PP0008

prime_field_polynomial_normalization_idempotent

Reducing a normalized polynomial again leaves every coefficient unchanged, not necessarily its raw code.

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. ∀ d. ∀ e. ∀ f. ∀ g. ∀ l. FpCoefficientReduction(p,b,c,d,e,l) → FpCoefficientReduction(p,d,e,f,g,l) → BetaPrefixEqual(d,e,f,g,l)

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

Definition DAG

BetaPrefixEqual(b,c,d,e,l) · 1FpCoefficientReduction(p,b,c,d,e,l) · 2

Actual proof prerequisites

prime_field_polynomial_normalization_functionalprime_field_polynomial_normalization_reflexiveprime_field_polynomial_normalization_bounded
Original expanded first-order statement 
forall p b c d e f g l. (forall pfp_index_idempotent_first. (exists pfa_gap_idempotent_firstindex. pfa_gap_idempotent_firstindex + S (pfp_index_idempotent_first) = (l)) -> exists pfp_source_idempotent_first pfp_residue_idempotent_first. ((((exists ff_h_pfp_idempotent_firstsource. ff_h_pfp_idempotent_firstsource + S (pfp_source_idempotent_first) = S ((S (pfp_index_idempotent_first)) * c)) /\ exists ff_q_pfp_idempotent_firstsource. b = ff_q_pfp_idempotent_firstsource * S ((S (pfp_index_idempotent_first)) * c) + (pfp_source_idempotent_first))) /\ (((((exists ff_h_pfp_idempotent_firsttarget. ff_h_pfp_idempotent_firsttarget + S (pfp_residue_idempotent_first) = S ((S (pfp_index_idempotent_first)) * e)) /\ exists ff_q_pfp_idempotent_firsttarget. d = ff_q_pfp_idempotent_firsttarget * S ((S (pfp_index_idempotent_first)) * e) + (pfp_residue_idempotent_first))) /\ ((((exists pfa_gap_idempotent_firstresiduebound. pfa_gap_idempotent_firstresiduebound + S (pfp_residue_idempotent_first) = (p)) /\ ((exists pfa_offset_left_idempotent_firstresiduecongruence pfa_offset_right_idempotent_firstresiduecongruence. (pfp_source_idempotent_first) + (p) * pfa_offset_left_idempotent_firstresiduecongruence = (pfp_residue_idempotent_first) + (p) * pfa_offset_right_idempotent_firstresiduecongruence))))))))) -> (forall pfp_index_idempotent_second. (exists pfa_gap_idempotent_secondindex. pfa_gap_idempotent_secondindex + S (pfp_index_idempotent_second) = (l)) -> exists pfp_source_idempotent_second pfp_residue_idempotent_second. ((((exists ff_h_pfp_idempotent_secondsource. ff_h_pfp_idempotent_secondsource + S (pfp_source_idempotent_second) = S ((S (pfp_index_idempotent_second)) * e)) /\ exists ff_q_pfp_idempotent_secondsource. d = ff_q_pfp_idempotent_secondsource * S ((S (pfp_index_idempotent_second)) * e) + (pfp_source_idempotent_second))) /\ (((((exists ff_h_pfp_idempotent_secondtarget. ff_h_pfp_idempotent_secondtarget + S (pfp_residue_idempotent_second) = S ((S (pfp_index_idempotent_second)) * g)) /\ exists ff_q_pfp_idempotent_secondtarget. f = ff_q_pfp_idempotent_secondtarget * S ((S (pfp_index_idempotent_second)) * g) + (pfp_residue_idempotent_second))) /\ ((((exists pfa_gap_idempotent_secondresiduebound. pfa_gap_idempotent_secondresiduebound + S (pfp_residue_idempotent_second) = (p)) /\ ((exists pfa_offset_left_idempotent_secondresiduecongruence pfa_offset_right_idempotent_secondresiduecongruence. (pfp_source_idempotent_second) + (p) * pfa_offset_left_idempotent_secondresiduecongruence = (pfp_residue_idempotent_second) + (p) * pfa_offset_right_idempotent_secondresiduecongruence))))))))) -> (forall mdr_i_pfp_idempotent_result mdr_a_pfp_idempotent_result. (exists mdr_gap_pfp_idempotent_resultb. mdr_gap_pfp_idempotent_resultb + S (mdr_i_pfp_idempotent_result) = (l)) -> (((exists ff_h_mdr_pfp_idempotent_resulto. ff_h_mdr_pfp_idempotent_resulto + S (mdr_a_pfp_idempotent_result) = S ((S (mdr_i_pfp_idempotent_result)) * e)) /\ exists ff_q_mdr_pfp_idempotent_resulto. d = ff_q_mdr_pfp_idempotent_resulto * S ((S (mdr_i_pfp_idempotent_result)) * e) + (mdr_a_pfp_idempotent_result))) -> (((exists ff_h_mdr_pfp_idempotent_resultn. ff_h_mdr_pfp_idempotent_resultn + S (mdr_a_pfp_idempotent_result) = S ((S (mdr_i_pfp_idempotent_result)) * g)) /\ exists ff_q_mdr_pfp_idempotent_resultn. f = ff_q_mdr_pfp_idempotent_resultn * S ((S (mdr_i_pfp_idempotent_result)) * g) + (mdr_a_pfp_idempotent_result))))

