FP0021

prime_field_zero_has_no_multiplicative_inverse

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 has no product equal to one; this follows from multiplication, not from the inverse definition's guard.

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. ∀ b. Prime(p) → ¬FpMul(p,0,b,1)

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

Definition DAG

FpMul(p,a,b,c) · 2Prime(p) · 1

Actual proof prerequisites

prime_field_multiply_zero_leftprime_field_multiply_functionalsucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall p b. (~((p) = 1) /\ forall pfa_factor_left_zero_inverse_domain pfa_factor_right_zero_inverse_domain. (p) = pfa_factor_left_zero_inverse_domain * pfa_factor_right_zero_inverse_domain -> pfa_factor_left_zero_inverse_domain = 1 \/ pfa_factor_right_zero_inverse_domain = 1) -> ~(((exists pfa_gap_zero_inverseleft. pfa_gap_zero_inverseleft + S (0) = (p)) /\ (((exists pfa_gap_zero_inverseright. pfa_gap_zero_inverseright + S (b) = (p)) /\ ((((exists pfa_gap_zero_inverseresultbound. pfa_gap_zero_inverseresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_zero_inverseresultcongruence pfa_offset_right_zero_inverseresultcongruence. ((0) * (b)) + (p) * pfa_offset_left_zero_inverseresultcongruence = (1) + (p) * pfa_offset_right_zero_inverseresultcongruence)))))))))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 5 reading checkpoints · 2 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 functionalprime field multiply zero left
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro hp
  4. L4
    intro hm
02Establish hcopyL5–6

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hcopy : FpMul(p,0,b,1)Definitions: FpMul(p,0,b,1)Original native command in the exact edition
  2. L6
    exact hm
03Separate the logical casesL7–8

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

  1. L7
    cases hcopy
  2. L8
    cases hcopy_right
04Establish hbadL9–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field multiply functional.

  1. L9
    have hbad : 1 = 0
  2. L10
    specialize prime_field_multiply_functional (p)
  3. L11
    specialize prime_field_multiply_functional (0)
  4. L12
    specialize prime_field_multiply_functional (b)
  5. L13
    specialize prime_field_multiply_functional (1)
  6. L14
    specialize prime_field_multiply_functional (0)
  7. L15
    apply prime_field_multiply_functional
  8. L16
    exact hm
  9. L17
    specialize prime_field_multiply_zero_left (p)
  10. L18
    specialize prime_field_multiply_zero_left (b)
05Use earlier factsL19–24

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

  1. L19
    apply prime_field_multiply_zero_left
  2. L20
    exact hp
  3. L21
    exact hcopy_right_left
  4. L22
    specialize succ_ne_zero (0)
  5. L23
    apply succ_ne_zero
  6. L24
    exact hbad

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro hp
  4. 0004intro hm
  5. 0005have hcopy : FpMul(p,0,b,1)
  6. 0006exact hm
  7. 0007cases hcopy
  8. 0008cases hcopy_right
  9. 0009have hbad : 1 = 0
  10. 0010specialize prime_field_multiply_functional (p)
  11. 0011specialize prime_field_multiply_functional (0)
  12. 0012specialize prime_field_multiply_functional (b)
  13. 0013specialize prime_field_multiply_functional (1)
  14. 0014specialize prime_field_multiply_functional (0)
  15. 0015apply prime_field_multiply_functional
  16. 0016exact hm
  17. 0017specialize prime_field_multiply_zero_left (p)
  18. 0018specialize prime_field_multiply_zero_left (b)
  19. 0019apply prime_field_multiply_zero_left
  20. 0020exact hp
  21. 0021exact hcopy_right_left
  22. 0022specialize succ_ne_zero (0)
  23. 0023apply succ_ne_zero
  24. 0024exact hbad