DF0010

dirichlet_grid_middle_factor_equation

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

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.

Every grid, slice, row sum and intermediate table is constructed. Retained cells have witnessed n=(a*e)*c and value F(a)*(H(e)*G(c)). The flat endpoint is unused. Table associativity includes N=0 and compares only positive values, not encodings. Full G009 remains broader.

Exact theorem in conservative defined notation

∀ n. ∀ a. ∀ e. ∀ c. ∀ q. ¬a = 0 → n = a · q → n = a · e · c → q = e · c

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

Definition DAG

none

Actual proof prerequisites

mul_left_cancel_nonzero · checked external prerequisitemul_assoc · checked external prerequisite
Original expanded first-order statement 
forall n a e c q. ~(a=0) -> n=a*q -> n=(a*e)*c -> q=e*c

Complete tactic proof in conservative notation

All 19 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

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.

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 assumptionsL1–8

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 defined 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