KoAT2 Proof Output

Initial Problem

Start: eval_t08_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_t08_bb0_in, eval_t08_bb1_in, eval_t08_bb2_in, eval_t08_bb3_in, eval_t08_bb4_in, eval_t08_bb5_in, eval_t08_start, eval_t08_stop
Transitions:
t₁: eval_t08_bb0_in(X₀, X₁, X₂, X₃) → eval_t08_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_t08_bb1_in(X₀, X₁, X₂, X₃) → eval_t08_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃
t₃: eval_t08_bb1_in(X₀, X₁, X₂, X₃) → eval_t08_bb3_in(X₀, X₀, X₂, X₃) :|: X₃ ≤ X₀
t₄: eval_t08_bb2_in(X₀, X₁, X₂, X₃) → eval_t08_bb1_in(1+X₀, X₁, X₂, X₃)
t₅: eval_t08_bb3_in(X₀, X₁, X₂, X₃) → eval_t08_bb4_in(X₀, X₁, X₂, X₃) :|: 3 ≤ X₁
t₆: eval_t08_bb3_in(X₀, X₁, X₂, X₃) → eval_t08_bb5_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2
t₇: eval_t08_bb4_in(X₀, X₁, X₂, X₃) → eval_t08_bb3_in(X₀, X₁-3, X₂, X₃)
t₈: eval_t08_bb5_in(X₀, X₁, X₂, X₃) → eval_t08_stop(X₀, X₁, X₂, X₃)
t₀: eval_t08_start(X₀, X₁, X₂, X₃) → eval_t08_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location eval_t08_bb3_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 2 ∧ X₁ ≤ X₀ for location eval_t08_bb5_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 2 ∧ X₁ ≤ X₀ for location eval_t08_stop

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location eval_t08_bb4_in

Found invariant X₂ ≤ X₀ for location eval_t08_bb1_in

Found invariant 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location eval_t08_bb2_in

Problem after Preprocessing

Start: eval_t08_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_t08_bb0_in, eval_t08_bb1_in, eval_t08_bb2_in, eval_t08_bb3_in, eval_t08_bb4_in, eval_t08_bb5_in, eval_t08_start, eval_t08_stop
Transitions:
t₁: eval_t08_bb0_in(X₀, X₁, X₂, X₃) → eval_t08_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_t08_bb1_in(X₀, X₁, X₂, X₃) → eval_t08_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀
t₃: eval_t08_bb1_in(X₀, X₁, X₂, X₃) → eval_t08_bb3_in(X₀, X₀, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀
t₄: eval_t08_bb2_in(X₀, X₁, X₂, X₃) → eval_t08_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀
t₅: eval_t08_bb3_in(X₀, X₁, X₂, X₃) → eval_t08_bb4_in(X₀, X₁, X₂, X₃) :|: 3 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₆: eval_t08_bb3_in(X₀, X₁, X₂, X₃) → eval_t08_bb5_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2 ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₇: eval_t08_bb4_in(X₀, X₁, X₂, X₃) → eval_t08_bb3_in(X₀, X₁-3, X₂, X₃) :|: 3 ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₈: eval_t08_bb5_in(X₀, X₁, X₂, X₃) → eval_t08_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2 ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₀: eval_t08_start(X₀, X₁, X₂, X₃) → eval_t08_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_t08_bb1_in(X₀, X₁, X₂, X₃) → eval_t08_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_t08_bb1_in: [X₃-X₀]
• eval_t08_bb2_in: [X₃-1-X₀]

MPRF for transition t₄: eval_t08_bb2_in(X₀, X₁, X₂, X₃) → eval_t08_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_t08_bb1_in: [X₃-X₀]
• eval_t08_bb2_in: [X₃-X₀]

MPRF for transition t₅: eval_t08_bb3_in(X₀, X₁, X₂, X₃) → eval_t08_bb4_in(X₀, X₁, X₂, X₃) :|: 3 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ of depth 1:

new bound:

3⋅X₂+X₃+2 {O(n)}

MPRF:

• eval_t08_bb3_in: [X₁-2]
• eval_t08_bb4_in: [X₁-5]

MPRF for transition t₇: eval_t08_bb4_in(X₀, X₁, X₂, X₃) → eval_t08_bb3_in(X₀, X₁-3, X₂, X₃) :|: 3 ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ of depth 1:

new bound:

3⋅X₂+X₃+2 {O(n)}

MPRF:

• eval_t08_bb3_in: [X₁-2]
• eval_t08_bb4_in: [X₁-2]

All Bounds

Timebounds

Overall timebound:4⋅X₃+8⋅X₂+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₂+X₃ {O(n)}
t₅: 3⋅X₂+X₃+2 {O(n)}
t₆: 1 {O(1)}
t₇: 3⋅X₂+X₃+2 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+8⋅X₂+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₂+X₃ {O(n)}
t₅: 3⋅X₂+X₃+2 {O(n)}
t₆: 1 {O(1)}
t₇: 3⋅X₂+X₃+2 {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 2⋅X₂+X₃ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 3⋅X₂+X₃ {O(n)}
t₃, X₁: 3⋅X₂+X₃ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 2⋅X₂+X₃ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 3⋅X₂+X₃ {O(n)}
t₅, X₁: 3⋅X₂+X₃ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₆, X₀: 2⋅X₃+6⋅X₂ {O(n)}
t₆, X₁: 2⋅X₃+6⋅X₂ {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₇, X₀: 3⋅X₂+X₃ {O(n)}
t₇, X₁: 3⋅X₂+X₃ {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₈, X₀: 2⋅X₃+6⋅X₂ {O(n)}
t₈, X₁: 2⋅X₃+6⋅X₂ {O(n)}
t₈, X₂: 4⋅X₂ {O(n)}
t₈, X₃: 4⋅X₃ {O(n)}