PX0012

prime_field_polynomial_equivalent_implies_equal_same_length

At a common annotated length, formal coefficient equivalence gives the exact inherited decoded-prefix equality.

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. PolynomialEquivalent(b,c,L,d,e,L) → BetaPrefixEqual(b,c,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) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

add_comm · checked external prerequisitebeta_at_exists · checked external prerequisite
Original expanded first-order statement 
forall b c d e L. (forall pfrep_power_equivalent_to_equal_input pfrep_left_equivalent_to_equal_input pfrep_right_equivalent_to_equal_input. ((exists pfrep_position_equivalent_to_equal_inputfirst. ((pfrep_position_equivalent_to_equal_inputfirst+S (pfrep_power_equivalent_to_equal_input)=(L)) /\ ((((exists ff_h_pfp_equivalent_to_equal_inputfirstentry. ff_h_pfp_equivalent_to_equal_inputfirstentry + S (pfrep_left_equivalent_to_equal_input) = S ((S (pfrep_position_equivalent_to_equal_inputfirst)) * c)) /\ exists ff_q_pfp_equivalent_to_equal_inputfirstentry. b = ff_q_pfp_equivalent_to_equal_inputfirstentry * S ((S (pfrep_position_equivalent_to_equal_inputfirst)) * c) + (pfrep_left_equivalent_to_equal_input)))))) \/ (((exists pfrep_gap_equivalent_to_equal_inputfirstoutside. pfrep_gap_equivalent_to_equal_inputfirstoutside+(L)=(pfrep_power_equivalent_to_equal_input)) /\ (((pfrep_left_equivalent_to_equal_input)=0))))) -> ((exists pfrep_position_equivalent_to_equal_inputsecond. ((pfrep_position_equivalent_to_equal_inputsecond+S (pfrep_power_equivalent_to_equal_input)=(L)) /\ ((((exists ff_h_pfp_equivalent_to_equal_inputsecondentry. ff_h_pfp_equivalent_to_equal_inputsecondentry + S (pfrep_right_equivalent_to_equal_input) = S ((S (pfrep_position_equivalent_to_equal_inputsecond)) * e)) /\ exists ff_q_pfp_equivalent_to_equal_inputsecondentry. d = ff_q_pfp_equivalent_to_equal_inputsecondentry * S ((S (pfrep_position_equivalent_to_equal_inputsecond)) * e) + (pfrep_right_equivalent_to_equal_input)))))) \/ (((exists pfrep_gap_equivalent_to_equal_inputsecondoutside. pfrep_gap_equivalent_to_equal_inputsecondoutside+(L)=(pfrep_power_equivalent_to_equal_input)) /\ (((pfrep_right_equivalent_to_equal_input)=0))))) -> pfrep_left_equivalent_to_equal_input=pfrep_right_equivalent_to_equal_input) -> (forall mdr_i_pfp_equivalent_to_equal_result mdr_a_pfp_equivalent_to_equal_result. (exists mdr_gap_pfp_equivalent_to_equal_resultb. mdr_gap_pfp_equivalent_to_equal_resultb + S (mdr_i_pfp_equivalent_to_equal_result) = (L)) -> (((exists ff_h_mdr_pfp_equivalent_to_equal_resulto. ff_h_mdr_pfp_equivalent_to_equal_resulto + S (mdr_a_pfp_equivalent_to_equal_result) = S ((S (mdr_i_pfp_equivalent_to_equal_result)) * c)) /\ exists ff_q_mdr_pfp_equivalent_to_equal_resulto. b = ff_q_mdr_pfp_equivalent_to_equal_resulto * S ((S (mdr_i_pfp_equivalent_to_equal_result)) * c) + (mdr_a_pfp_equivalent_to_equal_result))) -> (((exists ff_h_mdr_pfp_equivalent_to_equal_resultn. ff_h_mdr_pfp_equivalent_to_equal_resultn + S (mdr_a_pfp_equivalent_to_equal_result) = S ((S (mdr_i_pfp_equivalent_to_equal_result)) * e)) /\ exists ff_q_mdr_pfp_equivalent_to_equal_resultn. d = ff_q_mdr_pfp_equivalent_to_equal_resultn * S ((S (mdr_i_pfp_equivalent_to_equal_result)) * e) + (mdr_a_pfp_equivalent_to_equal_result))))

Complete tactic proof in conservative notation

All 39 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

39 script commands · 16 reading checkpoints · 3 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 d
  4. L4
    intro e
  5. L5
    intro L
  6. L6
    intro he
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro hi
  10. L10
    intro ha
02Separate the logical casesL11–11

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

  1. L11
    cases hi
03Establish hsL12–15

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have hs : i+S x=L
  2. L13
    trans x+S i
  3. L14
    simp [add_comm]
  4. L15
    exact hi_witness
04Establish hvL16–20

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

  1. L16
    have hv : ∃ r. BetaAt(d,e,i,r)Definitions: BetaAt(d,e,i,r)Original native command in the exact edition
  2. L17
    specialize beta_at_exists (d)
  3. L18
    specialize beta_at_exists (e)
  4. L19
    specialize beta_at_exists (i)
  5. L20
    apply beta_at_exists
05Separate the logical casesL21–21

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

  1. L21
    cases hv
06Establish heqL22–26

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

  1. L22
    have heq : a=x1
  2. L23
    specialize he (x)
  3. L24
    specialize he (a)
  4. L25
    specialize he (x1)
  5. L26
    apply he
07Separate the logical casesL27–27

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

  1. L27
    left
08Construct an explicit witnessL28–28

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

  1. L28
    exists i
09Separate the logical casesL29–29

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

  1. L29
    split
10Use earlier factsL30–31

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

  1. L30
    exact hs
  2. L31
    exact ha
11Separate the logical casesL32–32

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

  1. L32
    left
12Construct an explicit witnessL33–33

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

  1. L33
    exists i
13Separate the logical casesL34–34

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

  1. L34
    split
14Use earlier factsL35–36

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

  1. L35
    exact hs
  2. L36
    exact hv_witness
15Calculate and transport equalitiesL37–38

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L37
    rewrite heq
  2. L38
    rewrite heq
16Use earlier factsL39–39

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

  1. L39
    exact hv_witness

Library-wide reading audit

Original defined command ledger · 39 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro L
  6. 0006intro he
  7. 0007intro i
  8. 0008intro a
  9. 0009intro hi
  10. 0010intro ha
  11. 0011cases hi
  12. 0012have hs : i+S x=L
  13. 0013trans x+S i
  14. 0014simp [add_comm]
  15. 0015exact hi_witness
  16. 0016have hv : ∃ r. BetaAt(d,e,i,r)
  17. 0017specialize beta_at_exists (d)
  18. 0018specialize beta_at_exists (e)
  19. 0019specialize beta_at_exists (i)
  20. 0020apply beta_at_exists
  21. 0021cases hv
  22. 0022have heq : a=x1
  23. 0023specialize he (x)
  24. 0024specialize he (a)
  25. 0025specialize he (x1)
  26. 0026apply he
  27. 0027left
  28. 0028exists i
  29. 0029split
  30. 0030exact hs
  31. 0031exact ha
  32. 0032left
  33. 0033exists i
  34. 0034split
  35. 0035exact hs
  36. 0036exact hv_witness
  37. 0037rewrite heq
  38. 0038rewrite heq
  39. 0039exact hv_witness