PX0033

polynomial_quotient_length_bounds

The constructed quotient prefix fits in the input and its length plus the divisor degree covers every input coefficient.

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) → Le(q,L) ∧ Le(L,q + d)

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

Definition DAG

PolynomialQuotientLength(L,d,q) · 1Le(a,b) · 2

Actual proof prerequisites

zero_le · checked external prerequisitezero_add · checked external prerequisitele_add_right · checked external prerequisitele_refl · checked external prerequisite
Original expanded first-order statement 
forall L d q. (((((q)=0) /\ ((exists pfc_gap_division_length_bound_sourceshort. pfc_gap_division_length_bound_sourceshort+(L)=(d))))) \/ (((~((q)=0)) /\ (((q)+(d)=(L)))))) -> ((exists pfc_gap_division_length_quotient_bound. pfc_gap_division_length_quotient_bound+(q)=(L)) /\ ((exists pfc_gap_division_length_cover. pfc_gap_division_length_cover+(L)=(q+d))))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 10 reading checkpoints · 1 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 assumptionsL1–4

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
02Separate the logical casesL5–7

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

  1. L5
    cases h
  2. L6
    cases h_left
  3. L7
    split
03Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite h_left_left
04Use earlier factsL9–10

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

  1. L9
    specialize zero_le (L)
  2. L10
    apply zero_le
05Establish heqL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply zero add.

  1. L11
    have heq : q+d=d
  2. L12
    rewrite h_left_left
  3. L13
    apply zero_add
  4. L14
    rewrite heq
  5. L15
    exact h_left_right
06Separate the logical casesL16–17

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

  1. L16
    cases h_right
  2. L17
    split
07Calculate and transport equalitiesL18–18

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

  1. L18
    rewrite <- h_right_right
08Use earlier factsL19–21

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

  1. L19
    specialize le_add_right (q)
  2. L20
    specialize le_add_right (d)
  3. L21
    apply le_add_right
09Calculate and transport equalitiesL22–22

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

  1. L22
    rewrite h_right_right
10Use earlier factsL23–24

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

  1. L23
    specialize le_refl (L)
  2. L24
    apply le_refl

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro L
  2. 0002intro d
  3. 0003intro q
  4. 0004intro h
  5. 0005cases h
  6. 0006cases h_left
  7. 0007split
  8. 0008rewrite h_left_left
  9. 0009specialize zero_le (L)
  10. 0010apply zero_le
  11. 0011have heq : q+d=d
  12. 0012rewrite h_left_left
  13. 0013apply zero_add
  14. 0014rewrite heq
  15. 0015exact h_left_right
  16. 0016cases h_right
  17. 0017split
  18. 0018rewrite <- h_right_right
  19. 0019specialize le_add_right (q)
  20. 0020specialize le_add_right (d)
  21. 0021apply le_add_right
  22. 0022rewrite h_right_right
  23. 0023specialize le_refl (L)
  24. 0024apply le_refl