EU000F

euler_unit_product_factor_unit_value

At a genuine unit index the weighted factor is the index itself.

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

∀ m. ∀ i. ∀ v. Coprime(i,m) → UnitProductFactor(m,i,v) → v = i

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

Definition DAG

UnitProductFactor(m,i,v) · 1Coprime(a,b) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall m i v. (forall eut_divisor_eu_factor_unit. (exists eut_left_eu_factor_unit. (i) = eut_divisor_eu_factor_unit * eut_left_eu_factor_unit) -> (exists eut_right_eu_factor_unit. (m) = eut_divisor_eu_factor_unit * eut_right_eu_factor_unit) -> eut_divisor_eu_factor_unit = 1) -> ((((forall eut_divisor_eu_factor_unit_coprime. (exists eut_left_eu_factor_unit_coprime. (i) = eut_divisor_eu_factor_unit_coprime * eut_left_eu_factor_unit_coprime) -> (exists eut_right_eu_factor_unit_coprime. (m) = eut_divisor_eu_factor_unit_coprime * eut_right_eu_factor_unit_coprime) -> eut_divisor_eu_factor_unit_coprime = 1) /\ (v)=(i)) \/ (~(forall eut_divisor_eu_factor_unit_coprime. (exists eut_left_eu_factor_unit_coprime. (i) = eut_divisor_eu_factor_unit_coprime * eut_left_eu_factor_unit_coprime) -> (exists eut_right_eu_factor_unit_coprime. (m) = eut_divisor_eu_factor_unit_coprime * eut_right_eu_factor_unit_coprime) -> eut_divisor_eu_factor_unit_coprime = 1) /\ (v)=1))) -> v=i

Complete tactic proof in conservative notation

All 12 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

12 script commands · 5 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 assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro i
  3. L3
    intro v
  4. L4
    intro hc
  5. L5
    intro hf
02Separate the logical casesL6–7

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

  1. L6
    cases hf
  2. L7
    cases hf_left
03Use earlier factsL8–8

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

  1. L8
    exact hf_left_right
04Separate the logical casesL9–10

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

  1. L9
    cases hf_right
  2. L10
    exfalso
05Use earlier factsL11–12

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

  1. L11
    apply hf_right_left
  2. L12
    exact hc

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro m
  2. 0002intro i
  3. 0003intro v
  4. 0004intro hc
  5. 0005intro hf
  6. 0006cases hf
  7. 0007cases hf_left
  8. 0008exact hf_left_right
  9. 0009cases hf_right
  10. 0010exfalso
  11. 0011apply hf_right_left
  12. 0012exact hc