PG003B

prime_field_polynomial_common_representatives_symmetric

Swapping the two original inputs and their actual representatives preserves the common-representation graph.

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. CommonRepresentatives(ab,ac,L,bb,bc,M,ub,uc,vb,vc,K) → CommonRepresentatives(bb,bc,M,ab,ac,L,vb,vc,ub,uc,K)

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

Definition DAG

CommonRepresentatives(ab,ac,L,bb,bc,M,ub,uc,vb,vc,K) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall ab ac L bb bc M ub uc vb vc K. (((forall pfrep_power_common_symmetric_old_left pfrep_left_common_symmetric_old_left pfrep_right_common_symmetric_old_left. ((exists pfrep_position_common_symmetric_old_leftfirst. ((pfrep_position_common_symmetric_old_leftfirst+S (pfrep_power_common_symmetric_old_left)=(L)) /\ ((((exists ff_h_pfp_common_symmetric_old_leftfirstentry. ff_h_pfp_common_symmetric_old_leftfirstentry + S (pfrep_left_common_symmetric_old_left) = S ((S (pfrep_position_common_symmetric_old_leftfirst)) * ac)) /\ exists ff_q_pfp_common_symmetric_old_leftfirstentry. ab = ff_q_pfp_common_symmetric_old_leftfirstentry * S ((S (pfrep_position_common_symmetric_old_leftfirst)) * ac) + (pfrep_left_common_symmetric_old_left)))))) \/ (((exists pfrep_gap_common_symmetric_old_leftfirstoutside. pfrep_gap_common_symmetric_old_leftfirstoutside+(L)=(pfrep_power_common_symmetric_old_left)) /\ (((pfrep_left_common_symmetric_old_left)=0))))) -> ((exists pfrep_position_common_symmetric_old_leftsecond. ((pfrep_position_common_symmetric_old_leftsecond+S (pfrep_power_common_symmetric_old_left)=(K)) /\ ((((exists ff_h_pfp_common_symmetric_old_leftsecondentry. ff_h_pfp_common_symmetric_old_leftsecondentry + S (pfrep_right_common_symmetric_old_left) = S ((S (pfrep_position_common_symmetric_old_leftsecond)) * uc)) /\ exists ff_q_pfp_common_symmetric_old_leftsecondentry. ub = ff_q_pfp_common_symmetric_old_leftsecondentry * S ((S (pfrep_position_common_symmetric_old_leftsecond)) * uc) + (pfrep_right_common_symmetric_old_left)))))) \/ (((exists pfrep_gap_common_symmetric_old_leftsecondoutside. pfrep_gap_common_symmetric_old_leftsecondoutside+(K)=(pfrep_power_common_symmetric_old_left)) /\ (((pfrep_right_common_symmetric_old_left)=0))))) -> pfrep_left_common_symmetric_old_left=pfrep_right_common_symmetric_old_left) /\ ((forall pfrep_power_common_symmetric_old_right pfrep_left_common_symmetric_old_right pfrep_right_common_symmetric_old_right. ((exists pfrep_position_common_symmetric_old_rightfirst. ((pfrep_position_common_symmetric_old_rightfirst+S (pfrep_power_common_symmetric_old_right)=(M)) /\ ((((exists ff_h_pfp_common_symmetric_old_rightfirstentry. ff_h_pfp_common_symmetric_old_rightfirstentry + S (pfrep_left_common_symmetric_old_right) = S ((S (pfrep_position_common_symmetric_old_rightfirst)) * bc)) /\ exists ff_q_pfp_common_symmetric_old_rightfirstentry. bb = ff_q_pfp_common_symmetric_old_rightfirstentry * S ((S (pfrep_position_common_symmetric_old_rightfirst)) * bc) + (pfrep_left_common_symmetric_old_right)))))) \/ (((exists pfrep_gap_common_symmetric_old_rightfirstoutside. pfrep_gap_common_symmetric_old_rightfirstoutside+(M)=(pfrep_power_common_symmetric_old_right)) /\ (((pfrep_left_common_symmetric_old_right)=0))))) -> ((exists pfrep_position_common_symmetric_old_rightsecond. ((pfrep_position_common_symmetric_old_rightsecond+S (pfrep_power_common_symmetric_old_right)=(K)) /\ ((((exists ff_h_pfp_common_symmetric_old_rightsecondentry. ff_h_pfp_common_symmetric_old_rightsecondentry + S (pfrep_right_common_symmetric_old_right) = S ((S (pfrep_position_common_symmetric_old_rightsecond)) * vc)) /\ exists ff_q_pfp_common_symmetric_old_rightsecondentry. vb = ff_q_pfp_common_symmetric_old_rightsecondentry * S ((S (pfrep_position_common_symmetric_old_rightsecond)) * vc) + (pfrep_right_common_symmetric_old_right)))))) \/ (((exists pfrep_gap_common_symmetric_old_rightsecondoutside. pfrep_gap_common_symmetric_old_rightsecondoutside+(K)=(pfrep_power_common_symmetric_old_right)) /\ (((pfrep_right_common_symmetric_old_right)=0))))) -> pfrep_left_common_symmetric_old_right=pfrep_right_common_symmetric_old_right)))) -> (((forall pfrep_power_common_symmetric_new_left pfrep_left_common_symmetric_new_left pfrep_right_common_symmetric_new_left. ((exists