FP0016

prime_field_multiply_one_right

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.

Natural one is the actual multiplicative identity, including at p=2.

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

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. ∀ a. Prime(p) → Lt(a,p) → FpMul(p,a,1,a)

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

Definition DAG

FpMul(p,a,b,c) · 1Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_two_le · checked external prerequisiteprime_field_mod_of_equalmul_one · checked external prerequisite
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_multiply_one_domain pfa_factor_right_multiply_one_domain. (p) = pfa_factor_left_multiply_one_domain * pfa_factor_right_multiply_one_domain -> pfa_factor_left_multiply_one_domain = 1 \/ pfa_factor_right_multiply_one_domain = 1) -> (exists pfa_gap_multiply_one_bound. pfa_gap_multiply_one_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_oneleft. pfa_gap_multiply_oneleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_oneright. pfa_gap_multiply_oneright + S (1) = (p)) /\ ((((exists pfa_gap_multiply_oneresultbound. pfa_gap_multiply_oneresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_multiply_oneresultcongruence pfa_offset_right_multiply_oneresultcongruence. ((a) * (1)) + (p) * pfa_offset_left_multiply_oneresultcongruence = (a) + (p) * pfa_offset_right_multiply_oneresultcongruence)))))))))

Complete tactic proof in conservative notation

All 18 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

18 script commands · 7 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 (1)
prime field mod of equal
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro hp
  4. L4
    intro ha
02Separate the logical casesL5–5

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

  1. L5
    split
03Use earlier factsL6–6

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

  1. L6
    exact ha
04Separate the logical casesL7–7

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

  1. L7
    split
05Use earlier factsL8–10

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

  1. L8
    specialize prime_two_le (p)
  2. L9
    apply prime_two_le
  3. L10
    exact hp
06Separate the logical casesL11–11

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

  1. L11
    split
07Use earlier factsL12–18

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

  1. L12
    exact ha
  2. L13
    specialize prime_field_mod_of_equal (p)
  3. L14
    specialize prime_field_mod_of_equal (a * 1)
  4. L15
    specialize prime_field_mod_of_equal (a)
  5. L16
    apply prime_field_mod_of_equal
  6. L17
    specialize mul_one (a)
  7. L18
    apply mul_one

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005split
  6. 0006exact ha
  7. 0007split
  8. 0008specialize prime_two_le (p)
  9. 0009apply prime_two_le
  10. 0010exact hp
  11. 0011split
  12. 0012exact ha
  13. 0013specialize prime_field_mod_of_equal (p)
  14. 0014specialize prime_field_mod_of_equal (a * 1)
  15. 0015specialize prime_field_mod_of_equal (a)
  16. 0016apply prime_field_mod_of_equal
  17. 0017specialize mul_one (a)
  18. 0018apply mul_one