BT00JB · Bertrand theorem

eisenstein_initial_segment_decoded_choice

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Every decoded bit recovers its exact threshold semantics.

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

∀ q. ∀ b. ∀ c. ∀ k. ∀ j. ∀ bit. (∀ x. Lt(x,k) → ∃ y. BetaAt(b,c,x,y) ∧ (y = 1 ∧ Lt(x,q) ∨ y = 0 ∧ Lt(q,S x))) → Lt(j,k) → BetaAt(b,c,j,bit) → bit = 1 ∧ Lt(j,q) ∨ bit = 0 ∧ Lt(q,S j)

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

PD0002 Lt PD0013 BetaAt

8 occurrences

In local proof propositions

PD0002 Lt PD0013 BetaAt

3 occurrences

Exact expanded native-PA statement
forall q b c k j bit. (forall eis_index_initial_segment_semantic_source. (exists eis_lt_gap_initial_segment_semantic_source_bound. eis_lt_gap_initial_segment_semantic_source_bound + S (eis_index_initial_segment_semantic_source) = k) -> exists eis_bit_initial_segment_semantic_source. ((((exists ff_h_eis_initial_segment_semantic_source_decoded. ff_h_eis_initial_segment_semantic_source_decoded + S (eis_bit_initial_segment_semantic_source) = S ((S (eis_index_initial_segment_semantic_source)) * c)) /\ exists ff_q_eis_initial_segment_semantic_source_decoded. b = ff_q_eis_initial_segment_semantic_source_decoded * S ((S (eis_index_initial_segment_semantic_source)) * c) + (eis_bit_initial_segment_semantic_source))) /\ (((eis_bit_initial_segment_semantic_source = 1 /\ (exists eis_le_gap_initial_segment_semantic_source_choice_inside. eis_le_gap_initial_segment_semantic_source_choice_inside + (S eis_index_initial_segment_semantic_source) = q)) \/ (eis_bit_initial_segment_semantic_source = 0 /\ (exists eis_lt_gap_initial_segment_semantic_source_choice_outside. eis_lt_gap_initial_segment_semantic_source_choice_outside + S (q) = S eis_index_initial_segment_semantic_source)))))) -> (exists eis_lt_gap_initial_segment_semantic_bound. eis_lt_gap_initial_segment_semantic_bound + S (j) = k) -> (((exists ff_h_initial_segment_semantic_entry. ff_h_initial_segment_semantic_entry + S (bit) = S ((S (j)) * c)) /\ exists ff_q_initial_segment_semantic_entry. b = ff_q_initial_segment_semantic_entry * S ((S (j)) * c) + (bit))) -> (((bit = 1 /\ (exists eis_le_gap_initial_segment_semantic_result_inside. eis_le_gap_initial_segment_semantic_result_inside + (S j) = q)) \/ (bit = 0 /\ (exists eis_lt_gap_initial_segment_semantic_result_outside. eis_lt_gap_initial_segment_semantic_result_outside + S (q) = S j))))

Proof neighborhood

Direct theorem prerequisites

BT0042 beta_at_unique

Direct theorem dependents

BT00JD eisenstein_initial_segment_bit_count_functional BT00SK power_quotient_successor_pointwise_add

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

27 script commands · 7 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 (1)
beta at unique
01Fix variables and assumptionsL1–9

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

  1. L1
    intro q
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro k
  5. L5
    intro j
  6. L6
    intro bit
  7. L7
    intro hprefix
  8. L8
    intro hj
  9. L9
    intro hentry
02Use earlier factsL10–10

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

  1. L10
    specialize hprefix j
03Establish hstoredL11–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprefix.

  1. L11
    have hstored : ∃ stored. BetaAt(b,c,j,stored) ∧ (stored = 1 ∧ Lt(j,q) ∨ stored = 0 ∧ Lt(q,S j))Definitions: BetaAt(b,c,j,stored)Lt(j,q)Lt(q,S j)Original native command in the exact edition
  2. L12
    apply hprefix
  3. L13
    exact hj
04Separate the logical casesL14–15

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

  1. L14
    cases hstored
  2. L15
    cases hstored_witness
05Establish heqL16–25

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L16
    have heq : x = bit
  2. L17
    specialize beta_at_unique b
  3. L18
    specialize beta_at_unique c
  4. L19
    specialize beta_at_unique j
  5. L20
    specialize beta_at_unique x
  6. L21
    specialize beta_at_unique bit
  7. L22
    apply beta_at_unique
  8. L23
    exact hstored_witness_left
  9. L24
    exact hentry
  10. L25
    rewrite heq at hstored_witness_right
06Calculate and transport equalitiesL26–26

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L26
    rewrite heq at hstored_witness_right
07Use earlier factsL27–27

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

  1. L27
    exact hstored_witness_right

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro q
  2. 0002intro b
  3. 0003intro c
  4. 0004intro k
  5. 0005intro j
  6. 0006intro bit
  7. 0007intro hprefix
  8. 0008intro hj
  9. 0009intro hentry
  10. 0010specialize hprefix j
  11. 0011have hstored : ∃ stored. BetaAt(b,c,j,stored) ∧ (stored = 1 ∧ Lt(j,q) ∨ stored = 0 ∧ Lt(q,S j))
    Exact native replay linehave hstored : exists stored. ((((exists ff_h_initial_segment_semantic_stored. ff_h_initial_segment_semantic_stored + S (stored) = S ((S (j)) * c)) /\ exists ff_q_initial_segment_semantic_stored. b = ff_q_initial_segment_semantic_stored * S ((S (j)) * c) + (stored))) /\ (((stored = 1 /\ (exists eis_le_gap_initial_segment_semantic_stored_choice_inside. eis_le_gap_initial_segment_semantic_stored_choice_inside + (S j) = q)) \/ (stored = 0 /\ (exists eis_lt_gap_initial_segment_semantic_stored_choice_outside. eis_lt_gap_initial_segment_semantic_stored_choice_outside + S (q) = S j)))))
  12. 0012apply hprefix
  13. 0013exact hj
  14. 0014cases hstored
  15. 0015cases hstored_witness
  16. 0016have heq : x = bit
  17. 0017specialize beta_at_unique b
  18. 0018specialize beta_at_unique c
  19. 0019specialize beta_at_unique j
  20. 0020specialize beta_at_unique x
  21. 0021specialize beta_at_unique bit
  22. 0022apply beta_at_unique
  23. 0023exact hstored_witness_left
  24. 0024exact hentry
  25. 0025rewrite heq at hstored_witness_right
  26. 0026rewrite heq at hstored_witness_right
  27. 0027exact hstored_witness_right