EU0010

euler_unit_product_factor_nonunit_value

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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 expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 0 declared prerequisites and contains 12 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

EU0019 euler_nonunit_factor_unchanged_congruence

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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