FP0005

prime_field_residue_congruence_transport

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Balanced congruence transports canonical residues; no quotient oracle is assumed.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall p n m r. (exists pfa_offset_left_transport pfa_offset_right_transport. (n) + (p) * pfa_offset_left_transport = (m) + (p) * pfa_offset_right_transport) -> (((exists pfa_gap_transport_sourcebound. pfa_gap_transport_sourcebound + S (r) = (p)) /\ ((exists pfa_offset_left_transport_sourcecongruence pfa_offset_right_transport_sourcecongruence. (m) + (p) * pfa_offset_left_transport_sourcecongruence = (r) + (p) * pfa_offset_right_transport_sourcecongruence)))) -> (((exists pfa_gap_transport_resultbound. pfa_gap_transport_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_transport_resultcongruence pfa_offset_right_transport_resultcongruence. (n) + (p) * pfa_offset_left_transport_resultcongruence = (r) + (p) * pfa_offset_right_transport_resultcongruence))))

Constructive proof overview

Generated structural guide

Balanced congruence transports canonical residues; no quotient oracle is assumed.

The unchanged tactic script uses 1 declared prerequisite and contains 16 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mod_eq_trans Stable theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

16 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 hmod
  6. L6
    intro hr
02Separate the logical casesL7–8

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

  1. L7
    cases hr
  2. L8
    split
03Use earlier factsL9–16

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

  1. L9
    exact hr_left
  2. L10
    specialize mod_eq_trans (p)
  3. L11
    specialize mod_eq_trans (n)
  4. L12
    specialize mod_eq_trans (m)
  5. L13
    specialize mod_eq_trans (r)
  6. L14
    apply mod_eq_trans
  7. L15
    exact hmod
  8. L16
    exact hr_right

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro m
  4. 0004intro r
  5. 0005intro hmod
  6. 0006intro hr
  7. 0007cases hr
  8. 0008split
  9. 0009exact hr_left
  10. 0010specialize mod_eq_trans (p)
  11. 0011specialize mod_eq_trans (n)
  12. 0012specialize mod_eq_trans (m)
  13. 0013specialize mod_eq_trans (r)
  14. 0014apply mod_eq_trans
  15. 0015exact hmod
  16. 0016exact hr_right