PG0037

prime_field_polynomial_common_representatives_transport

Independent formal recodings of the original inputs preserve the same actual common representatives, without asserting a padding length inequality.

Alpha v34 checked-use · first admitted v34 · 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.

All products and sums use actual beta-coded coefficients; equality is formal coefficient equivalence, not equality of evaluations or raw codes. The zero gcd is included. Uniqueness is up to formal equivalence, not unique Bézout coefficients. This proves the polynomial gcd component, not G091 irreducible-polynomial existence or arbitrary prime-power-field construction.

Exact theorem in conservative defined notation

∀ ab. ∀ ac. ∀ L. ∀ bb. ∀ bc. ∀ M. ∀ ub. ∀ uc. ∀ vb. ∀ vc. ∀ K. ∀ db. ∀ dc. ∀ J. ∀ eb. ∀ ec. ∀ N. PolynomialEquivalent(db,dc,J,ab,ac,L) → PolynomialEquivalent(eb,ec,N,bb,bc,M) → CommonRepresentatives(ab,ac,L,bb,bc,M,ub,uc,vb,vc,K) → CommonRepresentatives(db,dc,J,eb,ec,N,ub,uc,vb,vc,K)

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

Definition DAG

PolynomialEquivalent(b,c,L,d,e,M) · 2CommonRepresentatives(ab,ac,L,bb,bc,M,ub,uc,vb,vc,K) · 2

