EU001D

euler_coprime_weighted_product_cancel

Cancel the proved coprime finite product in a balanced congruence, including the valid modulus-one case.

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. ∀ P. ∀ w. ¬m = 0 → Coprime(P,m) → ModEq(m,w · P,P) → ModEq(m,w,1)

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

Definition DAG

Coprime(a,b) · 1ModEq(m,a,b) · 2

Actual proof prerequisites

mod_eq_cancel_coprime · checked external prerequisitemul_comm · checked external prerequisitemul_one · checked external prerequisite
Original expanded first-order statement 
forall m P w. ~(m=0) -> (forall eut_divisor_eu_cancel_product. (exists eut_left_eu_cancel_product. (P) = eut_divisor_eu_cancel_product * eut_left_eu_cancel_product) -> (exists eut_right_eu_cancel_product. (m) = eut_divisor_eu_cancel_product * eut_right_eu_cancel_product) -> eut_divisor_eu_cancel_product = 1) -> (exists eu_mod_left_cancel_balance eu_mod_right_cancel_balance. (w*P) + (m) * eu_mod_left_cancel_balance = (P) + (m) * eu_mod_right_cancel_balance) -> (exists eu_mod_left_cancel_result eu_mod_right_cancel_result. (w) + (m) * eu_mod_left_cancel_result = (1) + (m) * eu_mod_right_cancel_result)

Complete tactic proof in conservative notation

All 23 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

23 script commands · 4 reading checkpoints · 2 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 m
  2. L2
    intro P
  3. L3
    intro w
  4. L4
    intro hm
  5. L5
    intro hP
  6. L6
    intro hmod
02Use earlier factsL7–13

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

  1. L7
    specialize mod_eq_cancel_coprime (m)
  2. L8
    specialize mod_eq_cancel_coprime (P)
  3. L9
    specialize mod_eq_cancel_coprime (w)
  4. L10
    specialize mod_eq_cancel_coprime (1)
  5. L11
    apply mod_eq_cancel_coprime
  6. L12
    exact hm
  7. L13
    exact hP
03Establish hleftL14–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul comm.

  1. L14
    have hleft : P*w=w*P
  2. L15
    specialize mul_comm (P)
  3. L16
    specialize mul_comm (w)
  4. L17
    apply mul_comm
04Establish hrightL18–23

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul one.

  1. L18
    have hright : P*1=P
  2. L19
    specialize mul_one (P)
  3. L20
    apply mul_one
  4. L21
    rewrite hleft
  5. L22
    rewrite hright
  6. L23
    exact hmod

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro m
  2. 0002intro P
  3. 0003intro w
  4. 0004intro hm
  5. 0005intro hP
  6. 0006intro hmod
  7. 0007specialize mod_eq_cancel_coprime (m)
  8. 0008specialize mod_eq_cancel_coprime (P)
  9. 0009specialize mod_eq_cancel_coprime (w)
  10. 0010specialize mod_eq_cancel_coprime (1)
  11. 0011apply mod_eq_cancel_coprime
  12. 0012exact hm
  13. 0013exact hP
  14. 0014have hleft : P*w=w*P
  15. 0015specialize mul_comm (P)
  16. 0016specialize mul_comm (w)
  17. 0017apply mul_comm
  18. 0018have hright : P*1=P
  19. 0019specialize mul_one (P)
  20. 0020apply mul_one
  21. 0021rewrite hleft
  22. 0022rewrite hright
  23. 0023exact hmod