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(X₅, X₄, X₂, X₃, X₄, X₅) :|: 1+X₄ ≤ X₅
t₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₅, X₅, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₀
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, X₄, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1+X₀, 1+X₀, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₄
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_bb1_in→eval_foo_bb3_in]

Eliminate variables [X₂; X₃] that do not contribute to the problem

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_bb2_in

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

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

Found invariant X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ 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:
t₂₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₂, X₂, X₃) :|: 1+X₂ ≤ X₃
t₂₂: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₃, X₂, X₃) :|: X₃ ≤ X₂
t₂₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂
t₂₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂
t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₂, X₂, X₃) :|: X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ X₂
t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, 1+X₀, X₂, X₃) :|: 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ X₂
t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂
t₂₈: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [X₂-X₁]

MPRF for transition t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₂, X₂, X₃) :|: X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ X₂ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₁]

MPRF for transition t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, 1+X₀, X₂, X₃) :|: 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ X₂ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₂-X₀]
• eval_foo_bb2_in: [1+X₂-X₁]

All Bounds

Timebounds

Overall timebound:3⋅X₂+3⋅X₃+8 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: X₂+X₃+1 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+X₃+1 {O(n)}
t₂₆: X₂+X₃+1 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+3⋅X₃+8 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: X₂+X₃+1 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+X₃+1 {O(n)}
t₂₆: X₂+X₃+1 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Sizebounds

t₂₁, X₀: X₃ {O(n)}
t₂₁, X₁: X₂ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₃ {O(n)}
t₂₂, X₁: X₃ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: 2⋅X₃+X₂+2 {O(n)}
t₂₃, X₁: 3⋅X₃+X₂+4 {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₄, X₀: 3⋅X₃+X₂+3 {O(n)}
t₂₄, X₁: 2⋅X₂ {O(n)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₄, X₃: 2⋅X₃ {O(n)}
t₂₅, X₀: 2⋅X₃+X₂+3 {O(n)}
t₂₅, X₁: X₂ {O(n)}
t₂₅, X₂: X₂ {O(n)}
t₂₅, X₃: X₃ {O(n)}
t₂₆, X₀: 2⋅X₃+X₂+2 {O(n)}
t₂₆, X₁: 3⋅X₃+X₂+4 {O(n)}
t₂₆, X₂: X₂ {O(n)}
t₂₆, X₃: X₃ {O(n)}
t₂₇, X₀: 3⋅X₃+X₂+3 {O(n)}
t₂₇, X₁: 2⋅X₂ {O(n)}
t₂₇, X₂: 2⋅X₂ {O(n)}
t₂₇, X₃: 2⋅X₃ {O(n)}
t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}
t₂₈, X₃: X₃ {O(n)}