Actual proof prerequisites

prime_field_polynomial_equivalent_transitive · checked external prerequisite
Original expanded first-order statement 
forall ab ac L bb bc M ub uc vb vc K db dc J eb ec N. (forall pfrep_power_common_transport_left pfrep_left_common_transport_left pfrep_right_common_transport_left. ((exists pfrep_position_common_transport_leftfirst. ((pfrep_position_common_transport_leftfirst+S (pfrep_power_common_transport_left)=(J)) /\ ((((exists ff_h_pfp_common_transport_leftfirstentry. ff_h_pfp_common_transport_leftfirstentry + S (pfrep_left_common_transport_left) = S ((S (pfrep_position_common_transport_leftfirst)) * dc)) /\ exists ff_q_pfp_common_transport_leftfirstentry. db = ff_q_pfp_common_transport_leftfirstentry * S ((S (pfrep_position_common_transport_leftfirst)) * dc) + (pfrep_left_common_transport_left)))))) \/ (((exists pfrep_gap_common_transport_leftfirstoutside. pfrep_gap_common_transport_leftfirstoutside+(J)=(pfrep_power_common_transport_left)) /\ (((pfrep_left_common_transport_left)=0))))) -> ((exists pfrep_position_common_transport_leftsecond. ((pfrep_position_common_transport_leftsecond+S (pfrep_power_common_transport_left)=(L)) /\ ((((exists ff_h_pfp_common_transport_leftsecondentry. ff_h_pfp_common_transport_leftsecondentry + S (pfrep_right_common_transport_left) = S ((S (pfrep_position_common_transport_leftsecond)) * ac)) /\ exists ff_q_pfp_common_transport_leftsecondentry. ab = ff_q_pfp_common_transport_leftsecondentry * S ((S (pfrep_position_common_transport_leftsecond)) * ac) + (pfrep_right_common_transport_left)))))) \/ (((exists pfrep_gap_common_transport_leftsecondoutside. pfrep_gap_common_transport_leftsecondoutside+(L)=(pfrep_power_common_transport_left)) /\ (((pfrep_right_common_transport_left)=0))))) -> pfrep_left_common_transport_left=pfrep_right_common_transport_left) -> (forall pfrep_power_common_transport_right pfrep_left_common_transport_right pfrep_right_common_transport_right. ((exists pfrep_position_common_transport_rightfirst. ((pfrep_position_common_transport_rightfirst+S (pfrep_power_common_transport_right)=(N)) /\ ((((exists ff_h_pfp_common_transport_rightfirstentry. ff_h_pfp_common_transport_rightfirstentry + S (pfrep_left_common_transport_right) = S ((S (pfrep_position_common_transport_rightfirst)) * ec)) /\ exists ff_q_pfp_common_transport_rightfirstentry. eb = ff_q_pfp_common_transport_rightfirstentry * S ((S (pfrep_position_common_transport_rightfirst)) * ec) + (pfrep_left_common_transport_right)))))) \/ (((exists pfrep_gap_common_transport_rightfirstoutside. pfrep_gap_common_transport_rightfirstoutside+(N)=(pfrep_power_common_transport_right)) /\ (((pfrep_left_common_transport_right)=0))))) -> ((exists pfrep_position_common_transport_rightsecond. ((pfrep_position_common_transport_rightsecond+S (pfrep_power_common_transport_right)=(M)) /\ ((((exists ff_h_pfp_common_transport_rightsecondentry. ff_h_pfp_common_transport_rightsecondentry + S (pfrep_right_common_transport_right) = S ((S (pfrep_position_common_transport_rightsecond)) * bc)) /\ exists ff_q_pfp_common_transport_rightsecondentry. bb = ff_q_pfp_common_transport_rightsecondentry * S ((S (pfrep_position_common_transport_rightsecond)) * bc) + (pfrep_right_common_transport_right)))))) \/ (((exists pfrep_gap_common_transport_rightsecondoutside. pfrep_gap_common_transport_rightsecondoutside+(M)=(pfrep_power_common_transport_right)) /\ (((pfrep_right_common_transport_right)=0))))) -> pfrep_left_common_transport_right=pfrep_right_common_transport_right) -> (((forall pfrep_power_common_transport_old_left pfrep_left_common_transport_old_left pfrep_right_common_transport_old_left. ((exists pfrep_position_common_transport_old_leftfirst. ((pfrep_position_common_transport_old_leftfirst+S (pfrep_power_common_transport_old_left)=(L)) /\ ((((exists ff_h_pfp_common_transport_old_leftfirstentry. ff_h_pfp_common_transport_old_leftfirstentry + S (pfrep_left_common_transport_old_left) = S ((S (pfrep_position_common_transport_old_leftfirst)) * ac)) /\ exists ff_q_pfp_common_transport_old_leftfirstentry. ab = ff_q_pfp_common_transport_old_leftfirstentry * S ((S (pfrep_position_common_transport_old_leftfirst)) * ac) + (pfrep_left_common_transport_old_left)))))) \/ (((exists pfrep_gap_common_transport_old_leftfirstoutside. pfrep_gap_common_transport_old_leftfirstoutside+(L)=(pfrep_power_common_transport_old_left)) /\ (((pfrep_left_common_transport_old_left)=0))))) -> ((exists pfrep_position_common_transport_old_leftsecond. ((pfrep_position_common_transport_old_leftsecond+S (pfrep_power_common_transport_old_left)=(K)) /\ ((((exists ff_h_pfp_common_transport_old_leftsecondentry. ff_h_pfp_common_transport_old_leftsecondentry + S (pfrep_right_common_transport_old_left) = S ((S (pfrep_position_common_transport_old_leftsecond)) * uc)) /\ exists ff_q_pfp_common_transport_old_leftsecondentry. ub = ff_q_pfp_common_transport_old_leftsecondentry * S ((S (pfrep_position_common_transport_old_leftsecond)) * uc) + (pfrep_right_common_transport_old_left)))))) \/ (((exists pfrep_gap_common_transport_old_leftsecondoutside. pfrep_gap_common_transport_old_leftsecondoutside+(K)=(pfrep_power_common_transport_old_left)) /\ (((pfrep_right_common_transport_old_left)=0))))) -> pfrep_left_common_transport_old_left=pfrep_right_common_transport_old_left) /\ ((forall pfrep_power_common_transport_old_right