FP000B

prime_field_add_commutative

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.

Commutativity of actual canonical add, not an assumed table axiom.

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. ∀ b. ∀ c. FpAdd(p,a,b,c) → FpAdd(p,b,a,c)

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

Definition DAG

FpAdd(p,a,b,c) · 2

Actual proof prerequisites

add_comm · checked external prerequisiteprime_field_residue_input_equal
Original expanded first-order statement 
forall p a b c. (((exists pfa_gap_addcomm_sourceleft. pfa_gap_addcomm_sourceleft + S (a) = (p)) /\ (((exists pfa_gap_addcomm_sourceright. pfa_gap_addcomm_sourceright + S (b) = (p)) /\ ((((exists pfa_gap_addcomm_sourceresultbound. pfa_gap_addcomm_sourceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_sourceresultcongruence pfa_offset_right_addcomm_sourceresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addcomm_sourceresultcongruence = (c) + (p) * pfa_offset_right_addcomm_sourceresultcongruence))))))))) -> (((exists pfa_gap_addcomm_targetleft. pfa_gap_addcomm_targetleft + S (b) = (p)) /\ (((exists pfa_gap_addcomm_targetright. pfa_gap_addcomm_targetright + S (a) = (p)) /\ ((((exists pfa_gap_addcomm_targetresultbound. pfa_gap_addcomm_targetresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_targetresultcongruence pfa_offset_right_addcomm_targetresultcongruence. ((b) + (a)) + (p) * pfa_offset_left_addcomm_targetresultcongruence = (c) + (p) * pfa_offset_right_addcomm_targetresultcongruence)))))))))

Complete tactic proof in conservative notation

All 20 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

20 script commands · 5 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 (1)
prime field residue input equal
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 c
  5. L5
    intro h
02Separate the logical casesL6–8

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

  1. L6
    cases h
  2. L7
    cases h_right
  3. L8
    split
03Use earlier factsL9–9

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

  1. L9
    exact h_right_left
04Separate the logical casesL10–10

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

  1. L10
    split
05Use earlier factsL11–20

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

  1. L11
    exact h_left
  2. L12
    specialize prime_field_residue_input_equal (p)
  3. L13
    specialize prime_field_residue_input_equal (b + a)
  4. L14
    specialize prime_field_residue_input_equal (a + b)
  5. L15
    specialize prime_field_residue_input_equal (c)
  6. L16
    apply prime_field_residue_input_equal
  7. L17
    specialize add_comm (b)
  8. L18
    specialize add_comm (a)
  9. L19
    apply add_comm
  10. L20
    exact h_right_right

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro h
  6. 0006cases h
  7. 0007cases h_right
  8. 0008split
  9. 0009exact h_right_left
  10. 0010split
  11. 0011exact h_left
  12. 0012specialize prime_field_residue_input_equal (p)
  13. 0013specialize prime_field_residue_input_equal (b + a)
  14. 0014specialize prime_field_residue_input_equal (a + b)
  15. 0015specialize prime_field_residue_input_equal (c)
  16. 0016apply prime_field_residue_input_equal
  17. 0017specialize add_comm (b)
  18. 0018specialize add_comm (a)
  19. 0019apply add_comm
  20. 0020exact h_right_right