EU0007

euler_multiplier_prefix_empty

The empty multiplier prefix has no residue obligations.

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. UnitMultiplierPrefix(a,m,b,c,0)

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

Definition DAG

UnitMultiplierPrefix(a,m,b,c,l) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall a m b c. forall eu_index_map_empty. (exists eut_gap_eu_map_empty_index. eut_gap_eu_map_empty_index + S (eu_index_map_empty) = (0)) -> exists eu_residue_map_empty. (((exists fs_h_eu_map_empty_at. fs_h_eu_map_empty_at + S (eu_residue_map_empty) = S ((S (eu_index_map_empty)) * c)) /\ exists fs_q_eu_map_empty_at. b = fs_q_eu_map_empty_at * S ((S (eu_index_map_empty)) * c) + (eu_residue_map_empty))) /\ ((exists eut_gap_eu_map_empty_bound. eut_gap_eu_map_empty_bound + S (eu_residue_map_empty) = (m)) /\ (exists eu_mod_left_map_empty_mod eu_mod_right_map_empty_mod. ((a)*eu_index_map_empty) + (m) * eu_mod_left_map_empty_mod = (eu_residue_map_empty) + (m) * eu_mod_right_map_empty_mod))

Complete tactic proof in conservative notation

All 13 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

13 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.

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 assumptionsL1–6

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 i
  6. L6
    intro hi
02Separate the logical casesL7–7

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

  1. L7
    exfalso
03Use earlier factsL8–13

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

  1. L8
    specialize lt_not_le (i)
  2. L9
    specialize lt_not_le (0)
  3. L10
    apply lt_not_le
  4. L11
    exact hi
  5. L12
    specialize zero_le (i)
  6. L13
    apply zero_le

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro b
  4. 0004intro c
  5. 0005intro i
  6. 0006intro hi
  7. 0007exfalso
  8. 0008specialize lt_not_le (i)
  9. 0009specialize lt_not_le (0)
  10. 0010apply lt_not_le
  11. 0011exact hi
  12. 0012specialize zero_le (i)
  13. 0013apply zero_le