KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_start_bb1_in

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

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

Found invariant X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_bb2_in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb3_in

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

Problem after Preprocessing

MPRF for transition t₉: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀-X₁]
• eval_start_bb2_in: [X₀-1-X₁]
• eval_start_bb3_in: [X₀-X₁]
• eval_start_bb4_in: [X₀-X₁]

MPRF for transition t₁₁: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 1+X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀-X₁]
• eval_start_bb2_in: [X₀-X₁]
• eval_start_bb3_in: [X₀-X₁]
• eval_start_bb4_in: [X₀-X₁]

MPRF for transition t₁₀: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb3_in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀⋅X₀+2⋅X₀+2 {O(n^2)}

MPRF:

• eval_start_bb1_in: [2+X₂]
• eval_start_bb2_in: [2+X₂]
• eval_start_bb3_in: [1+X₂]
• eval_start_bb4_in: [1+X₂]

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

new bound:

X₀⋅X₀+X₀+1 {O(n^2)}

MPRF:

• eval_start_bb1_in: [1+X₂]
• eval_start_bb2_in: [1+X₂]
• eval_start_bb3_in: [1+X₂]
• eval_start_bb4_in: [X₂]

MPRF for transition t₁₄: eval_start_bb4_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀⋅X₀+X₀+1 {O(n^2)}

MPRF:

• eval_start_bb1_in: [1+X₂]
• eval_start_bb2_in: [1+X₂]
• eval_start_bb3_in: [1+X₂]
• eval_start_bb4_in: [1+X₂]

Cut unreachable locations [eval_start_bb3_in] from the program graph

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb4_in_v1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₁ for location eval_start_bb1_in

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb4_in_v2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb3_in_v2

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

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

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb1_in_v1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb3_in_v1

Found invariant X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_bb2_in

Analysing control-flow refined program

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₈₂: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂

MPRF for transition t₈₆: eval_start_bb1_in_v1(X₀, X₁, X₂) → eval_start_bb3_in_v2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_start_bb1_in_v1: [1+X₂]
• eval_start_bb3_in_v2: [X₂]
• eval_start_bb4_in_v2: [X₂]

MPRF for transition t₈₈: eval_start_bb3_in_v2(X₀, X₁, X₂) → eval_start_bb4_in_v2(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_start_bb1_in_v1: [1+X₂]
• eval_start_bb3_in_v2: [1+X₂]
• eval_start_bb4_in_v2: [X₂]

MPRF for transition t₈₉: eval_start_bb4_in_v2(X₀, X₁, X₂) → eval_start_bb1_in_v1(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_start_bb1_in_v1: [X₂]
• eval_start_bb3_in_v2: [X₂]
• eval_start_bb4_in_v2: [X₂]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

cfr-program:

Start: eval_start_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb1_in_v1, eval_start_bb2_in, eval_start_bb3_in_v1, eval_start_bb3_in_v2, eval_start_bb4_in_v1, eval_start_bb4_in_v2, eval_start_bb5_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂) → eval_start_1(X₀, X₁, X₂)
t₃: eval_start_1(X₀, X₁, X₂) → eval_start_2(X₀, X₁, X₂)
t₄: eval_start_2(X₀, X₁, X₂) → eval_start_3(X₀, X₁, X₂)
t₅: eval_start_3(X₀, X₁, X₂) → eval_start_4(X₀, X₁, X₂)
t₆: eval_start_4(X₀, X₁, X₂) → eval_start_5(X₀, X₁, X₂)
t₇: eval_start_5(X₀, X₁, X₂) → eval_start_6(X₀, X₁, X₂)
t₈: eval_start_6(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 0, 0)
t₁: eval_start_bb0_in(X₀, X₁, X₂) → eval_start_0(X₀, X₁, X₂)
t₉: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₈₂: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₈₁: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb3_in_v1(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₈₆: eval_start_bb1_in_v1(X₀, X₁, X₂) → eval_start_bb3_in_v2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₁₁: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 1+X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₈₄: eval_start_bb3_in_v1(X₀, X₁, X₂) → eval_start_bb4_in_v1(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₈₃: eval_start_bb3_in_v1(X₀, X₁, X₂) → eval_start_bb5_in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₈₈: eval_start_bb3_in_v2(X₀, X₁, X₂) → eval_start_bb4_in_v2(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₈₇: eval_start_bb3_in_v2(X₀, X₁, X₂) → eval_start_bb5_in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂
t₈₅: eval_start_bb4_in_v1(X₀, X₁, X₂) → eval_start_bb1_in_v1(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₈₉: eval_start_bb4_in_v2(X₀, X₁, X₂) → eval_start_bb1_in_v1(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁
t₁₅: eval_start_bb5_in(X₀, X₁, X₂) → eval_start_stop(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₀: eval_start_start(X₀, X₁, X₂) → eval_start_bb0_in(X₀, X₁, X₂)

