PX003C

polynomial_quotient_length_product

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

Alpha v34 checked-use · first admitted v33 · 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.

Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.

Exact theorem in conservative defined notation

∀ L. ∀ d. ∀ q. PolynomialQuotientLength(L,d,q) → ¬q = 0 → PolynomialProductLength(q,S d,L)

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

Definition DAG

PolynomialProductLength(L,M,N) · 1PolynomialQuotientLength(L,d,q) · 1

Actual proof prerequisites

succ_ne_zero · checked external prerequisite
Original expanded first-order 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))))))))

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 · 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.

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 assumptionsL1–5

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 defined 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