EU000A

euler_multiplier_prefix_entry

Every decoded map entry, not just its construction witness, has the required bound and balanced 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.

The exact G014 theorem covers m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem handles m=1 by congruence, not by asserting that one is a canonical remainder. Multiplicative-order and RSA statements are not claimed.

Exact theorem in conservative defined notation

∀ a. ∀ m. ∀ b. ∀ c. ∀ l. ∀ i. ∀ r. UnitMultiplierPrefix(a,m,b,c,l) → Lt(i,l) → BetaAt(b,c,i,r) → CanonicalModularResidue(m,a · i,r)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 2UnitMultiplierPrefix(a,m,b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall a m b c l i r. (forall eu_index_map_entry. (exists eut_gap_eu_map_entry_index. eut_gap_eu_map_entry_index + S (eu_index_map_entry) = (l)) -> exists eu_residue_map_entry. (((exists fs_h_eu_map_entry_at. fs_h_eu_map_entry_at + S (eu_residue_map_entry) = S ((S (eu_index_map_entry)) * c)) /\ exists fs_q_eu_map_entry_at. b = fs_q_eu_map_entry_at * S ((S (eu_index_map_entry)) * c) + (eu_residue_map_entry))) /\ ((exists eut_gap_eu_map_entry_bound. eut_gap_eu_map_entry_bound + S (eu_residue_map_entry) = (m)) /\ (exists eu_mod_left_map_entry_mod eu_mod_right_map_entry_mod. ((a)*eu_index_map_entry) + (m) * eu_mod_left_map_entry_mod = (eu_residue_map_entry) + (m) * eu_mod_right_map_entry_mod))) -> (exists eut_gap_eu_entry_index. eut_gap_eu_entry_index + S (i) = (l)) -> (((exists fs_h_eu_entry_given. fs_h_eu_entry_given + S (r) = S ((S (i)) * c)) /\ exists fs_q_eu_entry_given. b = fs_q_eu_entry_given * S ((S (i)) * c) + (r))) -> (exists eut_gap_eu_entry_bound. eut_gap_eu_entry_bound + S (r) = (m)) /\ (exists eu_mod_left_entry_mod eu_mod_right_entry_mod. (a*i) + (m) * eu_mod_left_entry_mod = (r) + (m) * eu_mod_right_entry_mod)

Complete tactic proof in conservative notation

All 28 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

28 script commands · 6 reading checkpoints · 2 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 assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro r
  8. L8
    intro h
  9. L9
    intro hi
  10. L10
    intro hr
02Establish hpL11–14

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

  1. L11
    have hp : ∃ v. BetaAt(b,c,i,v) ∧ CanonicalModularResidue(m,a · i,v)Definitions: BetaAt(b,c,i,v)CanonicalModularResidue(m,a · i,v)Original native command in the exact edition
  2. L12
    specialize h (i)
  3. L13
    apply h
  4. L14
    exact hi
03Separate the logical casesL15–16

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

  1. L15
    cases hp
  2. L16
    cases hp_witness
04Establish heL17–26

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

  1. L17
    have he : x=r
  2. L18
    specialize beta_at_unique (b)
  3. L19
    specialize beta_at_unique (c)
  4. L20
    specialize beta_at_unique (i)
  5. L21
    specialize beta_at_unique (x)
  6. L22
    specialize beta_at_unique (r)
  7. L23
    apply beta_at_unique
  8. L24
    exact hp_witness_left
  9. L25
    exact hr
  10. L26
    rewrite he at hp_witness_right
05Calculate and transport equalitiesL27–27

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

  1. L27
    rewrite he at hp_witness_right
06Use earlier factsL28–28

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

  1. L28
    exact hp_witness_right

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro i
  7. 0007intro r
  8. 0008intro h
  9. 0009intro hi
  10. 0010intro hr
  11. 0011have hp : ∃ v. BetaAt(b,c,i,v) ∧ CanonicalModularResidue(m,a · i,v)
  12. 0012specialize h (i)
  13. 0013apply h
  14. 0014exact hi
  15. 0015cases hp
  16. 0016cases hp_witness
  17. 0017have he : x=r
  18. 0018specialize beta_at_unique (b)
  19. 0019specialize beta_at_unique (c)
  20. 0020specialize beta_at_unique (i)
  21. 0021specialize beta_at_unique (x)
  22. 0022specialize beta_at_unique (r)
  23. 0023apply beta_at_unique
  24. 0024exact hp_witness_left
  25. 0025exact hr
  26. 0026rewrite he at hp_witness_right
  27. 0027rewrite he at hp_witness_right
  28. 0028exact hp_witness_right