EU0011

euler_unit_product_factor_coprime

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

Every weighted factor is coprime to the modulus, including the modulus-one zero factor.

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_coprime_coprime. (exists eut_left_eu_factor_coprime_coprime. (i) = eut_divisor_eu_factor_coprime_coprime * eut_left_eu_factor_coprime_coprime) -> (exists eut_right_eu_factor_coprime_coprime. (m) = eut_divisor_eu_factor_coprime_coprime * eut_right_eu_factor_coprime_coprime) -> eut_divisor_eu_factor_coprime_coprime = 1) /\ (v)=(i)) \/ (~(forall eut_divisor_eu_factor_coprime_coprime. (exists eut_left_eu_factor_coprime_coprime. (i) = eut_divisor_eu_factor_coprime_coprime * eut_left_eu_factor_coprime_coprime) -> (exists eut_right_eu_factor_coprime_coprime. (m) = eut_divisor_eu_factor_coprime_coprime * eut_right_eu_factor_coprime_coprime) -> eut_divisor_eu_factor_coprime_coprime = 1) /\ (v)=1))) -> (forall eut_divisor_eu_factor_result_coprime. (exists eut_left_eu_factor_result_coprime. (v) = eut_divisor_eu_factor_result_coprime * eut_left_eu_factor_result_coprime) -> (exists eut_right_eu_factor_result_coprime. (m) = eut_divisor_eu_factor_result_coprime * eut_right_eu_factor_result_coprime) -> eut_divisor_eu_factor_result_coprime = 1)

Constructive proof overview

Generated structural guide

Every weighted factor is coprime to the modulus, including the modulus-one zero factor.

The unchanged tactic script uses 1 declared prerequisite 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

coprime_one_left Stable theorem; checked-use authorized

Direct dependents

EU0017 euler_unit_product_coprime

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 · 7 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 assumptionsL1–4

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 hf
02Separate the logical casesL5–6

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

  1. L5
    cases hf
  2. L6
    cases hf_left
03Calculate and transport equalitiesL7–7

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

  1. L7
    rewrite hf_left_right
04Use earlier factsL8–8

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

  1. L8
    exact hf_left_left
05Separate the logical casesL9–9

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

  1. L9
    cases hf_right
06Calculate and transport equalitiesL10–10

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

  1. L10
    rewrite hf_right_right
07Use earlier factsL11–12

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

  1. L11
    specialize coprime_one_left (m)
  2. L12
    apply coprime_one_left

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro m
  2. 0002intro i
  3. 0003intro v
  4. 0004intro hf
  5. 0005cases hf
  6. 0006cases hf_left
  7. 0007rewrite hf_left_right
  8. 0008exact hf_left_left
  9. 0009cases hf_right
  10. 0010rewrite hf_right_right
  11. 0011specialize coprime_one_left (m)
  12. 0012apply coprime_one_left