pfrep_position_common_symmetric_new_leftfirst. ((pfrep_position_common_symmetric_new_leftfirst+S (pfrep_power_common_symmetric_new_left)=(M)) /\ ((((exists ff_h_pfp_common_symmetric_new_leftfirstentry. ff_h_pfp_common_symmetric_new_leftfirstentry + S (pfrep_left_common_symmetric_new_left) = S ((S (pfrep_position_common_symmetric_new_leftfirst)) * bc)) /\ exists ff_q_pfp_common_symmetric_new_leftfirstentry. bb = ff_q_pfp_common_symmetric_new_leftfirstentry * S ((S (pfrep_position_common_symmetric_new_leftfirst)) * bc) + (pfrep_left_common_symmetric_new_left)))))) \/ (((exists pfrep_gap_common_symmetric_new_leftfirstoutside. pfrep_gap_common_symmetric_new_leftfirstoutside+(M)=(pfrep_power_common_symmetric_new_left)) /\ (((pfrep_left_common_symmetric_new_left)=0))))) -> ((exists pfrep_position_common_symmetric_new_leftsecond. ((pfrep_position_common_symmetric_new_leftsecond+S (pfrep_power_common_symmetric_new_left)=(K)) /\ ((((exists ff_h_pfp_common_symmetric_new_leftsecondentry. ff_h_pfp_common_symmetric_new_leftsecondentry + S (pfrep_right_common_symmetric_new_left) = S ((S (pfrep_position_common_symmetric_new_leftsecond)) * vc)) /\ exists ff_q_pfp_common_symmetric_new_leftsecondentry. vb = ff_q_pfp_common_symmetric_new_leftsecondentry * S ((S (pfrep_position_common_symmetric_new_leftsecond)) * vc) + (pfrep_right_common_symmetric_new_left)))))) \/ (((exists pfrep_gap_common_symmetric_new_leftsecondoutside. pfrep_gap_common_symmetric_new_leftsecondoutside+(K)=(pfrep_power_common_symmetric_new_left)) /\ (((pfrep_right_common_symmetric_new_left)=0))))) -> pfrep_left_common_symmetric_new_left=pfrep_right_common_symmetric_new_left) /\ ((forall pfrep_power_common_symmetric_new_right pfrep_left_common_symmetric_new_right pfrep_right_common_symmetric_new_right. ((exists pfrep_position_common_symmetric_new_rightfirst. ((pfrep_position_common_symmetric_new_rightfirst+S (pfrep_power_common_symmetric_new_right)=(L)) /\ ((((exists ff_h_pfp_common_symmetric_new_rightfirstentry. ff_h_pfp_common_symmetric_new_rightfirstentry + S (pfrep_left_common_symmetric_new_right) = S ((S (pfrep_position_common_symmetric_new_rightfirst)) * ac)) /\ exists ff_q_pfp_common_symmetric_new_rightfirstentry. ab = ff_q_pfp_common_symmetric_new_rightfirstentry * S ((S (pfrep_position_common_symmetric_new_rightfirst)) * ac) + (pfrep_left_common_symmetric_new_right)))))) \/ (((exists pfrep_gap_common_symmetric_new_rightfirstoutside. pfrep_gap_common_symmetric_new_rightfirstoutside+(L)=(pfrep_power_common_symmetric_new_right)) /\ (((pfrep_left_common_symmetric_new_right)=0))))) -> ((exists pfrep_position_common_symmetric_new_rightsecond. ((pfrep_position_common_symmetric_new_rightsecond+S (pfrep_power_common_symmetric_new_right)=(K)) /\ ((((exists ff_h_pfp_common_symmetric_new_rightsecondentry. ff_h_pfp_common_symmetric_new_rightsecondentry + S (pfrep_right_common_symmetric_new_right) = S ((S (pfrep_position_common_symmetric_new_rightsecond)) * uc)) /\ exists ff_q_pfp_common_symmetric_new_rightsecondentry. ub = ff_q_pfp_common_symmetric_new_rightsecondentry * S ((S (pfrep_position_common_symmetric_new_rightsecond)) * uc) + (pfrep_right_common_symmetric_new_right)))))) \/ (((exists pfrep_gap_common_symmetric_new_rightsecondoutside. pfrep_gap_common_symmetric_new_rightsecondoutside+(K)=(pfrep_power_common_symmetric_new_right)) /\ (((pfrep_right_common_symmetric_new_right)=0))))) -> pfrep_left_common_symmetric_new_right=pfrep_right_common_symmetric_new_right))))

Complete tactic proof in conservative notation

All 16 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

16 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.

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 assumptionsL11–12

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

  1. L11
    intro K
  2. L12
    intro h
03Separate the logical casesL13–14

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

  1. L13
    cases h
  2. L14
    split
04Use earlier factsL15–16

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

  1. L15
    exact h_right
  2. L16
    exact h_left

Library-wide reading audit

Original defined command ledger · 16 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 h
  13. 0013cases h
  14. 0014split
  15. 0015exact h_right
  16. 0016exact h_left