PX000F

prime_field_polynomial_equivalent_symmetric

Formal coefficient equivalence is symmetric without choosing canonical raw beta codes.

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

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

Definition DAG

PolynomialEquivalent(b,c,L,d,e,M) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c L d e M. (forall pfrep_power_equivalent_symmetric_old pfrep_left_equivalent_symmetric_old pfrep_right_equivalent_symmetric_old. ((exists pfrep_position_equivalent_symmetric_oldfirst. ((pfrep_position_equivalent_symmetric_oldfirst+S (pfrep_power_equivalent_symmetric_old)=(L)) /\ ((((exists ff_h_pfp_equivalent_symmetric_oldfirstentry. ff_h_pfp_equivalent_symmetric_oldfirstentry + S (pfrep_left_equivalent_symmetric_old) = S ((S (pfrep_position_equivalent_symmetric_oldfirst)) * c)) /\ exists ff_q_pfp_equivalent_symmetric_oldfirstentry. b = ff_q_pfp_equivalent_symmetric_oldfirstentry * S ((S (pfrep_position_equivalent_symmetric_oldfirst)) * c) + (pfrep_left_equivalent_symmetric_old)))))) \/ (((exists pfrep_gap_equivalent_symmetric_oldfirstoutside. pfrep_gap_equivalent_symmetric_oldfirstoutside+(L)=(pfrep_power_equivalent_symmetric_old)) /\ (((pfrep_left_equivalent_symmetric_old)=0))))) -> ((exists pfrep_position_equivalent_symmetric_oldsecond. ((pfrep_position_equivalent_symmetric_oldsecond+S (pfrep_power_equivalent_symmetric_old)=(M)) /\ ((((exists ff_h_pfp_equivalent_symmetric_oldsecondentry. ff_h_pfp_equivalent_symmetric_oldsecondentry + S (pfrep_right_equivalent_symmetric_old) = S ((S (pfrep_position_equivalent_symmetric_oldsecond)) * e)) /\ exists ff_q_pfp_equivalent_symmetric_oldsecondentry. d = ff_q_pfp_equivalent_symmetric_oldsecondentry * S ((S (pfrep_position_equivalent_symmetric_oldsecond)) * e) + (pfrep_right_equivalent_symmetric_old)))))) \/ (((exists pfrep_gap_equivalent_symmetric_oldsecondoutside. pfrep_gap_equivalent_symmetric_oldsecondoutside+(M)=(pfrep_power_equivalent_symmetric_old)) /\ (((pfrep_right_equivalent_symmetric_old)=0))))) -> pfrep_left_equivalent_symmetric_old=pfrep_right_equivalent_symmetric_old) -> (forall pfrep_power_equivalent_symmetric_new pfrep_left_equivalent_symmetric_new pfrep_right_equivalent_symmetric_new. ((exists pfrep_position_equivalent_symmetric_newfirst. ((pfrep_position_equivalent_symmetric_newfirst+S (pfrep_power_equivalent_symmetric_new)=(M)) /\ ((((exists ff_h_pfp_equivalent_symmetric_newfirstentry. ff_h_pfp_equivalent_symmetric_newfirstentry + S (pfrep_left_equivalent_symmetric_new) = S ((S (pfrep_position_equivalent_symmetric_newfirst)) * e)) /\ exists ff_q_pfp_equivalent_symmetric_newfirstentry. d = ff_q_pfp_equivalent_symmetric_newfirstentry * S ((S (pfrep_position_equivalent_symmetric_newfirst)) * e) + (pfrep_left_equivalent_symmetric_new)))))) \/ (((exists pfrep_gap_equivalent_symmetric_newfirstoutside. pfrep_gap_equivalent_symmetric_newfirstoutside+(M)=(pfrep_power_equivalent_symmetric_new)) /\ (((pfrep_left_equivalent_symmetric_new)=0))))) -> ((exists pfrep_position_equivalent_symmetric_newsecond. ((pfrep_position_equivalent_symmetric_newsecond+S (pfrep_power_equivalent_symmetric_new)=(L)) /\ ((((exists ff_h_pfp_equivalent_symmetric_newsecondentry. ff_h_pfp_equivalent_symmetric_newsecondentry + S (pfrep_right_equivalent_symmetric_new) = S ((S (pfrep_position_equivalent_symmetric_newsecond)) * c)) /\ exists ff_q_pfp_equivalent_symmetric_newsecondentry. b = ff_q_pfp_equivalent_symmetric_newsecondentry * S ((S (pfrep_position_equivalent_symmetric_newsecond)) * c) + (pfrep_right_equivalent_symmetric_new)))))) \/ (((exists pfrep_gap_equivalent_symmetric_newsecondoutside. pfrep_gap_equivalent_symmetric_newsecondoutside+(L)=(pfrep_power_equivalent_symmetric_new)) /\ (((pfrep_right_equivalent_symmetric_new)=0))))) -> pfrep_left_equivalent_symmetric_new=pfrep_right_equivalent_symmetric_new)

Complete tactic proof in conservative notation

All 21 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

21 script commands · 3 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 assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro L
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro M
  7. L7
    intro he
  8. L8
    intro k
  9. L9
    intro a
  10. L10
    intro r
02Fix variables and assumptionsL11–12

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

  1. L11
    intro ha
  2. L12
    intro hr
03Establish heqL13–21

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

  1. L13
    have heq : r=a
  2. L14
    specialize he (k)
  3. L15
    specialize he (r)
  4. L16
    specialize he (a)
  5. L17
    apply he
  6. L18
    exact hr
  7. L19
    exact ha
  8. L20
    symm
  9. L21
    exact heq

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro d
  5. 0005intro e
  6. 0006intro M
  7. 0007intro he
  8. 0008intro k
  9. 0009intro a
  10. 0010intro r
  11. 0011intro ha
  12. 0012intro hr
  13. 0013have heq : r=a
  14. 0014specialize he (k)
  15. 0015specialize he (r)
  16. 0016specialize he (a)
  17. 0017apply he
  18. 0018exact hr
  19. 0019exact ha
  20. 0020symm
  21. 0021exact heq