FP0026

prime_field_no_zero_divisors

A zero canonical product has a zero factor, including the characteristic-two case.

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. ∀ b. Prime(p) → FpMul(p,a,b,0) → a = 0 ∨ b = 0

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

Definition DAG

FpMul(p,a,b,c) · 1Prime(p) · 1

Actual proof prerequisites

eq_decidable · checked external prerequisiteprime_field_multiply_cancel_nonzero_leftprime_field_multiply_zero_right
Original expanded first-order statement 
forall p a b. (~((p) = 1) /\ forall pfa_factor_left_no_zero_divisors_domain pfa_factor_right_no_zero_divisors_domain. (p) = pfa_factor_left_no_zero_divisors_domain * pfa_factor_right_no_zero_divisors_domain -> pfa_factor_left_no_zero_divisors_domain = 1 \/ pfa_factor_right_no_zero_divisors_domain = 1) -> (((exists pfa_gap_no_zero_divisors_productleft. pfa_gap_no_zero_divisors_productleft + S (a) = (p)) /\ (((exists pfa_gap_no_zero_divisors_productright. pfa_gap_no_zero_divisors_productright + S (b) = (p)) /\ ((((exists pfa_gap_no_zero_divisors_productresultbound. pfa_gap_no_zero_divisors_productresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_no_zero_divisors_productresultcongruence pfa_offset_right_no_zero_divisors_productresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_no_zero_divisors_productresultcongruence = (0) + (p) * pfa_offset_right_no_zero_divisors_productresultcongruence))))))))) -> a = 0 \/ b = 0

Complete tactic proof in conservative notation

All 28 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

28 script commands · 9 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 (2)
prime field multiply cancel nonzero leftprime field multiply zero right
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hp
  5. L5
    intro hm
02Establish hcasesL6–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eq decidable.

  1. L6
    have hcases : a = 0 \/ ~(a = 0)
  2. L7
    specialize eq_decidable (a)
  3. L8
    specialize eq_decidable (0)
  4. L9
    apply eq_decidable
03Separate the logical casesL10–11

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

  1. L10
    cases hcases
  2. L11
    left
04Use earlier factsL12–12

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

  1. L12
    exact hcases_left
05Separate the logical casesL13–13

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

  1. L13
    right
06Use earlier factsL14–23

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

  1. L14
    specialize prime_field_multiply_cancel_nonzero_left (p)
  2. L15
    specialize prime_field_multiply_cancel_nonzero_left (a)
  3. L16
    specialize prime_field_multiply_cancel_nonzero_left (b)
  4. L17
    specialize prime_field_multiply_cancel_nonzero_left (0)
  5. L18
    specialize prime_field_multiply_cancel_nonzero_left (0)
  6. L19
    apply prime_field_multiply_cancel_nonzero_left
  7. L20
    exact hp
  8. L21
    exact hcases_right
  9. L22
    exact hm
  10. L23
    specialize prime_field_multiply_zero_right (p)
07Use earlier factsL24–26

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

  1. L24
    specialize prime_field_multiply_zero_right (a)
  2. L25
    apply prime_field_multiply_zero_right
  3. L26
    exact hp
08Separate the logical casesL27–27

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

  1. L27
    cases hm
09Use earlier factsL28–28

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

  1. L28
    exact hm_left

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hp
  5. 0005intro hm
  6. 0006have hcases : a = 0 \/ ~(a = 0)
  7. 0007specialize eq_decidable (a)
  8. 0008specialize eq_decidable (0)
  9. 0009apply eq_decidable
  10. 0010cases hcases
  11. 0011left
  12. 0012exact hcases_left
  13. 0013right
  14. 0014specialize prime_field_multiply_cancel_nonzero_left (p)
  15. 0015specialize prime_field_multiply_cancel_nonzero_left (a)
  16. 0016specialize prime_field_multiply_cancel_nonzero_left (b)
  17. 0017specialize prime_field_multiply_cancel_nonzero_left (0)
  18. 0018specialize prime_field_multiply_cancel_nonzero_left (0)
  19. 0019apply prime_field_multiply_cancel_nonzero_left
  20. 0020exact hp
  21. 0021exact hcases_right
  22. 0022exact hm
  23. 0023specialize prime_field_multiply_zero_right (p)
  24. 0024specialize prime_field_multiply_zero_right (a)
  25. 0025apply prime_field_multiply_zero_right
  26. 0026exact hp
  27. 0027cases hm
  28. 0028exact hm_left