BT008A · Bertrand theorem

beta_sum_exists

Stable checked-use theorem · independently kernel verified

Every decoded beta prefix has a relational finite sum.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

∀ b. ∀ c. ∀ l. ∃ n. Sum(b,c,l,n)

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0015 Sum

1 occurrences

In local proof propositions

PD0002 Lt PD0013 BetaAt

6 occurrences

Exact expanded native-PA statement
forall b c l. exists n. (exists ff_u_x ff_v_x. ((((exists ff_h_x_start. ff_h_x_start + S (0) = S ((S (0)) * ff_v_x)) /\ exists ff_q_x_start. ff_u_x = ff_q_x_start * S ((S (0)) * ff_v_x) + (0))) /\ ((((exists ff_h_x_terminal. ff_h_x_terminal + S (n) = S ((S (l)) * ff_v_x)) /\ exists ff_q_x_terminal. ff_u_x = ff_q_x_terminal * S ((S (l)) * ff_v_x) + (n))) /\ forall ff_i_x. (exists ff_lt_x_bound. ff_lt_x_bound + S ff_i_x = l) -> exists ff_a_x ff_r_x ff_s_x. ((((exists ff_h_x_summand. ff_h_x_summand + S (ff_a_x) = S ((S (ff_i_x)) * c)) /\ exists ff_q_x_summand. b = ff_q_x_summand * S ((S (ff_i_x)) * c) + (ff_a_x))) /\ ((((exists ff_h_x_partial. ff_h_x_partial + S (ff_r_x) = S ((S (ff_i_x)) * ff_v_x)) /\ exists ff_q_x_partial. ff_u_x = ff_q_x_partial * S ((S (ff_i_x)) * ff_v_x) + (ff_r_x))) /\ ((((exists ff_h_x_successor. ff_h_x_successor + S (ff_s_x) = S ((S (S ff_i_x)) * ff_v_x)) /\ exists ff_q_x_successor. ff_u_x = ff_q_x_successor * S ((S (S ff_i_x)) * ff_v_x) + (ff_s_x))) /\ ff_s_x = ff_r_x + ff_a_x))))))

Proof neighborhood

Direct theorem prerequisites

BT0089 beta_prefix_sum_trace_exists BT0041 beta_at_exists

Direct theorem dependents

BT008L bit_count_exists BT00S2 prime_legendre_sum_exists BT00ST legendre_sum_zero_extended_prefix BT00XR legendre_sum_extended_prefix_exists

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

25 script commands · 10 reading checkpoints · 2 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
beta prefix sum trace existsbeta at exists
01Fix variables and assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
02Establish htraceL4–8

Establish this local claim before using it. It is not an additional assumption.

  1. L4
    have htrace : ∃ fs_u_exists_trace. ∃ fs_v_exists_trace. BetaAt(fs_u_exists_trace,fs_v_exists_trace,0,0) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. BetaAt(b,c,x,y) ∧ (BetaAt(fs_u_exists_trace,fs_v_exists_trace,x,z) ∧ (BetaAt(fs_u_exists_trace,fs_v_exists_trace,S x,n) ∧ n = z + y)))Definitions: BetaAt(fs_u_exists_trace,fs_v_exists_trace,0,0)Lt(x,l)BetaAt(b,c,x,y)BetaAt(fs_u_exists_trace,fs_v_exists_trace,x,z)BetaAt(fs_u_exists_trace,fs_v_exists_trace,S x,n)Original native command in the exact edition
  2. L5
    specialize beta_prefix_sum_trace_exists b
  3. L6
    specialize beta_prefix_sum_trace_exists c
  4. L7
    specialize beta_prefix_sum_trace_exists l
  5. L8
    exact beta_prefix_sum_trace_exists
03Separate the logical casesL9–11

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases htrace
  2. L10
    cases htrace_witness
  3. L11
    cases htrace_witness_witness
04Establish hterminalL12–16

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have hterminal : ∃ n. BetaAt(x,x1,l,n)Definitions: BetaAt(x,x1,l,n)Original native command in the exact edition
  2. L13
    specialize beta_at_exists x
  3. L14
    specialize beta_at_exists x1
  4. L15
    specialize beta_at_exists l
  5. L16
    exact beta_at_exists
05Separate the logical casesL17–17

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L17
    cases hterminal
06Construct an explicit witnessL18–20

Supply the displayed value, then prove that it has the required property.

