FP0036

prime_field_inverse_table_exists

Construct every entry of the finite inverse table from primality alone.

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. Prime(p) → ∃ x. ∃ y. FpInvPrefix(p,x,y,p)

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

Definition DAG

FpInvPrefix(p,b,c,l) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_inverse_prefix_choiceprime_field_zero_extended_inverse_exists
Original expanded first-order statement 
forall p. (~((p) = 1) /\ forall pfa_factor_left_inversetable_domain pfa_factor_right_inversetable_domain. (p) = pfa_factor_left_inversetable_domain * pfa_factor_right_inversetable_domain -> pfa_factor_left_inversetable_domain = 1 \/ pfa_factor_right_inversetable_domain = 1) -> exists b c. (forall pft_index_inversetable_result. (exists pfa_gap_inversetable_resultprefix. pfa_gap_inversetable_resultprefix + S (pft_index_inversetable_result) = (p)) -> exists pft_value_inversetable_result. (((((exists ff_h_pft_inversetable_resultpointentry. ff_h_pft_inversetable_resultpointentry + S (pft_value_inversetable_result) = S ((S (pft_index_inversetable_result)) * c)) /\ exists ff_q_pft_inversetable_resultpointentry. b = ff_q_pft_inversetable_resultpointentry * S ((S (pft_index_inversetable_result)) * c) + (pft_value_inversetable_result))) /\ ((((exists pfa_gap_inversetable_resultpointvalueinput. pfa_gap_inversetable_resultpointvalueinput + S (pft_index_inversetable_result) = (p)) /\ (((exists pfa_gap_inversetable_resultpointvalueoutput. pfa_gap_inversetable_resultpointvalueoutput + S (pft_value_inversetable_result) = (p)) /\ ((((pft_index_inversetable_result) = 0 /\ (pft_value_inversetable_result) = 0) \/ (((~((pft_index_inversetable_result) = 0)) /\ ((((exists pfa_gap_inversetable_resultpointvaluenonzeromultiplicationleft. pfa_gap_inversetable_resultpointvaluenonzeromultiplicationleft + S (pft_index_inversetable_result) = (p)) /\ (((exists pfa_gap_inversetable_resultpointvaluenonzeromultiplicationright. pfa_gap_inversetable_resultpointvaluenonzeromultiplicationright + S (pft_value_inversetable_result) = (p)) /\ ((((exists pfa_gap_inversetable_resultpointvaluenonzeromultiplicationresultbound. pfa_gap_inversetable_resultpointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inversetable_resultpointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inversetable_resultpointvaluenonzeromultiplicationresultcongruence. ((pft_index_inversetable_result) * (pft_value_inversetable_result)) + (p) * pfa_offset_left_inversetable_resultpointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inversetable_resultpointvaluenonzeromultiplicationresultcongruence))))))))))))))))))))))

Complete tactic proof in conservative notation

All 12 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

12 script commands · 4 reading checkpoints · 0 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 (2)
prime field inverse prefix choiceprime field zero extended inverse exists
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hp
02Use earlier factsL3–5

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

  1. L3
    specialize prime_field_inverse_prefix_choice (p)
  2. L4
    specialize prime_field_inverse_prefix_choice (p)
  3. L5
    apply prime_field_inverse_prefix_choice
03Fix variables and assumptionsL6–7

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

  1. L6
    intro i
  2. L7
    intro hi
04Use earlier factsL8–12

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

  1. L8
    specialize prime_field_zero_extended_inverse_exists (p)
  2. L9
    specialize prime_field_zero_extended_inverse_exists (i)
  3. L10
    apply prime_field_zero_extended_inverse_exists
  4. L11
    exact hp
  5. L12
    exact hi

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003specialize prime_field_inverse_prefix_choice (p)
  4. 0004specialize prime_field_inverse_prefix_choice (p)
  5. 0005apply prime_field_inverse_prefix_choice
  6. 0006intro i
  7. 0007intro hi
  8. 0008specialize prime_field_zero_extended_inverse_exists (p)
  9. 0009specialize prime_field_zero_extended_inverse_exists (i)
  10. 0010apply prime_field_zero_extended_inverse_exists
  11. 0011exact hp
  12. 0012exact hi