KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4 for location eval_wcet2_7

Found invariant X₂ ≤ X₀ for location eval_wcet2_bb1_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 for location eval_wcet2_bb4_in

Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location eval_wcet2_bb5_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 for location eval_wcet2_bb2_in

Found invariant X₃ ≤ 9 ∧ X₂+X₃ ≤ 13 ∧ X₃ ≤ 6+X₀ ∧ X₀+X₃ ≤ 13 ∧ 0 ≤ X₃ ∧ X₂ ≤ 4+X₃ ∧ 3 ≤ X₀+X₃ ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location eval_wcet2_bb3_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 4 ∧ 1+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 9 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₁ ≤ 1+X₀ ∧ X₀+X₁ ≤ 9 ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 4 for location eval_wcet2_8

Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location eval_wcet2_stop

Problem after Preprocessing

Start: eval_wcet2_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_wcet2_0, eval_wcet2_1, eval_wcet2_2, eval_wcet2_3, eval_wcet2_7, eval_wcet2_8, 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:
t₂: eval_wcet2_0(X₀, X₁, X₂, X₃) → eval_wcet2_1(X₀, X₁, X₂, X₃)
t₃: eval_wcet2_1(X₀, X₁, X₂, X₃) → eval_wcet2_2(X₀, X₁, X₂, X₃)
t₄: eval_wcet2_2(X₀, X₁, X₂, X₃) → eval_wcet2_3(X₀, X₁, X₂, X₃)
t₅: eval_wcet2_3(X₀, X₁, X₂, X₃) → eval_wcet2_bb1_in(X₂, X₁, X₂, X₃)
t₁₃: eval_wcet2_7(X₀, X₁, X₂, X₃) → eval_wcet2_8(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 9 ∧ X₁+X₂ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 1+X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₁₄: eval_wcet2_8(X₀, X₁, X₂, X₃) → eval_wcet2_bb1_in(X₁, X₁, X₂, X₃) :|: X₀+X₁ ≤ 9 ∧ X₁+X₂ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 1+X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₁: eval_wcet2_bb0_in(X₀, X₁, X₂, X₃) → eval_wcet2_0(X₀, X₁, X₂, X₃)
t₆: eval_wcet2_bb1_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb2_in(X₀, X₁, X₂, 0) :|: X₀ ≤ 4 ∧ X₂ ≤ X₀
t₇: eval_wcet2_bb1_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb5_in(X₀, X₁, X₂, X₃) :|: 5 ≤ X₀ ∧ X₂ ≤ X₀
t₈: eval_wcet2_bb2_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ 9 ∧ 3 ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₉: eval_wcet2_bb2_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2 ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₁₀: eval_wcet2_bb2_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb4_in(X₀, X₁, X₂, X₃) :|: 10 ≤ X₃ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₁₁: eval_wcet2_bb3_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb2_in(X₀, X₁, X₂, 1+X₃) :|: X₀+X₃ ≤ 13 ∧ X₂+X₃ ≤ 13 ∧ X₃ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₃ ≤ 6+X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₁₂: eval_wcet2_bb4_in(X₀, X₁, X₂, X₃) → eval_wcet2_7(X₀, 1+X₀, X₂, X₃) :|: X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃
t₁₅: eval_wcet2_bb5_in(X₀, X₁, X₂, X₃) → eval_wcet2_stop(X₀, X₁, X₂, X₃) :|: 5 ≤ X₀ ∧ X₂ ≤ X₀
t₀: eval_wcet2_start(X₀, X₁, X₂, X₃) → eval_wcet2_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₆: eval_wcet2_bb1_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb2_in(X₀, X₁, X₂, 0) :|: X₀ ≤ 4 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_wcet2_7: [5-X₁]
• eval_wcet2_8: [5-X₁]
• 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 t₉: eval_wcet2_bb2_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2 ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+3 {O(n)}

MPRF:

• eval_wcet2_7: [3-X₁]
• eval_wcet2_8: [3-X₁]
• eval_wcet2_bb1_in: [3-X₀]
• eval_wcet2_bb2_in: [3-X₀]
• eval_wcet2_bb3_in: [3-X₀]
• eval_wcet2_bb4_in: [2-X₀]

