PA008D · theorem

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.

Statement with defined notation

∀ r. ∀ s. ∀ n. ∀ p. ∀ a. (∀ x. Lt(x,n) → ∃ y. Lt(y,n) ∧ (BetaAt(r,s,x,y) ∧ ModEq(p,a · S x,S y))) → BoundedPrefix(r,s,n)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0008 ModEq PD0013 BetaAt PD0024 BoundedPrefix

5 occurrences

In local proof propositions

PD0002 Lt PD0008 ModEq PD0013 BetaAt

3 occurrences

Exact expanded native-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)))

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

PA008I prime_mul_residue_reindex_exists

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

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.

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 assumptionsL1–8

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: 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)Original native command in the exact edition
  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 defined 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 : ∃ 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))
    Exact native replay linehave 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