MC0001

prime_toggle_square_quotient_divides

Cancel one actual nonzero factor in a witnessed square divisor; the quotient is genuinely divisible by p.

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 input n is positive. Signed code 2 denotes +1, so the actual sum is +1 at n=1 and zero for n>1. The positive-values result permits arbitrary F(0), which the divisor mask excludes. Prime-square multiples contribute zero. Full G007 inversion is established in its separate family.

Exact theorem in conservative defined notation

∀ p. ∀ d. ∀ q. ¬p = 0 → d = p · q → Dvd(p · p,d) → Dvd(p,q)

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

Definition DAG

Dvd(d,n) · 2

Actual proof prerequisites

mul_left_cancel_nonzero · checked external prerequisitemul_assoc · checked external prerequisite
Original expanded first-order statement 
forall p d q. ~(p=0) -> d=p*q -> (exists pvs_factor_square_input. (d) = (p*p) * pvs_factor_square_input) -> (exists pvs_factor_square_output. (q) = (p) * pvs_factor_square_output)

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 · 8 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 assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro d
  3. L3
    intro q
  4. L4
    intro hp
  5. L5
    intro heq
  6. L6
    intro hs
02Separate the logical casesL7–7

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

  1. L7
    cases hs
03Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists x
04Use earlier factsL9–13

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

  1. L9
    specialize mul_left_cancel_nonzero (p)
  2. L10
    specialize mul_left_cancel_nonzero (q)
  3. L11
    specialize mul_left_cancel_nonzero (p*x)
  4. L12
    apply mul_left_cancel_nonzero
  5. L13
    exact hp
05Calculate and transport equalitiesL14–15

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

  1. L14
    trans d
  2. L15
    symm
06Use earlier factsL16–16

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

  1. L16
    exact heq
07Calculate and transport equalitiesL17–17

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

  1. L17
    trans (p*p)*x
08Use earlier factsL18–19

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

  1. L18
    exact hs_witness
  2. L19
    apply mul_assoc

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro d
  3. 0003intro q
  4. 0004intro hp
  5. 0005intro heq
  6. 0006intro hs
  7. 0007cases hs
  8. 0008exists x
  9. 0009specialize mul_left_cancel_nonzero (p)
  10. 0010specialize mul_left_cancel_nonzero (q)
  11. 0011specialize mul_left_cancel_nonzero (p*x)
  12. 0012apply mul_left_cancel_nonzero
  13. 0013exact hp
  14. 0014trans d
  15. 0015symm
  16. 0016exact heq
  17. 0017trans (p*p)*x
  18. 0018exact hs_witness
  19. 0019apply mul_assoc