PP0012

prime_field_polynomial_add_commutative

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Coefficient addition commutes for actual finite tables, including the empty table.

Exact expanded first-order arithmetic statement

forall p ab ac bb bc cb cc l. (forall pfp_index_add_comm_old. (exists pfa_gap_add_comm_oldindex. pfa_gap_add_comm_oldindex + S (pfp_index_add_comm_old) = (l)) -> exists pfp_left_add_comm_old pfp_right_add_comm_old pfp_value_add_comm_old. ((((exists ff_h_pfp_add_comm_oldleft. ff_h_pfp_add_comm_oldleft + S (pfp_left_add_comm_old) = S ((S (pfp_index_add_comm_old)) * ac)) /\ exists ff_q_pfp_add_comm_oldleft. ab = ff_q_pfp_add_comm_oldleft * S ((S (pfp_index_add_comm_old)) * ac) + (pfp_left_add_comm_old))) /\ (((((exists ff_h_pfp_add_comm_oldright. ff_h_pfp_add_comm_oldright + S (pfp_right_add_comm_old) = S ((S (pfp_index_add_comm_old)) * bc)) /\ exists ff_q_pfp_add_comm_oldright. bb = ff_q_pfp_add_comm_oldright * S ((S (pfp_index_add_comm_old)) * bc) + (pfp_right_add_comm_old))) /\ (((((exists ff_h_pfp_add_comm_oldtarget. ff_h_pfp_add_comm_oldtarget + S (pfp_value_add_comm_old) = S ((S (pfp_index_add_comm_old)) * cc)) /\ exists ff_q_pfp_add_comm_oldtarget. cb = ff_q_pfp_add_comm_oldtarget * S ((S (pfp_index_add_comm_old)) * cc) + (pfp_value_add_comm_old))) /\ ((((exists pfa_gap_add_comm_oldoperationleft. pfa_gap_add_comm_oldoperationleft + S (pfp_left_add_comm_old) = (p)) /\ (((exists pfa_gap_add_comm_oldoperationright. pfa_gap_add_comm_oldoperationright + S (pfp_right_add_comm_old) = (p)) /\ ((((exists pfa_gap_add_comm_oldoperationresultbound. pfa_gap_add_comm_oldoperationresultbound + S (pfp_value_add_comm_old) = (p)) /\ ((exists pfa_offset_left_add_comm_oldoperationresultcongruence pfa_offset_right_add_comm_oldoperationresultcongruence. ((pfp_left_add_comm_old) + (pfp_right_add_comm_old)) + (p) * pfa_offset_left_add_comm_oldoperationresultcongruence = (pfp_value_add_comm_old) + (p) * pfa_offset_right_add_comm_oldoperationresultcongruence)))))))))))))))) -> (forall pfp_index_add_comm_new. (exists pfa_gap_add_comm_newindex. pfa_gap_add_comm_newindex + S (pfp_index_add_comm_new) = (l)) -> exists pfp_left_add_comm_new pfp_right_add_comm_new pfp_value_add_comm_new. ((((exists ff_h_pfp_add_comm_newleft. ff_h_pfp_add_comm_newleft + S (pfp_left_add_comm_new) = S ((S (pfp_index_add_comm_new)) * bc)) /\ exists ff_q_pfp_add_comm_newleft. bb = ff_q_pfp_add_comm_newleft * S ((S (pfp_index_add_comm_new)) * bc) + (pfp_left_add_comm_new))) /\ (((((exists ff_h_pfp_add_comm_newright. ff_h_pfp_add_comm_newright + S (pfp_right_add_comm_new) = S ((S (pfp_index_add_comm_new)) * ac)) /\ exists ff_q_pfp_add_comm_newright. ab = ff_q_pfp_add_comm_newright * S ((S (pfp_index_add_comm_new)) * ac) + (pfp_right_add_comm_new))) /\ (((((exists ff_h_pfp_add_comm_newtarget. ff_h_pfp_add_comm_newtarget + S (pfp_value_add_comm_new) = S ((S (pfp_index_add_comm_new)) * cc)) /\ exists ff_q_pfp_add_comm_newtarget. cb = ff_q_pfp_add_comm_newtarget * S ((S (pfp_index_add_comm_new)) * cc) + (pfp_value_add_comm_new))) /\ ((((exists pfa_gap_add_comm_newoperationleft. pfa_gap_add_comm_newoperationleft + S (pfp_left_add_comm_new) = (p)) /\ (((exists pfa_gap_add_comm_newoperationright. pfa_gap_add_comm_newoperationright + S (pfp_right_add_comm_new) = (p)) /\ ((((exists pfa_gap_add_comm_newoperationresultbound. pfa_gap_add_comm_newoperationresultbound + S (pfp_value_add_comm_new) = (p)) /\ ((exists pfa_offset_left_add_comm_newoperationresultcongruence pfa_offset_right_add_comm_newoperationresultcongruence. ((pfp_left_add_comm_new) + (pfp_right_add_comm_new)) + (p) * pfa_offset_left_add_comm_newoperationresultcongruence = (pfp_value_add_comm_new) + (p) * pfa_offset_right_add_comm_newoperationresultcongruence))))))))))))))))

Constructive proof overview

Generated structural guide

Coefficient addition commutes for actual finite tables, including the empty table.

