EU0010

euler_unit_product_factor_nonunit_value

A nonunit index contributes exactly one, not a zero or an assumed cancellable residue.

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 = 1

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_nonunit. (exists eut_left_eu_factor_nonunit. (i) = eut_divisor_eu_factor_nonunit * eut_left_eu_factor_nonunit) -> (exists eut_right_eu_factor_nonunit. (m) = eut_divisor_eu_factor_nonunit * eut_right_eu_factor_nonunit) -> eut_divisor_eu_factor_nonunit = 1) -> ((((forall eut_divisor_eu_factor_nonunit_coprime. (exists eut_left_eu_factor_nonunit_coprime. (i) = eut_divisor_eu_factor_nonunit_coprime * eut_left_eu_factor_nonunit_coprime) -> (exists eut_right_eu_factor_nonunit_coprime. (m) = eut_divisor_eu_factor_nonunit_coprime * eut_right_eu_factor_nonunit_coprime) -> eut_divisor_eu_factor_nonunit_coprime = 1) /\ (v)=(i)) \/ (~(forall eut_divisor_eu_factor_nonunit_coprime. (exists eut_left_eu_factor_nonunit_coprime. (i) = eut_divisor_eu_factor_nonunit_coprime * eut_left_eu_factor_nonunit_coprime) -> (exists eut_right_eu_factor_nonunit_coprime. (m) = eut_divisor_eu_factor_nonunit_coprime * eut_right_eu_factor_nonunit_coprime) -> eut_divisor_eu_factor_nonunit_coprime = 1) /\ (v)=1))) -> v=1

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 casesL6–8

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

  1. L6
    cases hf
  2. L7
    cases hf_left
  3. L8
    exfalso
03Use earlier factsL9–10

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

  1. L9
    apply hc
  2. L10
    exact hf_left_left
04Separate the logical casesL11–11

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

  1. L11
    cases hf_right
05Use earlier factsL12–12

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

  1. L12
    exact hf_right_right

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. 0008exfalso
  9. 0009apply hc
  10. 0010exact hf_left_left
  11. 0011cases hf_right
  12. 0012exact hf_right_right