FP0004

prime_field_residue_input_equal

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

Equality transports the dividend of an actual residue graph.

Exact expanded first-order arithmetic 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))))

Constructive proof overview

Generated structural guide

Equality transports the dividend of an actual residue graph.

The unchanged tactic script uses 0 declared prerequisites and contains 8 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

none

Direct dependents

FP000B prime_field_add_commutative FP000F prime_field_multiply_commutative FP0010 prime_field_add_associative FP0011 prime_field_multiply_associative FP0012 prime_field_left_distributive FP0013 prime_field_right_distributive FP004F prime_field_unit_trace_residue

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

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.

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 exact 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