Complete tactic proof in conservative notation

All 33 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

33 script commands · 4 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.

Named ingredients (3)
prime field polynomial normalization functionalprime field polynomial normalization reflexiveprime field polynomial normalization bounded
01Fix variables and assumptionsL1–10

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 d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro l
  9. L9
    intro hfirst
  10. L10
    intro hsecond
02Use earlier factsL11–20

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

  1. L11
    specialize prime_field_polynomial_normalization_functional (p)
  2. L12
    specialize prime_field_polynomial_normalization_functional (d)
  3. L13
    specialize prime_field_polynomial_normalization_functional (e)
  4. L14
    specialize prime_field_polynomial_normalization_functional (d)
  5. L15
    specialize prime_field_polynomial_normalization_functional (e)
  6. L16
    specialize prime_field_polynomial_normalization_functional (f)
  7. L17
    specialize prime_field_polynomial_normalization_functional (g)
  8. L18
    specialize prime_field_polynomial_normalization_functional (l)
  9. L19
    apply prime_field_polynomial_normalization_functional
  10. L20
    specialize prime_field_polynomial_normalization_reflexive (p)
03Use earlier factsL21–30

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

  1. L21
    specialize prime_field_polynomial_normalization_reflexive (d)
  2. L22
    specialize prime_field_polynomial_normalization_reflexive (e)
  3. L23
    specialize prime_field_polynomial_normalization_reflexive (l)
  4. L24
    apply prime_field_polynomial_normalization_reflexive
  5. L25
    specialize prime_field_polynomial_normalization_bounded (p)
  6. L26
    specialize prime_field_polynomial_normalization_bounded (b)
  7. L27
    specialize prime_field_polynomial_normalization_bounded (c)
  8. L28
    specialize prime_field_polynomial_normalization_bounded (d)
  9. L29
    specialize prime_field_polynomial_normalization_bounded (e)
  10. L30
    specialize prime_field_polynomial_normalization_bounded (l)
04Use earlier factsL31–33

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

  1. L31
    apply prime_field_polynomial_normalization_bounded
  2. L32
    exact hfirst
  3. L33
    exact hsecond

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro l
  9. 0009intro hfirst
  10. 0010intro hsecond
  11. 0011specialize prime_field_polynomial_normalization_functional (p)
  12. 0012specialize prime_field_polynomial_normalization_functional (d)
  13. 0013specialize prime_field_polynomial_normalization_functional (e)
  14. 0014specialize prime_field_polynomial_normalization_functional (d)
  15. 0015specialize prime_field_polynomial_normalization_functional (e)
  16. 0016specialize prime_field_polynomial_normalization_functional (f)
  17. 0017specialize prime_field_polynomial_normalization_functional (g)
  18. 0018specialize prime_field_polynomial_normalization_functional (l)
  19. 0019apply prime_field_polynomial_normalization_functional
  20. 0020specialize prime_field_polynomial_normalization_reflexive (p)
  21. 0021specialize prime_field_polynomial_normalization_reflexive (d)
  22. 0022specialize prime_field_polynomial_normalization_reflexive (e)
  23. 0023specialize prime_field_polynomial_normalization_reflexive (l)
  24. 0024apply prime_field_polynomial_normalization_reflexive
  25. 0025specialize prime_field_polynomial_normalization_bounded (p)
  26. 0026specialize prime_field_polynomial_normalization_bounded (b)
  27. 0027specialize prime_field_polynomial_normalization_bounded (c)
  28. 0028specialize prime_field_polynomial_normalization_bounded (d)
  29. 0029specialize prime_field_polynomial_normalization_bounded (e)
  30. 0030specialize prime_field_polynomial_normalization_bounded (l)
  31. 0031apply prime_field_polynomial_normalization_bounded
  32. 0032exact hfirst
  33. 0033exact hsecond