BT008Y · Bertrand theorem

factorial_exists

Stable checked-use theorem · independently kernel verified

Every natural has a beta-coded relational factorial value.

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

∀ n. ∃ z. Factorial(n,z)

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

PD0023 Factorial

1 occurrences

In local proof propositions

PD0018 Range

1 occurrences

Exact expanded native-PA statement
forall n. exists z. (exists ff_b_exists ff_c_exists. ((forall ff_i_exists_range. (exists ff_lt_exists_range_bound. ff_lt_exists_range_bound + S ff_i_exists_range = n) -> (((exists ff_h_exists_range_decoded. ff_h_exists_range_decoded + S (1 + ff_i_exists_range) = S ((S (ff_i_exists_range)) * ff_c_exists)) /\ exists ff_q_exists_range_decoded. ff_b_exists = ff_q_exists_range_decoded * S ((S (ff_i_exists_range)) * ff_c_exists) + (1 + ff_i_exists_range)))) /\ (exists ff_u_exists_product ff_v_exists_product. ((((exists ff_h_exists_product_start. ff_h_exists_product_start + S (1) = S ((S (0)) * ff_v_exists_product)) /\ exists ff_q_exists_product_start. ff_u_exists_product = ff_q_exists_product_start * S ((S (0)) * ff_v_exists_product) + (1))) /\ ((((exists ff_h_exists_product_terminal. ff_h_exists_product_terminal + S (z) = S ((S (n)) * ff_v_exists_product)) /\ exists ff_q_exists_product_terminal. ff_u_exists_product = ff_q_exists_product_terminal * S ((S (n)) * ff_v_exists_product) + (z))) /\ forall ff_i_exists_product. (exists ff_lt_exists_product_bound. ff_lt_exists_product_bound + S ff_i_exists_product = n) -> exists ff_p_exists_product ff_r_exists_product ff_s_exists_product. ((((exists ff_h_exists_product_factor. ff_h_exists_product_factor + S (ff_p_exists_product) = S ((S (ff_i_exists_product)) * ff_c_exists)) /\ exists ff_q_exists_product_factor. ff_b_exists = ff_q_exists_product_factor * S ((S (ff_i_exists_product)) * ff_c_exists) + (ff_p_exists_product))) /\ ((((exists ff_h_exists_product_partial. ff_h_exists_product_partial + S (ff_r_exists_product) = S ((S (ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_partial. ff_u_exists_product = ff_q_exists_product_partial * S ((S (ff_i_exists_product)) * ff_v_exists_product) + (ff_r_exists_product))) /\ ((((exists ff_h_exists_product_successor. ff_h_exists_product_successor + S (ff_s_exists_product) = S ((S (S ff_i_exists_product)) * ff_v_exists_product)) /\ exists ff_q_exists_product_successor. ff_u_exists_product = ff_q_exists_product_successor * S ((S (S ff_i_exists_product)) * ff_v_exists_product) + (ff_s_exists_product))) /\ ff_s_exists_product = ff_r_exists_product * ff_p_exists_product))))))))

Proof neighborhood

Direct theorem prerequisites

BT0086 beta_range_exists BT005F beta_product_exists

Direct theorem dependents

BT00RM factorial_valuation_exists BT00VC choose_prime_divides_between BT00VX central_binom_prime_divisor_le_double

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

21 script commands · 10 reading checkpoints · 1 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 range existsbeta product exists
01Fix variables and assumptionsL1–1

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

  1. L1
    intro n
02Establish hrangeL2–5

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

  1. L2
    have hrange : ∃ b. ∃ c. Range(b,c,1,n)Definitions: Range(b,c,1,n)Original native command in the exact edition
  2. L3
    specialize beta_range_exists 1
  3. L4
    specialize beta_range_exists n
  4. L5
    exact beta_range_exists
03Separate the logical casesL6–7

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

  1. L6
    cases hrange
  2. L7
    cases hrange_witness
04Use earlier factsL8–10

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

  1. L8
    specialize beta_product_exists x
  2. L9
    specialize beta_product_exists x1
  3. L10
    specialize beta_product_exists n
05Separate the logical casesL11–13

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

  1. L11
    cases beta_product_exists
  2. L12
    cases beta_product_exists_witness
  3. L13
    cases beta_product_exists_witness_witness
06Construct an explicit witnessL14–16

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

  1. L14
    exists x2
  2. L15
    exists x
  3. L16
    exists x1
07Separate the logical casesL17–17

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

  1. L17
    split
08Use earlier factsL18–18

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

  1. L18
    exact hrange_witness_witness
09Construct an explicit witnessL19–20

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

  1. L19
    exists x3
  2. L20
    exists x4
10Use earlier factsL21–21

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

  1. L21
    exact beta_product_exists_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro n
  2. 0002have hrange : ∃ b. ∃ c. Range(b,c,1,n)
    Exact native replay linehave hrange : exists b c. (forall i. (exists h. h + S i = n) -> ((exists h. h + S (1 + i) = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + (1 + i)))
  3. 0003specialize beta_range_exists 1
  4. 0004specialize beta_range_exists n
  5. 0005exact beta_range_exists
  6. 0006cases hrange
  7. 0007cases hrange_witness
  8. 0008specialize beta_product_exists x
  9. 0009specialize beta_product_exists x1
  10. 0010specialize beta_product_exists n
  11. 0011cases beta_product_exists
  12. 0012cases beta_product_exists_witness
  13. 0013cases beta_product_exists_witness_witness
  14. 0014exists x2
  15. 0015exists x
  16. 0016exists x1
  17. 0017split
  18. 0018exact hrange_witness_witness
  19. 0019exists x3
  20. 0020exists x4
  21. 0021exact beta_product_exists_witness_witness_witness