PX0011

prime_field_polynomial_equal_implies_equivalent

The inherited same-length decoded equality implies formal polynomial equivalence.

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

∀ b. ∀ c. ∀ d. ∀ e. ∀ L. BetaPrefixEqual(b,c,d,e,L) → PolynomialEquivalent(b,c,L,d,e,L)

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

Definition DAG

BetaPrefixEqual(b,c,d,e,l) · 1PolynomialEquivalent(b,c,L,d,e,M) · 1

Actual proof prerequisites

prime_field_polynomial_power_coefficient_functionalprime_field_polynomial_power_coefficient_transport
Original expanded first-order statement 
forall b c d e L. (forall mdr_i_pfp_equal_to_equivalent_input mdr_a_pfp_equal_to_equivalent_input. (exists mdr_gap_pfp_equal_to_equivalent_inputb. mdr_gap_pfp_equal_to_equivalent_inputb + S (mdr_i_pfp_equal_to_equivalent_input) = (L)) -> (((exists ff_h_mdr_pfp_equal_to_equivalent_inputo. ff_h_mdr_pfp_equal_to_equivalent_inputo + S (mdr_a_pfp_equal_to_equivalent_input) = S ((S (mdr_i_pfp_equal_to_equivalent_input)) * c)) /\ exists ff_q_mdr_pfp_equal_to_equivalent_inputo. b = ff_q_mdr_pfp_equal_to_equivalent_inputo * S ((S (mdr_i_pfp_equal_to_equivalent_input)) * c) + (mdr_a_pfp_equal_to_equivalent_input))) -> (((exists ff_h_mdr_pfp_equal_to_equivalent_inputn. ff_h_mdr_pfp_equal_to_equivalent_inputn + S (mdr_a_pfp_equal_to_equivalent_input) = S ((S (mdr_i_pfp_equal_to_equivalent_input)) * e)) /\ exists ff_q_mdr_pfp_equal_to_equivalent_inputn. d = ff_q_mdr_pfp_equal_to_equivalent_inputn * S ((S (mdr_i_pfp_equal_to_equivalent_input)) * e) + (mdr_a_pfp_equal_to_equivalent_input)))) -> (forall pfrep_power_equal_to_equivalent_result pfrep_left_equal_to_equivalent_result pfrep_right_equal_to_equivalent_result. ((exists pfrep_position_equal_to_equivalent_resultfirst. ((pfrep_position_equal_to_equivalent_resultfirst+S (pfrep_power_equal_to_equivalent_result)=(L)) /\ ((((exists ff_h_pfp_equal_to_equivalent_resultfirstentry. ff_h_pfp_equal_to_equivalent_resultfirstentry + S (pfrep_left_equal_to_equivalent_result) = S ((S (pfrep_position_equal_to_equivalent_resultfirst)) * c)) /\ exists ff_q_pfp_equal_to_equivalent_resultfirstentry. b = ff_q_pfp_equal_to_equivalent_resultfirstentry * S ((S (pfrep_position_equal_to_equivalent_resultfirst)) * c) + (pfrep_left_equal_to_equivalent_result)))))) \/ (((exists pfrep_gap_equal_to_equivalent_resultfirstoutside. pfrep_gap_equal_to_equivalent_resultfirstoutside+(L)=(pfrep_power_equal_to_equivalent_result)) /\ (((pfrep_left_equal_to_equivalent_result)=0))))) -> ((exists pfrep_position_equal_to_equivalent_resultsecond. ((pfrep_position_equal_to_equivalent_resultsecond+S (pfrep_power_equal_to_equivalent_result)=(L)) /\ ((((exists ff_h_pfp_equal_to_equivalent_resultsecondentry. ff_h_pfp_equal_to_equivalent_resultsecondentry + S (pfrep_right_equal_to_equivalent_result) = S ((S (pfrep_position_equal_to_equivalent_resultsecond)) * e)) /\ exists ff_q_pfp_equal_to_equivalent_resultsecondentry. d = ff_q_pfp_equal_to_equivalent_resultsecondentry * S ((S (pfrep_position_equal_to_equivalent_resultsecond)) * e) + (pfrep_right_equal_to_equivalent_result)))))) \/ (((exists pfrep_gap_equal_to_equivalent_resultsecondoutside. pfrep_gap_equal_to_equivalent_resultsecondoutside+(L)=(pfrep_power_equal_to_equivalent_result)) /\ (((pfrep_right_equal_to_equivalent_result)=0))))) -> pfrep_left_equal_to_equivalent_result=pfrep_right_equal_to_equivalent_result)

Complete tactic proof in conservative notation

All 29 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

29 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 (2)
prime field polynomial power coefficient functionalprime field polynomial power coefficient transport
01Fix variables and assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro L
  6. L6
    intro he
  7. L7
    intro k
  8. L8
    intro a
  9. L9
    intro r
  10. L10
    intro ha
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hr
03Use earlier factsL12–21

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

  1. L12
    specialize prime_field_polynomial_power_coefficient_functional (d)
  2. L13
    specialize prime_field_polynomial_power_coefficient_functional (e)
  3. L14
    specialize prime_field_polynomial_power_coefficient_functional (L)
  4. L15
    specialize prime_field_polynomial_power_coefficient_functional (k)
  5. L16
    specialize prime_field_polynomial_power_coefficient_functional (a)
  6. L17
    specialize prime_field_polynomial_power_coefficient_functional (r)
  7. L18
    apply prime_field_polynomial_power_coefficient_functional
  8. L19
    specialize prime_field_polynomial_power_coefficient_transport (b)
  9. L20
    specialize prime_field_polynomial_power_coefficient_transport (c)
  10. L21
    specialize prime_field_polynomial_power_coefficient_transport (d)
04Use earlier factsL22–29

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

  1. L22
    specialize prime_field_polynomial_power_coefficient_transport (e)
  2. L23
    specialize prime_field_polynomial_power_coefficient_transport (L)
  3. L24
    specialize prime_field_polynomial_power_coefficient_transport (k)
  4. L25
    specialize prime_field_polynomial_power_coefficient_transport (a)
  5. L26
    apply prime_field_polynomial_power_coefficient_transport
  6. L27
    exact he
  7. L28
    exact ha
  8. L29
    exact hr

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro L
  6. 0006intro he
  7. 0007intro k
  8. 0008intro a
  9. 0009intro r
  10. 0010intro ha
  11. 0011intro hr
  12. 0012specialize prime_field_polynomial_power_coefficient_functional (d)
  13. 0013specialize prime_field_polynomial_power_coefficient_functional (e)
  14. 0014specialize prime_field_polynomial_power_coefficient_functional (L)
  15. 0015specialize prime_field_polynomial_power_coefficient_functional (k)
  16. 0016specialize prime_field_polynomial_power_coefficient_functional (a)
  17. 0017specialize prime_field_polynomial_power_coefficient_functional (r)
  18. 0018apply prime_field_polynomial_power_coefficient_functional
  19. 0019specialize prime_field_polynomial_power_coefficient_transport (b)
  20. 0020specialize prime_field_polynomial_power_coefficient_transport (c)
  21. 0021specialize prime_field_polynomial_power_coefficient_transport (d)
  22. 0022specialize prime_field_polynomial_power_coefficient_transport (e)
  23. 0023specialize prime_field_polynomial_power_coefficient_transport (L)
  24. 0024specialize prime_field_polynomial_power_coefficient_transport (k)
  25. 0025specialize prime_field_polynomial_power_coefficient_transport (a)
  26. 0026apply prime_field_polynomial_power_coefficient_transport
  27. 0027exact he
  28. 0028exact ha
  29. 0029exact hr