DF0010

dirichlet_grid_middle_factor_equation

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

A positive first factor cancels from the actual nested factor equations, identifying the inner convolution quotient.

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 n a e c q. ~(a=0) -> n=a*q -> n=(a*e)*c -> q=e*c

Constructive proof overview

Generated structural guide

A positive first factor cancels from the actual nested factor equations, identifying the inner convolution quotient.

The unchanged tactic script uses 2 declared prerequisites and contains 19 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mul_left_cancel_nonzero Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized

Direct dependents

DF0011 dirichlet_grid_entry_from_convolution_entry

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

19 script commands · 6 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 assumptionsL1–8

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

  1. L1
    intro n
  2. L2
    intro a
  3. L3
    intro e
  4. L4
    intro c
  5. L5
    intro q
  6. L6
    intro ha
  7. L7
    intro hq
  8. L8
    intro hc
02Use earlier factsL9–13

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

  1. L9
    specialize mul_left_cancel_nonzero (a)
  2. L10
    specialize mul_left_cancel_nonzero (q)
  3. L11
    specialize mul_left_cancel_nonzero (e*c)
  4. L12
    apply mul_left_cancel_nonzero
  5. L13
    exact ha
03Calculate and transport equalitiesL14–15

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

  1. L14
    trans n
  2. L15
    symm
04Use earlier factsL16–16

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

  1. L16
    exact hq
05Calculate and transport equalitiesL17–17

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

  1. L17
    trans (a*e)*c
06Use earlier factsL18–19

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

  1. L18
    exact hc
  2. L19
    apply mul_assoc

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro e
  4. 0004intro c
  5. 0005intro q
  6. 0006intro ha
  7. 0007intro hq
  8. 0008intro hc
  9. 0009specialize mul_left_cancel_nonzero (a)
  10. 0010specialize mul_left_cancel_nonzero (q)
  11. 0011specialize mul_left_cancel_nonzero (e*c)
  12. 0012apply mul_left_cancel_nonzero
  13. 0013exact ha
  14. 0014trans n
  15. 0015symm
  16. 0016exact hq
  17. 0017trans (a*e)*c
  18. 0018exact hc
  19. 0019apply mul_assoc