FP0015

prime_field_add_zero_left

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.

Zero is also the left additive identity on every canonical representative.

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) → FpAdd(p,0,a,a)

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

Definition DAG

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

Actual proof prerequisites

prime_field_add_commutativeprime_field_add_zero_right
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_add_domain pfa_factor_right_zero_add_domain. (p) = pfa_factor_left_zero_add_domain * pfa_factor_right_zero_add_domain -> pfa_factor_left_zero_add_domain = 1 \/ pfa_factor_right_zero_add_domain = 1) -> (exists pfa_gap_zero_add_bound. pfa_gap_zero_add_bound + S (a) = (p)) -> (((exists pfa_gap_zero_addleft. pfa_gap_zero_addleft + S (0) = (p)) /\ (((exists pfa_gap_zero_addright. pfa_gap_zero_addright + S (a) = (p)) /\ ((((exists pfa_gap_zero_addresultbound. pfa_gap_zero_addresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_zero_addresultcongruence pfa_offset_right_zero_addresultcongruence. ((0) + (a)) + (p) * pfa_offset_left_zero_addresultcongruence = (a) + (p) * pfa_offset_right_zero_addresultcongruence)))))))))

Complete tactic proof in conservative notation

All 14 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

14 script commands · 2 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 add commutativeprime field add zero right
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
02Use earlier factsL5–14

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

  1. L5
    specialize prime_field_add_commutative (p)
  2. L6
    specialize prime_field_add_commutative (a)
  3. L7
    specialize prime_field_add_commutative (0)
  4. L8
    specialize prime_field_add_commutative (a)
  5. L9
    apply prime_field_add_commutative
  6. L10
    specialize prime_field_add_zero_right (p)
  7. L11
    specialize prime_field_add_zero_right (a)
  8. L12
    apply prime_field_add_zero_right
  9. L13
    exact hp
  10. L14
    exact ha

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005specialize prime_field_add_commutative (p)
  6. 0006specialize prime_field_add_commutative (a)
  7. 0007specialize prime_field_add_commutative (0)
  8. 0008specialize prime_field_add_commutative (a)
  9. 0009apply prime_field_add_commutative
  10. 0010specialize prime_field_add_zero_right (p)
  11. 0011specialize prime_field_add_zero_right (a)
  12. 0012apply prime_field_add_zero_right
  13. 0013exact hp
  14. 0014exact ha