pfrep_left_common_transport_old_right pfrep_right_common_transport_old_right. ((exists pfrep_position_common_transport_old_rightfirst. ((pfrep_position_common_transport_old_rightfirst+S (pfrep_power_common_transport_old_right)=(M)) /\ ((((exists ff_h_pfp_common_transport_old_rightfirstentry. ff_h_pfp_common_transport_old_rightfirstentry + S (pfrep_left_common_transport_old_right) = S ((S (pfrep_position_common_transport_old_rightfirst)) * bc)) /\ exists ff_q_pfp_common_transport_old_rightfirstentry. bb = ff_q_pfp_common_transport_old_rightfirstentry * S ((S (pfrep_position_common_transport_old_rightfirst)) * bc) + (pfrep_left_common_transport_old_right)))))) \/ (((exists pfrep_gap_common_transport_old_rightfirstoutside. pfrep_gap_common_transport_old_rightfirstoutside+(M)=(pfrep_power_common_transport_old_right)) /\ (((pfrep_left_common_transport_old_right)=0))))) -> ((exists pfrep_position_common_transport_old_rightsecond. ((pfrep_position_common_transport_old_rightsecond+S (pfrep_power_common_transport_old_right)=(K)) /\ ((((exists ff_h_pfp_common_transport_old_rightsecondentry. ff_h_pfp_common_transport_old_rightsecondentry + S (pfrep_right_common_transport_old_right) = S ((S (pfrep_position_common_transport_old_rightsecond)) * vc)) /\ exists ff_q_pfp_common_transport_old_rightsecondentry. vb = ff_q_pfp_common_transport_old_rightsecondentry * S ((S (pfrep_position_common_transport_old_rightsecond)) * vc) + (pfrep_right_common_transport_old_right)))))) \/ (((exists pfrep_gap_common_transport_old_rightsecondoutside. pfrep_gap_common_transport_old_rightsecondoutside+(K)=(pfrep_power_common_transport_old_right)) /\ (((pfrep_right_common_transport_old_right)=0))))) -> pfrep_left_common_transport_old_right=pfrep_right_common_transport_old_right)))) -> (((forall pfrep_power_common_transport_new_left pfrep_left_common_transport_new_left pfrep_right_common_transport_new_left. ((exists pfrep_position_common_transport_new_leftfirst. ((pfrep_position_common_transport_new_leftfirst+S (pfrep_power_common_transport_new_left)=(J)) /\ ((((exists ff_h_pfp_common_transport_new_leftfirstentry. ff_h_pfp_common_transport_new_leftfirstentry + S (pfrep_left_common_transport_new_left) = S ((S (pfrep_position_common_transport_new_leftfirst)) * dc)) /\ exists ff_q_pfp_common_transport_new_leftfirstentry. db = ff_q_pfp_common_transport_new_leftfirstentry * S ((S (pfrep_position_common_transport_new_leftfirst)) * dc) + (pfrep_left_common_transport_new_left)))))) \/ (((exists pfrep_gap_common_transport_new_leftfirstoutside. pfrep_gap_common_transport_new_leftfirstoutside+(J)=(pfrep_power_common_transport_new_left)) /\ (((pfrep_left_common_transport_new_left)=0))))) -> ((exists pfrep_position_common_transport_new_leftsecond. ((pfrep_position_common_transport_new_leftsecond+S (pfrep_power_common_transport_new_left)=(K)) /\ ((((exists ff_h_pfp_common_transport_new_leftsecondentry. ff_h_pfp_common_transport_new_leftsecondentry + S (pfrep_right_common_transport_new_left) = S ((S (pfrep_position_common_transport_new_leftsecond)) * uc)) /\ exists ff_q_pfp_common_transport_new_leftsecondentry. ub = ff_q_pfp_common_transport_new_leftsecondentry * S ((S (pfrep_position_common_transport_new_leftsecond)) * uc) + (pfrep_right_common_transport_new_left)))))) \/ (((exists pfrep_gap_common_transport_new_leftsecondoutside. pfrep_gap_common_transport_new_leftsecondoutside+(K)=(pfrep_power_common_transport_new_left)) /\ (((pfrep_right_common_transport_new_left)=0))))) -> pfrep_left_common_transport_new_left=pfrep_right_common_transport_new_left) /\ ((forall pfrep_power_common_transport_new_right pfrep_left_common_transport_new_right pfrep_right_common_transport_new_right. ((exists pfrep_position_common_transport_new_rightfirst. ((pfrep_position_common_transport_new_rightfirst+S (pfrep_power_common_transport_new_right)=(N)) /\ ((((exists ff_h_pfp_common_transport_new_rightfirstentry. ff_h_pfp_common_transport_new_rightfirstentry + S (pfrep_left_common_transport_new_right) = S ((S (pfrep_position_common_transport_new_rightfirst)) * ec)) /\ exists ff_q_pfp_common_transport_new_rightfirstentry. eb = ff_q_pfp_common_transport_new_rightfirstentry * S ((S (pfrep_position_common_transport_new_rightfirst)) * ec) + (pfrep_left_common_transport_new_right)))))) \/ (((exists pfrep_gap_common_transport_new_rightfirstoutside. pfrep_gap_common_transport_new_rightfirstoutside+(N)=(pfrep_power_common_transport_new_right)) /\ (((pfrep_left_common_transport_new_right)=0))))) -> ((exists pfrep_position_common_transport_new_rightsecond. ((pfrep_position_common_transport_new_rightsecond+S (pfrep_power_common_transport_new_right)=(K)) /\ ((((exists ff_h_pfp_common_transport_new_rightsecondentry. ff_h_pfp_common_transport_new_rightsecondentry + S (pfrep_right_common_transport_new_right) = S ((S (pfrep_position_common_transport_new_rightsecond)) * vc)) /\ exists ff_q_pfp_common_transport_new_rightsecondentry. vb = ff_q_pfp_common_transport_new_rightsecondentry * S ((S (pfrep_position_common_transport_new_rightsecond)) * vc) + (pfrep_right_common_transport_new_right)))))) \/ (((exists pfrep_gap_common_transport_new_rightsecondoutside. pfrep_gap_common_transport_new_rightsecondoutside+(K)=(pfrep_power_common_transport_new_right)) /\ (((pfrep_right_common_transport_new_right)=0))))) -> pfrep_left_common_transport_new_right=pfrep_right_common_transport_new_right))))

