PX0033

polynomial_quotient_length_bounds

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

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

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_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))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites and contains 24 exact native proof lines.

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

Proof neighborhood

Direct dependencies

zero_le Alpha theorem; checked-use authorized zero_add Alpha theorem; checked-use authorized le_add_right Alpha theorem; checked-use authorized le_refl Alpha theorem; checked-use authorized

Direct dependents

PX0037 prime_field_polynomial_division_quotient_data_exists PX003A prime_field_polynomial_division_remainder_degree

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

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.

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 exact 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