PP0006

prime_field_polynomial_normalization_reflexive

A table already consisting of canonical coefficients normalizes to itself.

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.

Length is representation length, not polynomial degree. Leading zeros and the empty zero polynomial are allowed; the canonical argument guard x<p also applies to the empty case. Evaluation is defined by actual field-operation steps, not an assumed residue invariant. Polynomial division, gcd, irreducibles and general prime-power extension fields remain open; this does not close G091.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ l. BetaPrefixInto(b,c,l,p) → FpCoefficientReduction(p,b,c,b,c,l)

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

Definition DAG

BetaPrefixInto(b,c,l,B) · 1FpCoefficientReduction(p,b,c,d,e,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

prime_field_residue_reflexive · checked external prerequisite
Original expanded first-order statement 
forall p b c l. (forall fom_index_pfp_reflexive_source. (exists fom_gap_pfp_reflexive_source_index_bound. fom_gap_pfp_reflexive_source_index_bound + S (fom_index_pfp_reflexive_source) = l) -> exists fom_value_pfp_reflexive_source. ((((exists fom_beta_height_pfp_reflexive_source_entry. fom_beta_height_pfp_reflexive_source_entry + S (fom_value_pfp_reflexive_source) = S ((S (fom_index_pfp_reflexive_source)) * c)) /\ exists fom_beta_quotient_pfp_reflexive_source_entry. b = fom_beta_quotient_pfp_reflexive_source_entry * S ((S (fom_index_pfp_reflexive_source)) * c) + (fom_value_pfp_reflexive_source))) /\ (exists fom_gap_pfp_reflexive_source_value_bound. fom_gap_pfp_reflexive_source_value_bound + S (fom_value_pfp_reflexive_source) = p))) -> (forall pfp_index_reflexive_result. (exists pfa_gap_reflexive_resultindex. pfa_gap_reflexive_resultindex + S (pfp_index_reflexive_result) = (l)) -> exists pfp_source_reflexive_result pfp_residue_reflexive_result. ((((exists ff_h_pfp_reflexive_resultsource. ff_h_pfp_reflexive_resultsource + S (pfp_source_reflexive_result) = S ((S (pfp_index_reflexive_result)) * c)) /\ exists ff_q_pfp_reflexive_resultsource. b = ff_q_pfp_reflexive_resultsource * S ((S (pfp_index_reflexive_result)) * c) + (pfp_source_reflexive_result))) /\ (((((exists ff_h_pfp_reflexive_resulttarget. ff_h_pfp_reflexive_resulttarget + S (pfp_residue_reflexive_result) = S ((S (pfp_index_reflexive_result)) * c)) /\ exists ff_q_pfp_reflexive_resulttarget. b = ff_q_pfp_reflexive_resulttarget * S ((S (pfp_index_reflexive_result)) * c) + (pfp_residue_reflexive_result))) /\ ((((exists pfa_gap_reflexive_resultresiduebound. pfa_gap_reflexive_resultresiduebound + S (pfp_residue_reflexive_result) = (p)) /\ ((exists pfa_offset_left_reflexive_resultresiduecongruence pfa_offset_right_reflexive_resultresiduecongruence. (pfp_source_reflexive_result) + (p) * pfa_offset_left_reflexive_resultresiduecongruence = (pfp_residue_reflexive_result) + (p) * pfa_offset_right_reflexive_resultresiduecongruence)))))))))

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 · 8 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 assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro h
  6. L6
    intro i
  7. L7
    intro hi
02Establish haL8–11

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

  1. L8
    have ha : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,p)Definitions: BetaAt(b,c,i,a)Lt(a,p)Original native command in the exact edition
  2. L9
    specialize h (i)
  3. L10
    apply h
  4. L11
    exact hi
03Separate the logical casesL12–13

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

  1. L12
    cases ha
  2. L13
    cases ha_witness
04Construct an explicit witnessL14–15

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

  1. L14
    exists x
  2. L15
    exists x
05Separate the logical casesL16–16

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

  1. L16
    split
06Use earlier factsL17–17

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

  1. L17
    exact ha_witness_left
07Separate the logical casesL18–18

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

  1. L18
    split
08Use earlier factsL19–23

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

  1. L19
    exact ha_witness_left
  2. L20
    specialize prime_field_residue_reflexive (p)
  3. L21
    specialize prime_field_residue_reflexive (x)
  4. L22
    apply prime_field_residue_reflexive
  5. L23
    exact ha_witness_right

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro h
  6. 0006intro i
  7. 0007intro hi
  8. 0008have ha : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,p)
  9. 0009specialize h (i)
  10. 0010apply h
  11. 0011exact hi
  12. 0012cases ha
  13. 0013cases ha_witness
  14. 0014exists x
  15. 0015exists x
  16. 0016split
  17. 0017exact ha_witness_left
  18. 0018split
  19. 0019exact ha_witness_left
  20. 0020specialize prime_field_residue_reflexive (p)
  21. 0021specialize prime_field_residue_reflexive (x)
  22. 0022apply prime_field_residue_reflexive
  23. 0023exact ha_witness_right