  1. L18
    exists x2
  2. L19
    exists x
  3. L20
    exists x1
07Separate the logical casesL21–21

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L21
    split
08Use earlier factsL22–22

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L22
    exact htrace_witness_witness_left
09Separate the logical casesL23–23

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L23
    split
10Use earlier factsL24–25

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L24
    exact hterminal_witness
  2. L25
    exact htrace_witness_witness_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have htrace : ∃ fs_u_exists_trace. ∃ fs_v_exists_trace. BetaAt(fs_u_exists_trace,fs_v_exists_trace,0,0) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. BetaAt(b,c,x,y) ∧ (BetaAt(fs_u_exists_trace,fs_v_exists_trace,x,z) ∧ (BetaAt(fs_u_exists_trace,fs_v_exists_trace,S x,n) ∧ n = z + y)))
    Exact native replay linehave htrace : exists fs_u_exists_trace fs_v_exists_trace. ((((exists fs_h_exists_trace_start. fs_h_exists_trace_start + S (0) = S ((S (0)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_start. fs_u_exists_trace = fs_q_exists_trace_start * S ((S (0)) * fs_v_exists_trace) + (0))) /\ forall fs_i_exists_trace_steps. (exists fs_lt_exists_trace_steps_bound. fs_lt_exists_trace_steps_bound + S fs_i_exists_trace_steps = l) -> exists fs_a_exists_trace_steps fs_r_exists_trace_steps fs_s_exists_trace_steps. ((((exists fs_h_exists_trace_steps_summand. fs_h_exists_trace_steps_summand + S (fs_a_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * c)) /\ exists fs_q_exists_trace_steps_summand. b = fs_q_exists_trace_steps_summand * S ((S (fs_i_exists_trace_steps)) * c) + (fs_a_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_partial. fs_h_exists_trace_steps_partial + S (fs_r_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_partial. fs_u_exists_trace = fs_q_exists_trace_steps_partial * S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_r_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_successor. fs_h_exists_trace_steps_successor + S (fs_s_exists_trace_steps) = S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_successor. fs_u_exists_trace = fs_q_exists_trace_steps_successor * S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_s_exists_trace_steps))) /\ fs_s_exists_trace_steps = fs_r_exists_trace_steps + fs_a_exists_trace_steps))))
  5. 0005specialize beta_prefix_sum_trace_exists b
  6. 0006specialize beta_prefix_sum_trace_exists c
  7. 0007specialize beta_prefix_sum_trace_exists l
  8. 0008exact beta_prefix_sum_trace_exists
  9. 0009cases htrace
  10. 0010cases htrace_witness
  11. 0011cases htrace_witness_witness
  12. 0012have hterminal : ∃ n. BetaAt(x,x1,l,n)
    Exact native replay linehave hterminal : exists n. ((exists fs_h_sum_terminal. fs_h_sum_terminal + S (n) = S ((S (l)) * x1)) /\ exists fs_q_sum_terminal. x = fs_q_sum_terminal * S ((S (l)) * x1) + (n))
  13. 0013specialize beta_at_exists x
  14. 0014specialize beta_at_exists x1
  15. 0015specialize beta_at_exists l
  16. 0016exact beta_at_exists
  17. 0017cases hterminal
  18. 0018exists x2
  19. 0019exists x
  20. 0020exists x1
  21. 0021split
  22. 0022exact htrace_witness_witness_left
  23. 0023split
  24. 0024exact hterminal_witness
  25. 0025exact htrace_witness_witness_right