FP0026

prime_field_no_zero_divisors

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.

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

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

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

eq_decidable Alpha theorem; checked-use authorized FP0025 prime_field_multiply_cancel_nonzero_left FP0018 prime_field_multiply_zero_right

Direct dependents

FP002A prime_field_arithmetic_laws

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 exact 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