FP0014

prime_field_add_zero_right

Natural zero is the actual additive identity, not an arbitrary chosen code.

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. ∀ a. Prime(p) → Lt(a,p) → FpAdd(p,a,0,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_zero_below_primeprime_field_mod_of_equal
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_add_zero_domain pfa_factor_right_add_zero_domain. (p) = pfa_factor_left_add_zero_domain * pfa_factor_right_add_zero_domain -> pfa_factor_left_add_zero_domain = 1 \/ pfa_factor_right_add_zero_domain = 1) -> (exists pfa_gap_add_zero_bound. pfa_gap_add_zero_bound + S (a) = (p)) -> (((exists pfa_gap_add_zeroleft. pfa_gap_add_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_add_zeroright. pfa_gap_add_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_add_zeroresultbound. pfa_gap_add_zeroresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_add_zeroresultcongruence pfa_offset_right_add_zeroresultcongruence. ((a) + (0)) + (p) * pfa_offset_left_add_zeroresultcongruence = (a) + (p) * pfa_offset_right_add_zeroresultcongruence)))))))))

Complete tactic proof in conservative notation

All 17 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

17 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 (2)
prime field zero below primeprime 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_field_zero_below_prime (p)
  2. L9
    apply prime_field_zero_below_prime
  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 factsL12–17

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 + 0)
  4. L15
    specialize prime_field_mod_of_equal (a)
  5. L16
    apply prime_field_mod_of_equal
  6. L17
    apply PA3

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005split
  6. 0006exact ha
  7. 0007split
  8. 0008specialize prime_field_zero_below_prime (p)
  9. 0009apply prime_field_zero_below_prime
  10. 0010exact hp
  11. 0011split
  12. 0012exact ha
  13. 0013specialize prime_field_mod_of_equal (p)
  14. 0014specialize prime_field_mod_of_equal (a + 0)
  15. 0015specialize prime_field_mod_of_equal (a)
  16. 0016apply prime_field_mod_of_equal
  17. 0017apply PA3