BT0041

beta_at_exists

Stable · empty-context checked

Every Gödel-beta position has a bounded decoded residue.

Exact expanded PA statement

forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

Structural proof guide

Every Gödel-beta position has a bounded decoded residue.

Direct prerequisites: beta_modulus_nonzero, mul_comm, division_remainder_exists. The authored body proceeds by case analysis (3), intermediate claims (2).

Proof neighborhood

Direct dependencies

BT003Y beta_modulus_nonzero BT0006 mul_comm BT001P division_remainder_exists

Direct dependents

BT005A beta_exclusive_recode_congruence_step BT005E beta_prefix_product_trace_exists BT005F beta_product_exists BT0089 beta_prefix_sum_trace_exists BT008A beta_sum_exists BT00T4 beta_pascal_row_step_extend BT00T8 choose_exists

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 i
  4. 0004have hm0 : ~(S ((S i) * c) = 0)
  5. 0005specialize beta_modulus_nonzero c
  6. 0006specialize beta_modulus_nonzero i
  7. 0007exact beta_modulus_nonzero
  8. 0008specialize division_remainder_exists (S ((S i) * c))
  9. 0009specialize division_remainder_exists b
  10. 0010have hdiv : exists q r. b = S ((S i) * c) * q + r /\ exists h. h + S r = S ((S i) * c)
  11. 0011apply division_remainder_exists
  12. 0012exact hm0
  13. 0013cases hdiv
  14. 0014cases hdiv_witness
  15. 0015cases hdiv_witness_witness
  16. 0016exists x1
  17. 0017split
  18. 0018exact hdiv_witness_witness_right
  19. 0019exists x
  20. 0020trans S ((S i) * c) * x + x1
  21. 0021exact hdiv_witness_witness_left
  22. 0022congr
  23. 0023apply mul_comm
  24. 0024refl