The unchanged tactic script uses 1 declared prerequisite and contains 36 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

prime_field_add_commutative external actual_inherited_body_freshly_checked_in_complete_bundle; no checked-use authority

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

Read the argument

Proof checkpoints

36 script commands · 11 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro ab
  3. L3
    intro ac
  4. L4
    intro bb
  5. L5
    intro bc
  6. L6
    intro cb
  7. L7
    intro cc
  8. L8
    intro l
  9. L9
    intro h
  10. L10
    intro i
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hi
03Establish hvL12–15

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

  1. L12
    have hv : ∃ a. ∃ b. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,b) ∧ (BetaAt(cb,cc,i,r) ∧ FpAdd(p,a,b,r)))Definitions: FpAddBetaAt
  2. L13
    specialize h (i)
  3. L14
    apply h
  4. L15
    exact hi
04Separate the logical casesL16–21

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

  1. L16
    cases hv
  2. L17
    cases hv_witness
  3. L18
    cases hv_witness_witness
  4. L19
    cases hv_witness_witness_witness
  5. L20
    cases hv_witness_witness_witness_right
  6. L21
    cases hv_witness_witness_witness_right_right
05Construct an explicit witnessL22–24

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

  1. L22
    exists x1
  2. L23
    exists x
  3. L24
    exists x2
06Separate the logical casesL25–25

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

  1. L25
    split
07Use earlier factsL26–26

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

  1. L26
    exact hv_witness_witness_witness_right_left
08Separate the logical casesL27–27

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

  1. L27
    split
09Use earlier factsL28–28

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

  1. L28
    exact hv_witness_witness_witness_left
10Separate the logical casesL29–29

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

  1. L29
    split
11Use earlier factsL30–36

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

  1. L30
    exact hv_witness_witness_witness_right_right_left
  2. L31
    specialize prime_field_add_commutative (p)
  3. L32
    specialize prime_field_add_commutative (x)
  4. L33
    specialize prime_field_add_commutative (x1)
  5. L34
    specialize prime_field_add_commutative (x2)
  6. L35
    apply prime_field_add_commutative
  7. L36
    exact hv_witness_witness_witness_right_right_right

Library-wide reading audit

Original exact command ledger · 36 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro cb
  7. 0007intro cc
  8. 0008intro l
  9. 0009intro h
  10. 0010intro i
  11. 0011intro hi
  12. 0012have hv : exists a b r. ((((exists ff_h_pfp_add_comm_chosen_a. ff_h_pfp_add_comm_chosen_a + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_add_comm_chosen_a. ab = ff_q_pfp_add_comm_chosen_a * S ((S (i)) * ac) + (a))) /\ (((((exists ff_h_pfp_add_comm_chosen_b. ff_h_pfp_add_comm_chosen_b + S (b) = S ((S (i)) * bc)) /\ exists ff_q_pfp_add_comm_chosen_b. bb = ff_q_pfp_add_comm_chosen_b * S ((S (i)) * bc) + (b))) /\ (((((exists ff_h_pfp_add_comm_chosen_r. ff_h_pfp_add_comm_chosen_r + S (r) = S ((S (i)) * cc)) /\ exists ff_q_pfp_add_comm_chosen_r. cb = ff_q_pfp_add_comm_chosen_r * S ((S (i)) * cc) + (r))) /\ ((((exists pfa_gap_add_comm_operationleft. pfa_gap_add_comm_operationleft + S (a) = (p)) /\ (((exists pfa_gap_add_comm_operationright. pfa_gap_add_comm_operationright + S (b) = (p)) /\ ((((exists pfa_gap_add_comm_operationresultbound. pfa_gap_add_comm_operationresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_add_comm_operationresultcongruence pfa_offset_right_add_comm_operationresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_add_comm_operationresultcongruence = (r) + (p) * pfa_offset_right_add_comm_operationresultcongruence)))))))))))))))
  13. 0013specialize h (i)
  14. 0014apply h
  15. 0015exact hi
  16. 0016cases hv
  17. 0017cases hv_witness
  18. 0018cases hv_witness_witness
  19. 0019cases hv_witness_witness_witness
  20. 0020cases hv_witness_witness_witness_right
  21. 0021cases hv_witness_witness_witness_right_right
  22. 0022exists x1
  23. 0023exists x
  24. 0024exists x2
  25. 0025split
  26. 0026exact hv_witness_witness_witness_right_left
  27. 0027split
  28. 0028exact hv_witness_witness_witness_left
  29. 0029split
  30. 0030exact hv_witness_witness_witness_right_right_left
  31. 0031specialize prime_field_add_commutative (p)
  32. 0032specialize prime_field_add_commutative (x)
  33. 0033specialize prime_field_add_commutative (x1)
  34. 0034specialize prime_field_add_commutative (x2)
  35. 0035apply prime_field_add_commutative
  36. 0036exact hv_witness_witness_witness_right_right_right