FP0004

prime_field_residue_input_equal

Equality transports the dividend of an actual residue graph.

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. ∀ n. ∀ m. ∀ r. n = m → CanonicalModularResidue(p,m,r) → CanonicalModularResidue(p,n,r)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall p n m r. n = m -> (((exists pfa_gap_input_equal_sourcebound. pfa_gap_input_equal_sourcebound + S (r) = (p)) /\ ((exists pfa_offset_left_input_equal_sourcecongruence pfa_offset_right_input_equal_sourcecongruence. (m) + (p) * pfa_offset_left_input_equal_sourcecongruence = (r) + (p) * pfa_offset_right_input_equal_sourcecongruence)))) -> (((exists pfa_gap_input_equal_targetbound. pfa_gap_input_equal_targetbound + S (r) = (p)) /\ ((exists pfa_offset_left_input_equal_targetcongruence pfa_offset_right_input_equal_targetcongruence. (n) + (p) * pfa_offset_left_input_equal_targetcongruence = (r) + (p) * pfa_offset_right_input_equal_targetcongruence))))

Complete tactic proof in conservative notation

All 8 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

8 script commands · 3 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.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro m
  4. L4
    intro r
  5. L5
    intro heq
  6. L6
    intro hr
02Calculate and transport equalitiesL7–7

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L7
    rewrite heq
03Use earlier factsL8–8

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

  1. L8
    exact hr

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro m
  4. 0004intro r
  5. 0005intro heq
  6. 0006intro hr
  7. 0007rewrite heq
  8. 0008exact hr