MPRF for transition t₁₀: eval_wcet2_bb2_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb4_in(X₀, X₁, X₂, X₃) :|: 10 ≤ X₃ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_wcet2_7: [4-X₀]
• eval_wcet2_8: [5-X₁]
• 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 t₁₂: eval_wcet2_bb4_in(X₀, X₁, X₂, X₃) → eval_wcet2_7(X₀, 1+X₀, X₂, X₃) :|: X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_wcet2_7: [5-X₁]
• eval_wcet2_8: [5-X₁]
• 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 t₁₃: eval_wcet2_7(X₀, X₁, X₂, X₃) → eval_wcet2_8(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 9 ∧ X₁+X₂ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 1+X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_wcet2_7: [5-X₀]
• eval_wcet2_8: [4-X₀]
• 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 t₁₄: eval_wcet2_8(X₀, X₁, X₂, X₃) → eval_wcet2_bb1_in(X₁, X₁, X₂, X₃) :|: X₀+X₁ ≤ 9 ∧ X₁+X₂ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 5 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₁ ≤ 3+X₃ ∧ X₀ ≤ 2+X₃ ∧ X₂ ≤ 2+X₃ ∧ X₁ ≤ 1+X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_wcet2_7: [5-X₀]
• eval_wcet2_8: [5-X₀]
• 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 t₈: eval_wcet2_bb2_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ 9 ∧ 3 ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

10⋅X₂+60 {O(n)}

MPRF:

• eval_wcet2_7: [-X₃]
• eval_wcet2_8: [-X₃]
• eval_wcet2_bb1_in: [10]
• eval_wcet2_bb2_in: [10-X₃]
• eval_wcet2_bb3_in: [9-X₃]
• eval_wcet2_bb4_in: [-X₃]

MPRF for transition t₁₁: eval_wcet2_bb3_in(X₀, X₁, X₂, X₃) → eval_wcet2_bb2_in(X₀, X₁, X₂, 1+X₃) :|: X₀+X₃ ≤ 13 ∧ X₂+X₃ ≤ 13 ∧ X₃ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₃ ≤ 6+X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₃ ∧ X₂ ≤ 4 ∧ X₂ ≤ 4+X₃ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₃ of depth 1:

new bound:

10⋅X₂+60 {O(n)}

MPRF:

• eval_wcet2_7: [-X₃]
• eval_wcet2_8: [-X₃]
• eval_wcet2_bb1_in: [10]
• eval_wcet2_bb2_in: [10-X₃]
• eval_wcet2_bb3_in: [10-X₃]
• eval_wcet2_bb4_in: [-X₃]

All Bounds

Timebounds

Overall timebound:26⋅X₂+156 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: X₂+5 {O(n)}
t₇: 1 {O(1)}
t₈: 10⋅X₂+60 {O(n)}
t₉: X₂+3 {O(n)}
t₁₀: X₂+5 {O(n)}
t₁₁: 10⋅X₂+60 {O(n)}
t₁₂: X₂+5 {O(n)}
t₁₃: X₂+5 {O(n)}
t₁₄: X₂+5 {O(n)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 26⋅X₂+156 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: X₂+5 {O(n)}
t₇: 1 {O(1)}
t₈: 10⋅X₂+60 {O(n)}
t₉: X₂+3 {O(n)}
t₁₀: X₂+5 {O(n)}
t₁₁: 10⋅X₂+60 {O(n)}
t₁₂: X₂+5 {O(n)}
t₁₃: X₂+5 {O(n)}
t₁₄: X₂+5 {O(n)}
t₁₅: 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: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 2⋅X₂+9 {O(n)}
t₆, X₁: 2⋅X₂+X₁+15 {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: 0 {O(1)}
t₇, X₀: 3⋅X₂+9 {O(n)}
t₇, X₁: 2⋅X₂+X₁+15 {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: X₃+10 {O(n)}
t₈, X₀: 4 {O(1)}
t₈, X₁: 2⋅X₂+X₁+15 {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: 9 {O(1)}
t₉, X₀: 2⋅X₂+9 {O(n)}
t₉, X₁: 2⋅X₂+X₁+15 {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: 0 {O(1)}
t₁₀, X₀: 4 {O(1)}
t₁₀, X₁: 2⋅X₂+X₁+15 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: 10 {O(1)}
t₁₁, X₀: 4 {O(1)}
t₁₁, X₁: 2⋅X₂+X₁+15 {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: 10 {O(1)}
t₁₂, X₀: 2⋅X₂+9 {O(n)}
t₁₂, X₁: 2⋅X₂+15 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: 10 {O(1)}
t₁₃, X₀: 2⋅X₂+9 {O(n)}
t₁₃, X₁: 2⋅X₂+15 {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: 10 {O(1)}
t₁₄, X₀: 2⋅X₂+9 {O(n)}
t₁₄, X₁: 2⋅X₂+15 {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: 10 {O(1)}
t₁₅, X₀: 3⋅X₂+9 {O(n)}
t₁₅, X₁: 2⋅X₂+X₁+15 {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₅, X₃: X₃+10 {O(n)}