BT005I

beta_product_zero

Stable · empty-context checked

The product of an empty decoded prefix is one.

Exact expanded 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

Structural proof guide

The product of an empty decoded prefix is one.

Direct prerequisites: beta_at_unique. The authored body proceeds by case analysis (4).

Proof neighborhood

Direct dependencies

BT0042 beta_at_unique

Direct 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

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

  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