PG0036

prime_field_polynomial_common_representatives_same_length

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Two prefixes already at one length serve as their actual common representatives; no new beta encoding or prime premise is needed.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall ab ac bb bc K. (((forall pfrep_power_common_self_left pfrep_left_common_self_left pfrep_right_common_self_left. ((exists pfrep_position_common_self_leftfirst. ((pfrep_position_common_self_leftfirst+S (pfrep_power_common_self_left)=(K)) /\ ((((exists ff_h_pfp_common_self_leftfirstentry. ff_h_pfp_common_self_leftfirstentry + S (pfrep_left_common_self_left) = S ((S (pfrep_position_common_self_leftfirst)) * ac)) /\ exists ff_q_pfp_common_self_leftfirstentry. ab = ff_q_pfp_common_self_leftfirstentry * S ((S (pfrep_position_common_self_leftfirst)) * ac) + (pfrep_left_common_self_left)))))) \/ (((exists pfrep_gap_common_self_leftfirstoutside. pfrep_gap_common_self_leftfirstoutside+(K)=(pfrep_power_common_self_left)) /\ (((pfrep_left_common_self_left)=0))))) -> ((exists pfrep_position_common_self_leftsecond. ((pfrep_position_common_self_leftsecond+S (pfrep_power_common_self_left)=(K)) /\ ((((exists ff_h_pfp_common_self_leftsecondentry. ff_h_pfp_common_self_leftsecondentry + S (pfrep_right_common_self_left) = S ((S (pfrep_position_common_self_leftsecond)) * ac)) /\ exists ff_q_pfp_common_self_leftsecondentry. ab = ff_q_pfp_common_self_leftsecondentry * S ((S (pfrep_position_common_self_leftsecond)) * ac) + (pfrep_right_common_self_left)))))) \/ (((exists pfrep_gap_common_self_leftsecondoutside. pfrep_gap_common_self_leftsecondoutside+(K)=(pfrep_power_common_self_left)) /\ (((pfrep_right_common_self_left)=0))))) -> pfrep_left_common_self_left=pfrep_right_common_self_left) /\ ((forall pfrep_power_common_self_right pfrep_left_common_self_right pfrep_right_common_self_right. ((exists pfrep_position_common_self_rightfirst. ((pfrep_position_common_self_rightfirst+S (pfrep_power_common_self_right)=(K)) /\ ((((exists ff_h_pfp_common_self_rightfirstentry. ff_h_pfp_common_self_rightfirstentry + S (pfrep_left_common_self_right) = S ((S (pfrep_position_common_self_rightfirst)) * bc)) /\ exists ff_q_pfp_common_self_rightfirstentry. bb = ff_q_pfp_common_self_rightfirstentry * S ((S (pfrep_position_common_self_rightfirst)) * bc) + (pfrep_left_common_self_right)))))) \/ (((exists pfrep_gap_common_self_rightfirstoutside. pfrep_gap_common_self_rightfirstoutside+(K)=(pfrep_power_common_self_right)) /\ (((pfrep_left_common_self_right)=0))))) -> ((exists pfrep_position_common_self_rightsecond. ((pfrep_position_common_self_rightsecond+S (pfrep_power_common_self_right)=(K)) /\ ((((exists ff_h_pfp_common_self_rightsecondentry. ff_h_pfp_common_self_rightsecondentry + S (pfrep_right_common_self_right) = S ((S (pfrep_position_common_self_rightsecond)) * bc)) /\ exists ff_q_pfp_common_self_rightsecondentry. bb = ff_q_pfp_common_self_rightsecondentry * S ((S (pfrep_position_common_self_rightsecond)) * bc) + (pfrep_right_common_self_right)))))) \/ (((exists pfrep_gap_common_self_rightsecondoutside. pfrep_gap_common_self_rightsecondoutside+(K)=(pfrep_power_common_self_right)) /\ (((pfrep_right_common_self_right)=0))))) -> pfrep_left_common_self_right=pfrep_right_common_self_right))))

Constructive proof overview

Generated structural guide

Two prefixes already at one length serve as their actual common representatives; no new beta encoding or prime premise is needed.

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v34 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_field_polynomial_power_coefficient_functional Alpha theorem; checked-use authorized

Direct dependents

PG003E prime_field_polynomial_aligned_add_from_fixed

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

14 script commands · 3 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro ab
  2. L2
    intro ac
  3. L3
    intro bb
  4. L4
    intro bc
  5. L5
    intro K
02Separate the logical casesL6–6

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

  1. L6
    split
03Use earlier factsL7–14

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

  1. L7
    specialize prime_field_polynomial_power_coefficient_functional (ab)
  2. L8
    specialize prime_field_polynomial_power_coefficient_functional (ac)
  3. L9
    specialize prime_field_polynomial_power_coefficient_functional (K)
  4. L10
    apply prime_field_polynomial_power_coefficient_functional
  5. L11
    specialize prime_field_polynomial_power_coefficient_functional (bb)
  6. L12
    specialize prime_field_polynomial_power_coefficient_functional (bc)
  7. L13
    specialize prime_field_polynomial_power_coefficient_functional (K)
  8. L14
    apply prime_field_polynomial_power_coefficient_functional

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro bb
  4. 0004intro bc
  5. 0005intro K
  6. 0006split
  7. 0007specialize prime_field_polynomial_power_coefficient_functional (ab)
  8. 0008specialize prime_field_polynomial_power_coefficient_functional (ac)
  9. 0009specialize prime_field_polynomial_power_coefficient_functional (K)
  10. 0010apply prime_field_polynomial_power_coefficient_functional
  11. 0011specialize prime_field_polynomial_power_coefficient_functional (bb)
  12. 0012specialize prime_field_polynomial_power_coefficient_functional (bc)
  13. 0013specialize prime_field_polynomial_power_coefficient_functional (K)
  14. 0014apply prime_field_polynomial_power_coefficient_functional