FP004C

prime_field_cardinality_exists

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Exactly p canonical elements are witnessed by an actual finite bijection, not an external model count.

Exact expanded first-order arithmetic statement

forall p. exists b c. (((forall pff_enumeration_index_cardinality_existsenumeration. (exists pfa_gap_cardinality_existsenumerationbound. pfa_gap_cardinality_existsenumerationbound + S (pff_enumeration_index_cardinality_existsenumeration) = (p)) -> (((exists ff_h_pft_cardinality_existsenumerationentry. ff_h_pft_cardinality_existsenumerationentry + S (pff_enumeration_index_cardinality_existsenumeration) = S ((S (pff_enumeration_index_cardinality_existsenumeration)) * c)) /\ exists ff_q_pft_cardinality_existsenumerationentry. b = ff_q_pft_cardinality_existsenumerationentry * S ((S (pff_enumeration_index_cardinality_existsenumeration)) * c) + (pff_enumeration_index_cardinality_existsenumeration)))) /\ (((forall pff_cardinality_i_cardinality_exists pff_cardinality_a_cardinality_exists. (exists pfa_gap_cardinality_existsbounded_index. pfa_gap_cardinality_existsbounded_index + S (pff_cardinality_i_cardinality_exists) = (p)) -> (((exists ff_h_pft_cardinality_existsbounded_entry. ff_h_pft_cardinality_existsbounded_entry + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_i_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existsbounded_entry. b = ff_q_pft_cardinality_existsbounded_entry * S ((S (pff_cardinality_i_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists))) -> (exists pfa_gap_cardinality_existsbounded_value. pfa_gap_cardinality_existsbounded_value + S (pff_cardinality_a_cardinality_exists) = (p))) /\ (((forall pff_cardinality_i_cardinality_exists pff_cardinality_j_cardinality_exists pff_cardinality_a_cardinality_exists. (exists pfa_gap_cardinality_existsinjective_i. pfa_gap_cardinality_existsinjective_i + S (pff_cardinality_i_cardinality_exists) = (p)) -> (exists pfa_gap_cardinality_existsinjective_j. pfa_gap_cardinality_existsinjective_j + S (pff_cardinality_j_cardinality_exists) = (p)) -> (((exists ff_h_pft_cardinality_existsinjective_first. ff_h_pft_cardinality_existsinjective_first + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_i_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existsinjective_first. b = ff_q_pft_cardinality_existsinjective_first * S ((S (pff_cardinality_i_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists))) -> (((exists ff_h_pft_cardinality_existsinjective_second. ff_h_pft_cardinality_existsinjective_second + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_j_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existsinjective_second. b = ff_q_pft_cardinality_existsinjective_second * S ((S (pff_cardinality_j_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists))) -> pff_cardinality_i_cardinality_exists = pff_cardinality_j_cardinality_exists) /\ ((forall pff_cardinality_a_cardinality_exists. (exists pfa_gap_cardinality_existssurjective_value. pfa_gap_cardinality_existssurjective_value + S (pff_cardinality_a_cardinality_exists) = (p)) -> exists pff_cardinality_i_cardinality_exists. (exists pfa_gap_cardinality_existssurjective_index. pfa_gap_cardinality_existssurjective_index + S (pff_cardinality_i_cardinality_exists) = (p)) /\ (((exists ff_h_pft_cardinality_existssurjective_entry. ff_h_pft_cardinality_existssurjective_entry + S (pff_cardinality_a_cardinality_exists) = S ((S (pff_cardinality_i_cardinality_exists)) * c)) /\ exists ff_q_pft_cardinality_existssurjective_entry. b = ff_q_pft_cardinality_existssurjective_entry * S ((S (pff_cardinality_i_cardinality_exists)) * c) + (pff_cardinality_a_cardinality_exists)))))))))))

Constructive proof overview

Generated structural guide

Exactly p canonical elements are witnessed by an actual finite bijection, not an external model count.

The unchanged tactic script uses 2 declared prerequisites and contains 13 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

matrix_lattice_identity_selector_exists Alpha theorem; checked-use authorized FP004B prime_field_enumeration_is_bijection

Direct dependents

FP0057 prime_field_of_prime_order_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

Read the argument

Proof checkpoints

13 script commands · 5 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.

Named ingredients (1)
prime field enumeration is bijection

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–1

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

  1. L1
    intro p
02Establish heL2–4

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix lattice identity selector exists.

  1. L2
    have he : ∃ b. ∃ c. IdentityMatrixSelector(b,c,p)Definitions: IdentityMatrixSelector
  2. L3
    specialize matrix_lattice_identity_selector_exists (p)
  3. L4
    apply matrix_lattice_identity_selector_exists
03Separate the logical casesL5–6

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

  1. L5
    cases he
  2. L6
    cases he_witness
04Construct an explicit witnessL7–8

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

  1. L7
    exists x
  2. L8
    exists x1
05Use earlier factsL9–13

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

  1. L9
    specialize prime_field_enumeration_is_bijection (p)
  2. L10
    specialize prime_field_enumeration_is_bijection (x)
  3. L11
    specialize prime_field_enumeration_is_bijection (x1)
  4. L12
    apply prime_field_enumeration_is_bijection
  5. L13
    exact he_witness_witness

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro p
  2. 0002have he : exists b c. (forall pff_enumeration_index_cardinality_enumeration. (exists pfa_gap_cardinality_enumerationbound. pfa_gap_cardinality_enumerationbound + S (pff_enumeration_index_cardinality_enumeration) = (p)) -> (((exists ff_h_pft_cardinality_enumerationentry. ff_h_pft_cardinality_enumerationentry + S (pff_enumeration_index_cardinality_enumeration) = S ((S (pff_enumeration_index_cardinality_enumeration)) * c)) /\ exists ff_q_pft_cardinality_enumerationentry. b = ff_q_pft_cardinality_enumerationentry * S ((S (pff_enumeration_index_cardinality_enumeration)) * c) + (pff_enumeration_index_cardinality_enumeration))))
  3. 0003specialize matrix_lattice_identity_selector_exists (p)
  4. 0004apply matrix_lattice_identity_selector_exists
  5. 0005cases he
  6. 0006cases he_witness
  7. 0007exists x
  8. 0008exists x1
  9. 0009specialize prime_field_enumeration_is_bijection (p)
  10. 0010specialize prime_field_enumeration_is_bijection (x)
  11. 0011specialize prime_field_enumeration_is_bijection (x1)
  12. 0012apply prime_field_enumeration_is_bijection
  13. 0013exact he_witness_witness