FP004C

prime_field_cardinality_exists

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

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

This checkpoint constructs prime-order fields (k=1), with genuine finite arithmetic tables, cardinality, and characteristic. Inversion is proved only for nonzero elements; the table's zero entry is a zero-to-zero convention. G091 for every prime power p^k, with an irreducible polynomial of degree k, remains open. No extension-field construction or G091 closure is claimed.

Exact theorem in conservative defined notation

∀ p. ∃ b. ∃ c. FpCardinality(p,b,c)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

IdentityMatrixSelector(b,c,l) · 1FpCardinality(p,b,c) · 1

Actual proof prerequisites

matrix_lattice_identity_selector_exists · checked external prerequisiteprime_field_enumeration_is_bijection
Original expanded first-order 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)))))))))))

Complete tactic proof in conservative notation

All 13 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

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)
prime field enumeration is bijection
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(b,c,p)Original native command in the exact edition
  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 defined command ledger · 13 lines
  1. 0001intro p
  2. 0002have he : ∃ b. ∃ c. IdentityMatrixSelector(b,c,p)
  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