All Bounds

Timebounds

Overall timebound:6⋅X₀+18 {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: 1 {O(1)}
t₈: 1 {O(1)}
t₉: X₀ {O(n)}
t₁₁: X₀ {O(n)}
t₁₅: 1 {O(1)}
t₈₁: 1 {O(1)}
t₈₂: X₀+1 {O(n)}
t₈₃: 1 {O(1)}
t₈₄: 1 {O(1)}
t₈₅: 1 {O(1)}
t₈₆: X₀+1 {O(n)}
t₈₇: 1 {O(1)}
t₈₈: X₀+1 {O(n)}
t₈₉: X₀ {O(n)}

Costbounds

Overall costbound: 6⋅X₀+18 {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: 1 {O(1)}
t₈: 1 {O(1)}
t₉: X₀ {O(n)}
t₁₁: X₀ {O(n)}
t₁₅: 1 {O(1)}
t₈₁: 1 {O(1)}
t₈₂: X₀+1 {O(n)}
t₈₃: 1 {O(1)}
t₈₄: 1 {O(1)}
t₈₅: 1 {O(1)}
t₈₆: X₀+1 {O(n)}
t₈₇: 1 {O(1)}
t₈₈: X₀+1 {O(n)}
t₈₉: X₀ {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: 0 {O(1)}
t₈, X₂: 0 {O(1)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀ {O(n)}
t₉, X₂: X₀ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₀ {O(n)}
t₁₁, X₂: X₀ {O(n)}
t₁₅, X₀: 4⋅X₀ {O(n)}
t₁₅, X₁: X₀ {O(n)}
t₁₅, X₂: 0 {O(1)}
t₈₁, X₀: 2⋅X₀ {O(n)}
t₈₁, X₁: X₀ {O(n)}
t₈₁, X₂: X₀ {O(n)}
t₈₂, X₀: X₀ {O(n)}
t₈₂, X₁: X₀ {O(n)}
t₈₂, X₂: X₀ {O(n)}
t₈₃, X₀: 2⋅X₀ {O(n)}
t₈₃, X₁: 0 {O(1)}
t₈₃, X₂: 0 {O(1)}
t₈₄, X₀: 2⋅X₀ {O(n)}
t₈₄, X₁: X₀ {O(n)}
t₈₄, X₂: X₀ {O(n)}
t₈₅, X₀: 2⋅X₀ {O(n)}
t₈₅, X₁: X₀ {O(n)}
t₈₅, X₂: X₀ {O(n)}
t₈₆, X₀: 2⋅X₀ {O(n)}
t₈₆, X₁: X₀ {O(n)}
t₈₆, X₂: X₀ {O(n)}
t₈₇, X₀: 2⋅X₀ {O(n)}
t₈₇, X₁: X₀ {O(n)}
t₈₇, X₂: 0 {O(1)}
t₈₈, X₀: 2⋅X₀ {O(n)}
t₈₈, X₁: X₀ {O(n)}
t₈₈, X₂: X₀ {O(n)}
t₈₉, X₀: 2⋅X₀ {O(n)}
t₈₉, X₁: X₀ {O(n)}
t₈₉, X₂: X₀ {O(n)}