KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₁ ≤ X₀ for location eval_wcet2_bb1_in

Found invariant 0 ≤ X₂ ∧ X₁ ≤ 2+X₂ ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 for location eval_wcet2_bb4_in

Found invariant X₁ ≤ X₀ ∧ 5 ≤ X₀ for location eval_wcet2_bb5_in

Found invariant 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 for location eval_wcet2_bb2_in

Found invariant X₂ ≤ 9 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 6+X₀ ∧ X₀+X₂ ≤ 13 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location eval_wcet2_bb3_in

Found invariant X₁ ≤ X₀ ∧ 5 ≤ X₀ for location eval_wcet2_stop

Problem after Preprocessing

Start: eval_wcet2_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_wcet2_bb0_in, eval_wcet2_bb1_in, eval_wcet2_bb2_in, eval_wcet2_bb3_in, eval_wcet2_bb4_in, eval_wcet2_bb5_in, eval_wcet2_start, eval_wcet2_stop
Transitions:
t₁: eval_wcet2_bb0_in(X₀, X₁, X₂) → eval_wcet2_bb1_in(X₁, X₁, X₂)
t₂: eval_wcet2_bb1_in(X₀, X₁, X₂) → eval_wcet2_bb2_in(X₀, X₁, 0) :|: X₀ ≤ 4 ∧ X₁ ≤ X₀
t₃: eval_wcet2_bb1_in(X₀, X₁, X₂) → eval_wcet2_bb5_in(X₀, X₁, X₂) :|: 5 ≤ X₀ ∧ X₁ ≤ X₀
t₄: eval_wcet2_bb2_in(X₀, X₁, X₂) → eval_wcet2_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ 9 ∧ 3 ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₅: eval_wcet2_bb2_in(X₀, X₁, X₂) → eval_wcet2_bb4_in(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₆: eval_wcet2_bb2_in(X₀, X₁, X₂) → eval_wcet2_bb4_in(X₀, X₁, X₂) :|: 10 ≤ X₂ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₇: eval_wcet2_bb3_in(X₀, X₁, X₂) → eval_wcet2_bb2_in(X₀, X₁, 1+X₂) :|: X₀+X₂ ≤ 13 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 9 ∧ X₀+X₁ ≤ 8 ∧ X₂ ≤ 6+X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₈: eval_wcet2_bb4_in(X₀, X₁, X₂) → eval_wcet2_bb1_in(1+X₀, X₁, X₂) :|: X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₁ ≤ 4 ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 2+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂
t₉: eval_wcet2_bb5_in(X₀, X₁, X₂) → eval_wcet2_stop(X₀, X₁, X₂) :|: 5 ≤ X₀ ∧ X₁ ≤ X₀
t₀: eval_wcet2_start(X₀, X₁, X₂) → eval_wcet2_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂: eval_wcet2_bb1_in(X₀, X₁, X₂) → eval_wcet2_bb2_in(X₀, X₁, 0) :|: X₀ ≤ 4 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

• eval_wcet2_bb1_in: [5-X₀]
• eval_wcet2_bb2_in: [4-X₀]
• eval_wcet2_bb3_in: [4-X₀]
• eval_wcet2_bb4_in: [4-X₀]

MPRF for transition t₅: eval_wcet2_bb2_in(X₀, X₁, X₂) → eval_wcet2_bb4_in(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

• eval_wcet2_bb1_in: [5-X₀]
• eval_wcet2_bb2_in: [5-X₀]
• eval_wcet2_bb3_in: [5-X₀]
• eval_wcet2_bb4_in: [4-X₀]

MPRF for transition t₆: eval_wcet2_bb2_in(X₀, X₁, X₂) → eval_wcet2_bb4_in(X₀, X₁, X₂) :|: 10 ≤ X₂ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

• eval_wcet2_bb1_in: [5-X₀]
• eval_wcet2_bb2_in: [5-X₀]
• eval_wcet2_bb3_in: [5-X₀]
• eval_wcet2_bb4_in: [4-X₀]

MPRF for transition t₈: eval_wcet2_bb4_in(X₀, X₁, X₂) → eval_wcet2_bb1_in(1+X₀, X₁, X₂) :|: X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₁ ≤ 4 ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 2+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

• eval_wcet2_bb1_in: [5-X₀]
• eval_wcet2_bb2_in: [5-X₀]
• eval_wcet2_bb3_in: [5-X₀]
• eval_wcet2_bb4_in: [5-X₀]

MPRF for transition t₄: eval_wcet2_bb2_in(X₀, X₁, X₂) → eval_wcet2_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ 9 ∧ 3 ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

10⋅X₁+60 {O(n)}

MPRF:

• eval_wcet2_bb1_in: [10]
• eval_wcet2_bb2_in: [10-X₂]
• eval_wcet2_bb3_in: [9-X₂]
• eval_wcet2_bb4_in: [-X₂]

MPRF for transition t₇: eval_wcet2_bb3_in(X₀, X₁, X₂) → eval_wcet2_bb2_in(X₀, X₁, 1+X₂) :|: X₀+X₂ ≤ 13 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 9 ∧ X₀+X₁ ≤ 8 ∧ X₂ ≤ 6+X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

10⋅X₁+60 {O(n)}

MPRF:

• eval_wcet2_bb1_in: [10]
• eval_wcet2_bb2_in: [10-X₂]
• eval_wcet2_bb3_in: [10-X₂]
• eval_wcet2_bb4_in: [-X₂]

All Bounds

Timebounds

Overall timebound:24⋅X₁+144 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+5 {O(n)}
t₃: 1 {O(1)}
t₄: 10⋅X₁+60 {O(n)}
t₅: X₁+5 {O(n)}
t₆: X₁+5 {O(n)}
t₇: 10⋅X₁+60 {O(n)}
t₈: X₁+5 {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 24⋅X₁+144 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+5 {O(n)}
t₃: 1 {O(1)}
t₄: 10⋅X₁+60 {O(n)}
t₅: X₁+5 {O(n)}
t₆: X₁+5 {O(n)}
t₇: 10⋅X₁+60 {O(n)}
t₈: X₁+5 {O(n)}
t₉: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₁+9 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 0 {O(1)}
t₃, X₀: 3⋅X₁+9 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₂+10 {O(n)}
t₄, X₀: 4 {O(1)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 9 {O(1)}
t₅, X₀: 2⋅X₁+9 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 0 {O(1)}
t₆, X₀: 4 {O(1)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 10 {O(1)}
t₇, X₀: 4 {O(1)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 10 {O(1)}
t₈, X₀: 2⋅X₁+9 {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 10 {O(1)}
t₉, X₀: 3⋅X₁+9 {O(n)}
t₉, X₁: 2⋅X₁ {O(n)}
t₉, X₂: X₂+10 {O(n)}