PQ0012

prime_field_polynomial_subtract_recover_add

Relate the actual subtraction witnesses to the actual aligned B+R=A table; no algebraic identity is assumed.

Alpha v34 checked-use · first admitted v32 · 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 coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ p. ∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ rb. ∀ rc. ∀ l. FpCoefficientSubtraction(p,ab,ac,bb,bc,rb,rc,l) → FpPolyAdd(p,bb,bc,rb,rc,ab,ac,l)

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

Definition DAG

FpAdd(p,a,b,c) · 1FpPolyAdd(p,ab,ac,bb,bc,cb,cc,l) · 1FpCoefficientSubtraction(p,ab,ac,bb,bc,rb,rc,L) · 1BetaAt(b,c,i,x) · 3

Actual proof prerequisites

none
Original expanded first-order statement 
forall p ab ac bb bc rb rc l. (forall pfs_index_sub_bridge_old_recover_add. (exists pfa_gap_sub_bridge_old_recover_addindex. pfa_gap_sub_bridge_old_recover_addindex + S (pfs_index_sub_bridge_old_recover_add) = (l)) -> exists pfs_left_sub_bridge_old_recover_add pfs_right_sub_bridge_old_recover_add pfs_result_sub_bridge_old_recover_add. ((((exists ff_h_pfp_sub_bridge_old_recover_addleft. ff_h_pfp_sub_bridge_old_recover_addleft + S (pfs_left_sub_bridge_old_recover_add) = S ((S (pfs_index_sub_bridge_old_recover_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_old_recover_addleft. ab = ff_q_pfp_sub_bridge_old_recover_addleft * S ((S (pfs_index_sub_bridge_old_recover_add)) * ac) + (pfs_left_sub_bridge_old_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_recover_addright. ff_h_pfp_sub_bridge_old_recover_addright + S (pfs_right_sub_bridge_old_recover_add) = S ((S (pfs_index_sub_bridge_old_recover_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_old_recover_addright. bb = ff_q_pfp_sub_bridge_old_recover_addright * S ((S (pfs_index_sub_bridge_old_recover_add)) * bc) + (pfs_right_sub_bridge_old_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_recover_addresult. ff_h_pfp_sub_bridge_old_recover_addresult + S (pfs_result_sub_bridge_old_recover_add) = S ((S (pfs_index_sub_bridge_old_recover_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_old_recover_addresult. rb = ff_q_pfp_sub_bridge_old_recover_addresult * S ((S (pfs_index_sub_bridge_old_recover_add)) * rc) + (pfs_result_sub_bridge_old_recover_add))) /\ ((((exists pfa_gap_sub_bridge_old_recover_addoperationleft. pfa_gap_sub_bridge_old_recover_addoperationleft + S (pfs_right_sub_bridge_old_recover_add) = (p)) /\ (((exists pfa_gap_sub_bridge_old_recover_addoperationright. pfa_gap_sub_bridge_old_recover_addoperationright + S (pfs_result_sub_bridge_old_recover_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_old_recover_addoperationresultbound. pfa_gap_sub_bridge_old_recover_addoperationresultbound + S (pfs_left_sub_bridge_old_recover_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_old_recover_addoperationresultcongruence pfa_offset_right_sub_bridge_old_recover_addoperationresultcongruence. ((pfs_right_sub_bridge_old_recover_add) + (pfs_result_sub_bridge_old_recover_add)) + (p) * pfa_offset_left_sub_bridge_old_recover_addoperationresultcongruence = (pfs_left_sub_bridge_old_recover_add) + (p) * pfa_offset_right_sub_bridge_old_recover_addoperationresultcongruence)))))))))))))))) -> (forall pfp_index_sub_bridge_new_recover_add. (exists pfa_gap_sub_bridge_new_recover_addindex. pfa_gap_sub_bridge_new_recover_addindex + S (pfp_index_sub_bridge_new_recover_add) = (l)) -> exists pfp_left_sub_bridge_new_recover_add pfp_right_sub_bridge_new_recover_add pfp_value_sub_bridge_new_recover_add. ((((exists ff_h_pfp_sub_bridge_new_recover_addleft. ff_h_pfp_sub_bridge_new_recover_addleft + S (pfp_left_sub_bridge_new_recover_add) = S ((S (pfp_index_sub_bridge_new_recover_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_new_recover_addleft. bb = ff_q_pfp_sub_bridge_new_recover_addleft * S ((S (pfp_index_sub_bridge_new_recover_add)) * bc) + (pfp_left_sub_bridge_new_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_recover_addright. ff_h_pfp_sub_bridge_new_recover_addright + S (pfp_right_sub_bridge_new_recover_add) = S ((S (pfp_index_sub_bridge_new_recover_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_new_recover_addright. rb = ff_q_pfp_sub_bridge_new_recover_addright * S ((S (pfp_index_sub_bridge_new_recover_add)) * rc) + (pfp_right_sub_bridge_new_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_recover_addtarget. ff_h_pfp_sub_bridge_new_recover_addtarget + S (pfp_value_sub_bridge_new_recover_add) = S ((S (pfp_index_sub_bridge_new_recover_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_new_recover_addtarget. ab = ff_q_pfp_sub_bridge_new_recover_addtarget * S ((S (pfp_index_sub_bridge_new_recover_add)) * ac) + (pfp_value_sub_bridge_new_recover_add))) /\ ((((exists pfa_gap_sub_bridge_new_recover_addoperationleft. pfa_gap_sub_bridge_new_recover_addoperationleft + S (pfp_left_sub_bridge_new_recover_add) = (p)) /\ (((exists pfa_gap_sub_bridge_new_recover_addoperationright. pfa_gap_sub_bridge_new_recover_addoperationright + S (pfp_right_sub_bridge_new_recover_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_new_recover_addoperationresultbound. pfa_gap_sub_bridge_new_recover_addoperationresultbound + S (pfp_value_sub_bridge_new_recover_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_new_recover_addoperationresultcongruence pfa_offset_right_sub_bridge_new_recover_addoperationresultcongruence. ((pfp_left_sub_bridge_new_recover_add) + (pfp_right_sub_bridge_new_recover_add)) + (p) * pfa_offset_left_sub_bridge_new_recover_addoperationresultcongruence = (pfp_value_sub_bridge_new_recover_add) + (p) * pfa_offset_right_sub_bridge_new_recover_addoperationresultcongruence))))))))))))))))

