BT005I · Bertrand theorem

beta_product_zero

Stable checked-use theorem · independently kernel verified

The product of an empty decoded prefix is one.

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. ∀ n. Product(b,c,0,n) → n = 1

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

PD0014 Product

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + n) /\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S S i) * v)) /\ exists q. u = q * S ((S S i) * v) + s) /\ s = r * p)))))) -> n = 1

Proof neighborhood

Direct theorem prerequisites

BT0042 beta_at_unique

Direct theorem dependents

BT0081 pow_zero BT0091 factorial_zero BT00DH beta_product_pointwise_coprime BT00UC primorial_zero BT00UQ beta_product_prefix_suffix_split BT00VD beta_pairwise_coprime_product_divides_common_multiple BT00X9 beta_product_pointwise_le BT010U beta_product_all_one_exact

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

16 script commands · 3 reading checkpoints · 0 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 (1)
beta at unique
01Fix variables and assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro n
  4. L4
    intro hproduct
02Separate the logical casesL5–8

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

  1. L5
    cases hproduct
  2. L6
    cases hproduct_witness
  3. L7
    cases hproduct_witness_witness
  4. L8
    cases hproduct_witness_witness_right
03Use earlier factsL9–16

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

  1. L9
    specialize beta_at_unique x
  2. L10
    specialize beta_at_unique x1
  3. L11
    specialize beta_at_unique 0
  4. L12
    specialize beta_at_unique n
  5. L13
    specialize beta_at_unique 1
  6. L14
    apply beta_at_unique
  7. L15
    exact hproduct_witness_witness_right_left
  8. L16
    exact hproduct_witness_witness_left

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro n
  4. 0004intro hproduct
  5. 0005cases hproduct
  6. 0006cases hproduct_witness
  7. 0007cases hproduct_witness_witness
  8. 0008cases hproduct_witness_witness_right
  9. 0009specialize beta_at_unique x
  10. 0010specialize beta_at_unique x1
  11. 0011specialize beta_at_unique 0
  12. 0012specialize beta_at_unique n
  13. 0013specialize beta_at_unique 1
  14. 0014apply beta_at_unique
  15. 0015exact hproduct_witness_witness_right_left
  16. 0016exact hproduct_witness_witness_left