KoAT2 Proof Output

Initial Problem

Preprocessing

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_speed_pldi09_fig1_bb2_in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_speed_pldi09_fig1_bb1_in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb5_in

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb4_in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb3_in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_stop

Problem after Preprocessing

MPRF for transition t₂: eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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_speed_pldi09_fig1_bb1_in: [X₀-X₁]
• eval_speed_pldi09_fig1_bb2_in: [X₀-1-X₁]
• eval_speed_pldi09_fig1_bb3_in: [X₀-X₁]
• eval_speed_pldi09_fig1_bb4_in: [X₀-X₁]

MPRF for transition t₄: eval_speed_pldi09_fig1_bb2_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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_speed_pldi09_fig1_bb1_in: [X₀-X₁]
• eval_speed_pldi09_fig1_bb2_in: [X₀-X₁]
• eval_speed_pldi09_fig1_bb3_in: [X₀-X₁]
• eval_speed_pldi09_fig1_bb4_in: [X₀-X₁]

MPRF for transition t₃: eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb3_in(X₀, X₁, 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_speed_pldi09_fig1_bb1_in: [1+X₂]
• eval_speed_pldi09_fig1_bb2_in: [1+X₂]
• eval_speed_pldi09_fig1_bb3_in: [X₂]
• eval_speed_pldi09_fig1_bb4_in: [X₂]

MPRF for transition t₅: eval_speed_pldi09_fig1_bb3_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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_speed_pldi09_fig1_bb1_in: [1+X₂]
• eval_speed_pldi09_fig1_bb2_in: [1+X₂]
• eval_speed_pldi09_fig1_bb3_in: [1+X₂]
• eval_speed_pldi09_fig1_bb4_in: [X₂]

MPRF for transition t₇: eval_speed_pldi09_fig1_bb4_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀⋅X₀ {O(n^2)}

MPRF:

• eval_speed_pldi09_fig1_bb1_in: [X₂]
• eval_speed_pldi09_fig1_bb2_in: [X₂]
• eval_speed_pldi09_fig1_bb3_in: [X₂]
• eval_speed_pldi09_fig1_bb4_in: [X₂]

Cut unreachable locations [eval_speed_pldi09_fig1_bb3_in] from the program graph

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_speed_pldi09_fig1_bb2_in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₁ for location eval_speed_pldi09_fig1_bb1_in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb5_in

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb3_in_v1

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_stop

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb3_in_v2

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb4_in_v2

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb4_in_v1

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_pldi09_fig1_bb1_in_v1

Analysing control-flow refined program

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₄₇: eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂

MPRF for transition t₅₁: eval_speed_pldi09_fig1_bb1_in_v1(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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_speed_pldi09_fig1_bb1_in_v1: [1+X₂]
• eval_speed_pldi09_fig1_bb3_in_v2: [X₂]
• eval_speed_pldi09_fig1_bb4_in_v2: [X₂]

MPRF for transition t₅₃: eval_speed_pldi09_fig1_bb3_in_v2(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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_speed_pldi09_fig1_bb1_in_v1: [1+X₂]
• eval_speed_pldi09_fig1_bb3_in_v2: [1+X₂]
• eval_speed_pldi09_fig1_bb4_in_v2: [X₂]

MPRF for transition t₅₄: eval_speed_pldi09_fig1_bb4_in_v2(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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_speed_pldi09_fig1_bb1_in_v1: [X₂]
• eval_speed_pldi09_fig1_bb3_in_v2: [X₂]
• eval_speed_pldi09_fig1_bb4_in_v2: [X₂]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

cfr-program:

