KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_bb4_in, eval_start_bb5_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃) → eval_start_1(X₀, X₁, X₂, X₃)
t₃: eval_start_1(X₀, X₁, X₂, X₃) → eval_start_2(X₀, X₁, X₂, X₃)
t₄: eval_start_2(X₀, X₁, X₂, X₃) → eval_start_3(X₀, X₁, X₂, X₃)
t₅: eval_start_3(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₂, X₁, X₂, X₃)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₀, X₂, X₃) :|: X₃ ≤ X₀
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(1+X₀, X₁, X₂, X₃)
t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_bb4_in(X₀, X₁, X₂, X₃) :|: 3 ≤ X₁
t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_bb5_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2
t₁₁: eval_start_bb4_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₁-3, X₂, X₃)
t₁₂: eval_start_bb5_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃)
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₂ ≤ X₀ for location eval_start_bb1_in

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

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 2 ∧ X₁ ≤ X₀ for location eval_start_bb5_in

Found invariant 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location eval_start_bb2_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location eval_start_bb3_in

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location eval_start_bb4_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_bb4_in, eval_start_bb5_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃) → eval_start_1(X₀, X₁, X₂, X₃)
t₃: eval_start_1(X₀, X₁, X₂, X₃) → eval_start_2(X₀, X₁, X₂, X₃)
t₄: eval_start_2(X₀, X₁, X₂, X₃) → eval_start_3(X₀, X₁, X₂, X₃)
t₅: eval_start_3(X₀, X₁, X₂, X₃) → eval_start_bb1_in(X₂, X₁, X₂, X₃)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃) → eval_start_0(X₀, X₁, X₂, X₃)
t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₀, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀
t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀
t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_bb4_in(X₀, X₁, X₂, X₃) :|: 3 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_bb5_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2 ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₁₁: eval_start_bb4_in(X₀, X₁, X₂, X₃) → eval_start_bb3_in(X₀, X₁-3, X₂, X₃) :|: 3 ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₁₂: eval_start_bb5_in(X₀, X₁, X₂, X₃) → eval_start_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2 ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀
t₀: eval_start_start(X₀, X₁, X₂, X₃) → eval_start_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₆: eval_start_bb1_in(X₀, X₁, X₂, X₃) → eval_start_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_start_bb1_in: [X₃-X₀]
• eval_start_bb2_in: [X₃-1-X₀]

MPRF for transition t₈: eval_start_bb2_in(X₀, X₁, X₂, X₃) → eval_start_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_start_bb1_in: [X₃-X₀]
• eval_start_bb2_in: [X₃-X₀]

MPRF for transition t₉: eval_start_bb3_in(X₀, X₁, X₂, X₃) → eval_start_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_start_bb3_in: [X₁-2]
• eval_start_bb4_in: [X₁-5]

MPRF for transition t₁₁: eval_start_bb4_in(X₀, X₁, X₂, X₃) → eval_start_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_start_bb3_in: [X₁-2]
• eval_start_bb4_in: [X₁-2]

All Bounds

Timebounds

Overall timebound:4⋅X₃+8⋅X₂+13 {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+X₃ {O(n)}
t₇: 1 {O(1)}
t₈: X₂+X₃ {O(n)}
t₉: 3⋅X₂+X₃+2 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 3⋅X₂+X₃+2 {O(n)}
t₁₂: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+8⋅X₂+13 {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+X₃ {O(n)}
t₇: 1 {O(1)}
t₈: X₂+X₃ {O(n)}
t₉: 3⋅X₂+X₃+2 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 3⋅X₂+X₃+2 {O(n)}
t₁₂: 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+X₃ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: 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⋅X₂+X₃ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: 3⋅X₂+X₃ {O(n)}
t₉, X₁: 3⋅X₂+X₃ {O(n)}
t₉, X₂: 2⋅X₂ {O(n)}
t₉, X₃: 2⋅X₃ {O(n)}
t₁₀, X₀: 2⋅X₃+6⋅X₂ {O(n)}
t₁₀, X₁: 2⋅X₃+6⋅X₂ {O(n)}
t₁₀, X₂: 4⋅X₂ {O(n)}
t₁₀, X₃: 4⋅X₃ {O(n)}
t₁₁, X₀: 3⋅X₂+X₃ {O(n)}
t₁₁, X₁: 3⋅X₂+X₃ {O(n)}
t₁₁, X₂: 2⋅X₂ {O(n)}
t₁₁, X₃: 2⋅X₃ {O(n)}
t₁₂, X₀: 2⋅X₃+6⋅X₂ {O(n)}
t₁₂, X₁: 2⋅X₃+6⋅X₂ {O(n)}
t₁₂, X₂: 4⋅X₂ {O(n)}
t₁₂, X₃: 4⋅X₃ {O(n)}