}
t₁₈, X₁: 2⋅X₄+X₃ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 3⋅X₂+X₃ {O(n)}
t₁₉, X₁: 3⋅X₄+X₃ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: 2⋅X₂+X₃ {O(n)}
t₂₀, X₁: 2⋅X₄+X₃ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: 2⋅X₂+X₃ {O(n)}
t₂₁, X₁: 2⋅X₄+X₃ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: 3⋅X₂+X₃ {O(n)}
t₂₂, X₁: 3⋅X₄+X₃ {O(n)}
t₂₂, X₂: 2⋅X₂ {O(n)}
t₂₂, X₃: 2⋅X₃ {O(n)}
t₂₂, X₄: 2⋅X₄ {O(n)}
t₂₃, X₀: X₀ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₃, X₄: X₄ {O(n)}