PX003C

polynomial_quotient_length_product

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

A positive constructed quotient has exactly the proper product length L with the length-S d divisor.

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 L d q. (((((q)=0) /\ ((exists pfc_gap_division_product_length_sourceshort. pfc_gap_division_product_length_sourceshort+(L)=(d))))) \/ (((~((q)=0)) /\ (((q)+(d)=(L)))))) -> ~(q=0) -> (((((q)=0 \/ (S d)=0) /\ (((L)=0)))) \/ (((~((q)=0)) /\ (((~((S d)=0)) /\ (((q)+(S d)=S (L))))))))

Constructive proof overview

Generated structural guide

A positive constructed quotient has exactly the proper product length L with the length-S d divisor.

The unchanged tactic script uses 1 declared prerequisite and contains 23 exact native proof lines.

Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

succ_ne_zero Alpha theorem; checked-use authorized

Direct dependents

PX003D prime_field_polynomial_quotient_proper_product

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

23 script commands · 10 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 assumptionsL1–5

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

  1. L1
    intro L
  2. L2
    intro d
  3. L3
    intro q
  4. L4
    intro h
  5. L5
    intro hq
02Separate the logical casesL6–8

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

  1. L6
    cases h
  2. L7
    cases h_left
  3. L8
    exfalso
03Use earlier factsL9–10

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

  1. L9
    apply hq
  2. L10
    exact h_left_left
04Separate the logical casesL11–13

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

  1. L11
    cases h_right
  2. L12
    right
  3. L13
    split
05Use earlier factsL14–14

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

  1. L14
    exact hq
06Separate the logical casesL15–15

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

  1. L15
    split
07Fix variables and assumptionsL16–16

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

  1. L16
    intro hz
08Use earlier factsL17–19

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

  1. L17
    specialize succ_ne_zero (d)
  2. L18
    apply succ_ne_zero
  3. L19
    exact hz
09Calculate and transport equalitiesL20–22

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

  1. L20
    trans S (q+d)
  2. L21
    simp
  3. L22
    congr
10Use earlier factsL23–23

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

  1. L23
    exact h_right_right

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro L
  2. 0002intro d
  3. 0003intro q
  4. 0004intro h
  5. 0005intro hq
  6. 0006cases h
  7. 0007cases h_left
  8. 0008exfalso
  9. 0009apply hq
  10. 0010exact h_left_left
  11. 0011cases h_right
  12. 0012right
  13. 0013split
  14. 0014exact hq
  15. 0015split
  16. 0016intro hz
  17. 0017specialize succ_ne_zero (d)
  18. 0018apply succ_ne_zero
  19. 0019exact hz
  20. 0020trans S (q+d)
  21. 0021simp
  22. 0022congr
  23. 0023exact h_right_right