Start: eval_speed_pldi09_fig1_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_speed_pldi09_fig1_bb0_in, eval_speed_pldi09_fig1_bb1_in, eval_speed_pldi09_fig1_bb1_in_v1, eval_speed_pldi09_fig1_bb2_in, eval_speed_pldi09_fig1_bb3_in_v1, eval_speed_pldi09_fig1_bb3_in_v2, eval_speed_pldi09_fig1_bb4_in_v1, eval_speed_pldi09_fig1_bb4_in_v2, eval_speed_pldi09_fig1_bb5_in, eval_speed_pldi09_fig1_start, eval_speed_pldi09_fig1_stop
Transitions:
t₁: eval_speed_pldi09_fig1_bb0_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb1_in(X₀, 0, 0)
t₂: eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₄₇: eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb2_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₄₆: eval_speed_pldi09_fig1_bb1_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb3_in_v1(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₅₁: eval_speed_pldi09_fig1_bb1_in_v1(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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
t₄: eval_speed_pldi09_fig1_bb2_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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
t₄₉: eval_speed_pldi09_fig1_bb3_in_v1(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb4_in_v1(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₄₈: eval_speed_pldi09_fig1_bb3_in_v1(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb5_in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₅₃: eval_speed_pldi09_fig1_bb3_in_v2(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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
t₅₂: eval_speed_pldi09_fig1_bb3_in_v2(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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
t₅₀: eval_speed_pldi09_fig1_bb4_in_v1(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb1_in_v1(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₅₄: eval_speed_pldi09_fig1_bb4_in_v2(X₀, X₁, X₂) → eval_speed_pldi09_fig1_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
t₈: eval_speed_pldi09_fig1_bb5_in(X₀, X₁, X₂) → eval_speed_pldi09_fig1_stop(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₀: eval_speed_pldi09_fig1_start(X₀, X₁, X₂) → eval_speed_pldi09_fig1_bb0_in(X₀, X₁, X₂)

All Bounds

Timebounds

Overall timebound:6⋅X₀+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₄: X₀ {O(n)}
t₈: 1 {O(1)}
t₄₆: 1 {O(1)}
t₄₇: X₀+1 {O(n)}
t₄₈: 1 {O(1)}
t₄₉: 1 {O(1)}
t₅₀: 1 {O(1)}
t₅₁: X₀+1 {O(n)}
t₅₂: 1 {O(1)}
t₅₃: X₀+1 {O(n)}
t₅₄: X₀ {O(n)}

Costbounds

Overall costbound: 6⋅X₀+11 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₄: X₀ {O(n)}
t₈: 1 {O(1)}
t₄₆: 1 {O(1)}
t₄₇: X₀+1 {O(n)}
t₄₈: 1 {O(1)}
t₄₉: 1 {O(1)}
t₅₀: 1 {O(1)}
t₅₁: X₀+1 {O(n)}
t₅₂: 1 {O(1)}
t₅₃: X₀+1 {O(n)}
t₅₄: 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: 0 {O(1)}
t₁, X₂: 0 {O(1)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: X₀ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: X₀ {O(n)}
t₈, X₀: 4⋅X₀ {O(n)}
t₈, X₁: X₀ {O(n)}
t₈, X₂: 0 {O(1)}
t₄₆, X₀: 2⋅X₀ {O(n)}
t₄₆, X₁: X₀ {O(n)}
t₄₆, X₂: X₀ {O(n)}
t₄₇, X₀: X₀ {O(n)}
t₄₇, X₁: X₀ {O(n)}
t₄₇, X₂: X₀ {O(n)}
t₄₈, X₀: 2⋅X₀ {O(n)}
t₄₈, X₁: 0 {O(1)}
t₄₈, X₂: 0 {O(1)}
t₄₉, X₀: 2⋅X₀ {O(n)}
t₄₉, X₁: X₀ {O(n)}
t₄₉, X₂: X₀ {O(n)}
t₅₀, X₀: 2⋅X₀ {O(n)}
t₅₀, X₁: X₀ {O(n)}
t₅₀, X₂: X₀ {O(n)}
t₅₁, X₀: 2⋅X₀ {O(n)}
t₅₁, X₁: X₀ {O(n)}
t₅₁, X₂: X₀ {O(n)}
t₅₂, X₀: 2⋅X₀ {O(n)}
t₅₂, X₁: X₀ {O(n)}
t₅₂, X₂: 0 {O(1)}
t₅₃, X₀: 2⋅X₀ {O(n)}
t₅₃, X₁: X₀ {O(n)}
t₅₃, X₂: X₀ {O(n)}
t₅₄, X₀: 2⋅X₀ {O(n)}
t₅₄, X₁: X₀ {O(n)}
t₅₄, X₂: X₀ {O(n)}