Complete tactic proof in conservative notation

All 46 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

46 script commands · 6 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro ab
  2. L2
    intro ac
  3. L3
    intro L
  4. L4
    intro bb
  5. L5
    intro bc
  6. L6
    intro M
  7. L7
    intro ub
  8. L8
    intro uc
  9. L9
    intro vb
  10. L10
    intro vc
02Fix variables and assumptionsL11–20

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

  1. L11
    intro K
  2. L12
    intro db
  3. L13
    intro dc
  4. L14
    intro J
  5. L15
    intro eb
  6. L16
    intro ec
  7. L17
    intro N
  8. L18
    intro ha
  9. L19
    intro hb
  10. L20
    intro h
03Separate the logical casesL21–22

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

  1. L21
    cases h
  2. L22
    split
04Use earlier factsL23–32

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

  1. L23
    specialize prime_field_polynomial_equivalent_transitive (db)
  2. L24
    specialize prime_field_polynomial_equivalent_transitive (dc)
  3. L25
    specialize prime_field_polynomial_equivalent_transitive (J)
  4. L26
    specialize prime_field_polynomial_equivalent_transitive (ab)
  5. L27
    specialize prime_field_polynomial_equivalent_transitive (ac)
  6. L28
    specialize prime_field_polynomial_equivalent_transitive (L)
  7. L29
    specialize prime_field_polynomial_equivalent_transitive (ub)
  8. L30
    specialize prime_field_polynomial_equivalent_transitive (uc)
  9. L31
    specialize prime_field_polynomial_equivalent_transitive (K)
  10. L32
    apply prime_field_polynomial_equivalent_transitive
05Use earlier factsL33–42

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

  1. L33
    exact ha
  2. L34
    exact h_left
  3. L35
    specialize prime_field_polynomial_equivalent_transitive (eb)
  4. L36
    specialize prime_field_polynomial_equivalent_transitive (ec)
  5. L37
    specialize prime_field_polynomial_equivalent_transitive (N)
  6. L38
    specialize prime_field_polynomial_equivalent_transitive (bb)
  7. L39
    specialize prime_field_polynomial_equivalent_transitive (bc)
  8. L40
    specialize prime_field_polynomial_equivalent_transitive (M)
  9. L41
    specialize prime_field_polynomial_equivalent_transitive (vb)
  10. L42
    specialize prime_field_polynomial_equivalent_transitive (vc)
06Use earlier factsL43–46

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

  1. L43
    specialize prime_field_polynomial_equivalent_transitive (K)
  2. L44
    apply prime_field_polynomial_equivalent_transitive
  3. L45
    exact hb
  4. L46
    exact h_right

Library-wide reading audit

Original defined command ledger · 46 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro L
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro M
  7. 0007intro ub
  8. 0008intro uc
  9. 0009intro vb
  10. 0010intro vc
  11. 0011intro K
  12. 0012intro db
  13. 0013intro dc
  14. 0014intro J
  15. 0015intro eb
  16. 0016intro ec
  17. 0017intro N
  18. 0018intro ha
  19. 0019intro hb
  20. 0020intro h
  21. 0021cases h
  22. 0022split
  23. 0023specialize prime_field_polynomial_equivalent_transitive (db)
  24. 0024specialize prime_field_polynomial_equivalent_transitive (dc)
  25. 0025specialize prime_field_polynomial_equivalent_transitive (J)
  26. 0026specialize prime_field_polynomial_equivalent_transitive (ab)
  27. 0027specialize prime_field_polynomial_equivalent_transitive (ac)
  28. 0028specialize prime_field_polynomial_equivalent_transitive (L)
  29. 0029specialize prime_field_polynomial_equivalent_transitive (ub)
  30. 0030specialize prime_field_polynomial_equivalent_transitive (uc)
  31. 0031specialize prime_field_polynomial_equivalent_transitive (K)
  32. 0032apply prime_field_polynomial_equivalent_transitive
  33. 0033exact ha
  34. 0034exact h_left
  35. 0035specialize prime_field_polynomial_equivalent_transitive (eb)
  36. 0036specialize prime_field_polynomial_equivalent_transitive (ec)
  37. 0037specialize prime_field_polynomial_equivalent_transitive (N)
  38. 0038specialize prime_field_polynomial_equivalent_transitive (bb)
  39. 0039specialize prime_field_polynomial_equivalent_transitive (bc)
  40. 0040specialize prime_field_polynomial_equivalent_transitive (M)
  41. 0041specialize prime_field_polynomial_equivalent_transitive (vb)
  42. 0042specialize prime_field_polynomial_equivalent_transitive (vc)
  43. 0043specialize prime_field_polynomial_equivalent_transitive (K)
  44. 0044apply prime_field_polynomial_equivalent_transitive
  45. 0045exact hb
  46. 0046exact h_right