KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, 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_bb6_in, eval_start_bb7_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅)
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀
t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: X₀ ≤ 0
t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀-1, 2+X₁, X₂, X₃, X₄, X₅)
t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂
t₁₁: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb5_in(X₀, X₁, X₂, X₂, X₄, X₅) :|: X₂ ≤ 0
t₁₂: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅)
t₁₃: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₃
t₁₄: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb7_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ 0
t₁₅: eval_start_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb5_in(X₀, X₁, X₂, X₃-1, X₄, X₅)
t₁₆: eval_start_bb7_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₀+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_start_bb7_in

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ for location eval_start_bb1_in

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₀+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_start_stop

Found invariant 1 ≤ 0 for location eval_start_bb6_in

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₀+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ 0 for location eval_start_bb5_in

Found invariant X₅ ≤ X₁ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1 ≤ X₀ for location eval_start_bb2_in

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 0 for location eval_start_bb3_in

Found invariant X₅ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_start_bb4_in

Cut unsatisfiable transition [t₁₃: eval_start_bb5_in→eval_start_bb6_in; t₁₅: eval_start_bb6_in→eval_start_bb5_in]

Cut unreachable locations [eval_start_bb6_in] from the program graph

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, 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_bb7_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅)
t₃: eval_start_1(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅)
t₄: eval_start_2(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅)
t₅: eval_start_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅)
t₆: eval_start_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₁: eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_0(X₀, X₁, X₂, X₃, X₄, X₅)
t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₈: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀-1, 2+X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁
t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₅ ≤ X₁
t₁₁: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb5_in(X₀, X₁, X₂, X₂, X₄, X₅) :|: X₂ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₅ ≤ X₁
t₁₂: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₅ ≤ X₁
t₁₄: eval_start_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb7_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀+X₂ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₅ ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂+X₃ ≤ 0
t₁₆: eval_start_bb7_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀+X₂ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₅ ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂+X₃ ≤ 0 ∧ X₃ ≤ 0
t₀: eval_start_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₇: eval_start_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

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

MPRF for transition t₉: eval_start_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb1_in(X₀-1, 2+X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₅ ≤ X₁ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₀]

MPRF for transition t₁₀: eval_start_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₅ ≤ X₁ of depth 1:

new bound:

2⋅X₄+2⋅X₅ {O(n)}

MPRF:

• eval_start_bb3_in: [X₂]
• eval_start_bb4_in: [X₂-1]

MPRF for transition t₁₂: eval_start_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_start_bb3_in(X₀, X₁, X₂-1, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₅ ≤ X₁ of depth 1:

new bound:

2⋅X₄+2⋅X₅ {O(n)}

MPRF:

• eval_start_bb3_in: [X₂]
• eval_start_bb4_in: [X₂]

All Bounds

Timebounds

Overall timebound:4⋅X₅+6⋅X₄+11 {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: 1 {O(1)}
t₇: X₄ {O(n)}
t₈: 1 {O(1)}
t₉: X₄ {O(n)}
t₁₀: 2⋅X₄+2⋅X₅ {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 2⋅X₄+2⋅X₅ {O(n)}
t₁₄: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₅+6⋅X₄+11 {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: 1 {O(1)}
t₇: X₄ {O(n)}
t₈: 1 {O(1)}
t₉: X₄ {O(n)}
t₁₀: 2⋅X₄+2⋅X₅ {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 2⋅X₄+2⋅X₅ {O(n)}
t₁₄: 1 {O(1)}
t₁₆: 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)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {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)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₀: X₄ {O(n)}
t₆, X₁: X₅ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: 2⋅X₄+X₅ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: 2⋅X₄ {O(n)}
t₈, X₁: 2⋅X₄+2⋅X₅ {O(n)}
t₈, X₂: 2⋅X₄+2⋅X₅ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₈, X₄: 2⋅X₄ {O(n)}
t₈, X₅: 2⋅X₅ {O(n)}
t₉, X₀: X₄ {O(n)}
t₉, X₁: 2⋅X₄+X₅ {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₀: 2⋅X₄ {O(n)}
t₁₀, X₁: 2⋅X₄+2⋅X₅ {O(n)}
t₁₀, X₂: 2⋅X₄+2⋅X₅ {O(n)}
t₁₀, X₃: 2⋅X₃ {O(n)}
t₁₀, X₄: 2⋅X₄ {O(n)}
t₁₀, X₅: 2⋅X₅ {O(n)}
t₁₁, X₀: 4⋅X₄ {O(n)}
t₁₁, X₁: 4⋅X₄+4⋅X₅ {O(n)}
t₁₁, X₂: 4⋅X₄+4⋅X₅ {O(n)}
t₁₁, X₃: 4⋅X₄+4⋅X₅ {O(n)}
t₁₁, X₄: 4⋅X₄ {O(n)}
t₁₁, X₅: 4⋅X₅ {O(n)}
t₁₂, X₀: 2⋅X₄ {O(n)}
t₁₂, X₁: 2⋅X₄+2⋅X₅ {O(n)}
t₁₂, X₂: 2⋅X₄+2⋅X₅ {O(n)}
t₁₂, X₃: 2⋅X₃ {O(n)}
t₁₂, X₄: 2⋅X₄ {O(n)}
t₁₂, X₅: 2⋅X₅ {O(n)}
t₁₄, X₀: 4⋅X₄ {O(n)}
t₁₄, X₁: 4⋅X₄+4⋅X₅ {O(n)}
t₁₄, X₂: 4⋅X₄+4⋅X₅ {O(n)}
t₁₄, X₃: 4⋅X₄+4⋅X₅ {O(n)}
t₁₄, X₄: 4⋅X₄ {O(n)}
t₁₄, X₅: 4⋅X₅ {O(n)}
t₁₆, X₀: 4⋅X₄ {O(n)}
t₁₆, X₁: 4⋅X₄+4⋅X₅ {O(n)}
t₁₆, X₂: 4⋅X₄+4⋅X₅ {O(n)}
t₁₆, X₃: 4⋅X₄+4⋅X₅ {O(n)}
t₁₆, X₄: 4⋅X₄ {O(n)}
t₁₆, X₅: 4⋅X₅ {O(n)}