FP0001

prime_field_mod_of_equal

Equality gives genuine balanced modular congruence.

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. a = b → ModEq(p,a,b)

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

Definition DAG

ModEq(m,a,b) · 1

Actual proof prerequisites

mod_eq_refl · checked external prerequisite
Original expanded first-order statement 
forall p a b. a = b -> (exists pfa_offset_left_equal pfa_offset_right_equal. (a) + (p) * pfa_offset_left_equal = (b) + (p) * pfa_offset_right_equal)

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 assumptionsL1–4

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 heq
02Calculate and transport equalitiesL5–5

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

  1. L5
    rewrite heq
03Use earlier factsL6–8

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

  1. L6
    specialize mod_eq_refl (p)
  2. L7
    specialize mod_eq_refl (b)
  3. L8
    apply mod_eq_refl

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro heq
  5. 0005rewrite heq
  6. 0006specialize mod_eq_refl (p)
  7. 0007specialize mod_eq_refl (b)
  8. 0008apply mod_eq_refl