KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_wcet0_bb3_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 1+X₃ ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₁ for location eval_wcet0_bb4_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₁ for location eval_wcet0_0

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_wcet0_bb2_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₁ for location eval_wcet0_1

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 1 ≤ X₁ for location eval_wcet0_bb1_in

Problem after Preprocessing

Start: eval_wcet0_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_wcet0_0, eval_wcet0_1, eval_wcet0_bb0_in, eval_wcet0_bb1_in, eval_wcet0_bb2_in, eval_wcet0_bb3_in, eval_wcet0_bb4_in, eval_wcet0_bb5_in, eval_wcet0_start, eval_wcet0_stop
Transitions:
t₅: eval_wcet0_0(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_1(nondef.0, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₆: eval_wcet0_1(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₇: eval_wcet0_1(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₁: eval_wcet0_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb1_in(X₀, X₄, 0, X₃, X₄) :|: 1 ≤ X₄
t₂: eval_wcet0_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₃: eval_wcet0_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_0(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₈: eval_wcet0_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, 0, X₄) :|: X₄ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₉: eval_wcet0_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, 1+X₂, X₄) :|: 2+X₂ ≤ X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₁₀: eval_wcet0_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, 0, X₄) :|: X₂+X₄ ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₄
t₁₁: eval_wcet0_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, X₂-1, X₄) :|: 2 ≤ X₂+X₄ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₄
t₁₂: eval_wcet0_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb1_in(X₀, X₁-1, X₃, X₃, X₄) :|: 2 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 1+X₃ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₁₃: eval_wcet0_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 1+X₃ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄
t₁₄: eval_wcet0_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_wcet0_start(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₃: eval_wcet0_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_0(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁-1]
• eval_wcet0_1: [X₁-1]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁-1]
• eval_wcet0_bb3_in: [X₁-1]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₅: eval_wcet0_0(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_1(nondef.0, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁-1]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁-1]
• eval_wcet0_bb3_in: [X₁-1]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₆: eval_wcet0_1(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁-1]
• eval_wcet0_bb3_in: [X₁]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₇: eval_wcet0_1(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁-1]
• eval_wcet0_bb3_in: [X₁-1]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₈: eval_wcet0_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, 0, X₄) :|: X₄ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁]
• eval_wcet0_bb3_in: [X₁]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₉: eval_wcet0_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, 1+X₂, X₄) :|: 2+X₂ ≤ X₄ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁]
• eval_wcet0_bb3_in: [X₁]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₁₀: eval_wcet0_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, 0, X₄) :|: X₂+X₄ ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁]
• eval_wcet0_bb3_in: [X₁]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₁₁: eval_wcet0_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb4_in(X₀, X₁, X₂, X₂-1, X₄) :|: 2 ≤ X₂+X₄ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_wcet0_0: [X₁]
• eval_wcet0_1: [X₁]
• eval_wcet0_bb1_in: [X₁]
• eval_wcet0_bb2_in: [X₁]
• eval_wcet0_bb3_in: [X₁]
• eval_wcet0_bb4_in: [X₁-1]

MPRF for transition t₁₂: eval_wcet0_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_wcet0_bb1_in(X₀, X₁-1, X₃, X₃, X₄) :|: 2 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 1+X₃ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ of depth 1:

new bound:

2⋅X₄+1 {O(n)}

MPRF:

• eval_wcet0_0: [1+2⋅X₁]
• eval_wcet0_1: [1+2⋅X₁]
• eval_wcet0_bb1_in: [1+2⋅X₁]
• eval_wcet0_bb2_in: [1+2⋅X₁]
• eval_wcet0_bb3_in: [2⋅X₁]
• eval_wcet0_bb4_in: [2⋅X₁]

All Bounds

Timebounds

Overall timebound:10⋅X₄+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₄ {O(n)}
t₅: X₄ {O(n)}
t₆: X₄ {O(n)}
t₇: X₄ {O(n)}
t₈: X₄ {O(n)}
t₉: X₄ {O(n)}
t₁₀: X₄ {O(n)}
t₁₁: X₄ {O(n)}
t₁₂: 2⋅X₄+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}

Costbounds

Overall costbound: 10⋅X₄+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₄ {O(n)}
t₅: X₄ {O(n)}
t₆: X₄ {O(n)}
t₇: X₄ {O(n)}
t₈: X₄ {O(n)}
t₉: X₄ {O(n)}
t₁₀: X₄ {O(n)}
t₁₁: X₄ {O(n)}
t₁₂: 2⋅X₄+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₁: X₄ {O(n)}
t₃, X₂: 2⋅X₄ {O(n)}
t₃, X₃: 4⋅X₄+X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₅, X₁: X₄ {O(n)}
t₅, X₂: 2⋅X₄ {O(n)}
t₅, X₃: 4⋅X₄+X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₁: X₄ {O(n)}
t₆, X₂: 2⋅X₄ {O(n)}
t₆, X₃: 4⋅X₄+X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₇, X₁: X₄ {O(n)}
t₇, X₂: 2⋅X₄ {O(n)}
t₇, X₃: 4⋅X₄+X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₁: X₄ {O(n)}
t₈, X₂: 2⋅X₄ {O(n)}
t₈, X₃: 0 {O(1)}
t₈, X₄: X₄ {O(n)}
t₉, X₁: X₄ {O(n)}
t₉, X₂: 2⋅X₄ {O(n)}
t₉, X₃: 2⋅X₄ {O(n)}
t₉, X₄: X₄ {O(n)}
t₁₀, X₁: X₄ {O(n)}
t₁₀, X₂: 2⋅X₄ {O(n)}
t₁₀, X₃: 0 {O(1)}
t₁₀, X₄: X₄ {O(n)}
t₁₁, X₁: X₄ {O(n)}
t₁₁, X₂: 2⋅X₄ {O(n)}
t₁₁, X₃: 2⋅X₄ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₂, X₁: X₄ {O(n)}
t₁₂, X₂: 2⋅X₄ {O(n)}
t₁₂, X₃: 4⋅X₄ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₁: 1 {O(1)}
t₁₃, X₂: 8⋅X₄ {O(n)}
t₁₃, X₃: 4⋅X₄ {O(n)}
t₁₃, X₄: 4⋅X₄ {O(n)}
t₁₄, X₁: X₁+1 {O(n)}
t₁₄, X₂: 8⋅X₄+X₂ {O(n)}
t₁₄, X₃: 4⋅X₄+X₃ {O(n)}
t₁₄, X₄: 5⋅X₄ {O(n)}