PA008D

fermat_index_map_bounded

Alpha v34 checked-use theorem · independently closed; not Stable

The canonical multiplication-residue index map is bounded.

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 original first-admission records.

Exact expanded PA statement

forall r s n p a. (forall frm_index_bounded_map. (exists frm_gap_bounded_map_index_bound. frm_gap_bounded_map_index_bound + S frm_index_bounded_map = n) -> (exists frm_residue_bounded_map_result. (exists frm_gap_bounded_map_result_residue_bound. frm_gap_bounded_map_result_residue_bound + S frm_residue_bounded_map_result = n) /\ ((((exists ff_h_frm_bounded_map_result_decoded. ff_h_frm_bounded_map_result_decoded + S (frm_residue_bounded_map_result) = S ((S (frm_index_bounded_map)) * s)) /\ exists ff_q_frm_bounded_map_result_decoded. r = ff_q_frm_bounded_map_result_decoded * S ((S (frm_index_bounded_map)) * s) + (frm_residue_bounded_map_result))) /\ (exists frm_mod_left_bounded_map_result_congruence frm_mod_right_bounded_map_result_congruence. a * S frm_index_bounded_map + p * frm_mod_left_bounded_map_result_congruence = S frm_residue_bounded_map_result + p * frm_mod_right_bounded_map_result_congruence)))) -> (forall fp_i_bounded_result. (exists fp_gap_bounded_result_index. fp_gap_bounded_result_index + S fp_i_bounded_result = n) -> exists fp_value_bounded_result. ((((exists ff_h_bounded_result_entry. ff_h_bounded_result_entry + S (fp_value_bounded_result) = S ((S (fp_i_bounded_result)) * s)) /\ exists ff_q_bounded_result_entry. r = ff_q_bounded_result_entry * S ((S (fp_i_bounded_result)) * s) + (fp_value_bounded_result))) /\ (exists fp_gap_bounded_result_value. fp_gap_bounded_result_value + S fp_value_bounded_result = n)))

Structural proof guide

Generated structural guide

The canonical multiplication-residue index map is bounded.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by case analysis (3), intermediate claims (1).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA008I prime_mul_residue_reindex_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

19 script commands · 6 reading checkpoints · 1 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–8

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

  1. L1
    intro r
  2. L2
    intro s
  3. L3
    intro n
  4. L4
    intro p
  5. L5
    intro a
  6. L6
    intro hmap
  7. L7
    intro i
  8. L8
    intro hi
02Establish hentryL9–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hmap.

  1. L9
    have hentry : ∃ frm_residue_bounded_at_i. Lt(frm_residue_bounded_at_i,n) ∧ (BetaAt(r,s,i,frm_residue_bounded_at_i) ∧ ModEq(p,a · S i,S frm_residue_bounded_at_i))Definitions: LtModEqBetaAt
  2. L10
    specialize hmap i
  3. L11
    apply hmap
  4. L12
    exact hi
03Separate the logical casesL13–15

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

  1. L13
    cases hentry
  2. L14
    cases hentry_witness
  3. L15
    cases hentry_witness_right
04Construct an explicit witnessL16–16

Supply the displayed value, then prove that it has the required property.

  1. L16
    exists x
05Separate the logical casesL17–17

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

  1. L17
    split
06Use earlier factsL18–19

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

  1. L18
    exact hentry_witness_right_left
  2. L19
    exact hentry_witness_left

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro r
  2. 0002intro s
  3. 0003intro n
  4. 0004intro p
  5. 0005intro a
  6. 0006intro hmap
  7. 0007intro i
  8. 0008intro hi
  9. 0009have hentry : exists frm_residue_bounded_at_i. (exists frm_gap_bounded_at_i_residue_bound. frm_gap_bounded_at_i_residue_bound + S frm_residue_bounded_at_i = n) /\ ((((exists ff_h_frm_bounded_at_i_decoded. ff_h_frm_bounded_at_i_decoded + S (frm_residue_bounded_at_i) = S ((S (i)) * s)) /\ exists ff_q_frm_bounded_at_i_decoded. r = ff_q_frm_bounded_at_i_decoded * S ((S (i)) * s) + (frm_residue_bounded_at_i))) /\ (exists frm_mod_left_bounded_at_i_congruence frm_mod_right_bounded_at_i_congruence. a * S i + p * frm_mod_left_bounded_at_i_congruence = S frm_residue_bounded_at_i + p * frm_mod_right_bounded_at_i_congruence))
  10. 0010specialize hmap i
  11. 0011apply hmap
  12. 0012exact hi
  13. 0013cases hentry
  14. 0014cases hentry_witness
  15. 0015cases hentry_witness_right
  16. 0016exists x
  17. 0017split
  18. 0018exact hentry_witness_right_left
  19. 0019exact hentry_witness_left