Complete tactic proof in conservative notation

All 31 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

31 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.

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 p
  2. L2
    intro ab
  3. L3
    intro ac
  4. L4
    intro bb
  5. L5
    intro bc
  6. L6
    intro rb
  7. L7
    intro rc
  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(rb,rc,i,r) ∧ FpAdd(p,b,r,a)))Definitions: BetaAt(ab,ac,i,a)BetaAt(bb,bc,i,b)BetaAt(rb,rc,i,r)FpAdd(p,b,r,a)Original native command in the exact edition
  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 x2
  3. L24
    exists x
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_right_right_left
10Separate the logical casesL29–29

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

  1. L29
    split
11Use earlier factsL30–31

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

  1. L30
    exact hv_witness_witness_witness_left
  2. L31
    exact hv_witness_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro rb
  7. 0007intro rc
  8. 0008intro l
  9. 0009intro h
  10. 0010intro i
  11. 0011intro hi
  12. 0012have hv : ∃ a. ∃ b. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,b) ∧ (BetaAt(rb,rc,i,r) ∧ FpAdd(p,b,r,a)))
  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 x2
  24. 0024exists x
  25. 0025split
  26. 0026exact hv_witness_witness_witness_right_left
  27. 0027split
  28. 0028exact hv_witness_witness_witness_right_right_left
  29. 0029split
  30. 0030exact hv_witness_witness_witness_left
  31. 0031exact hv_witness_witness_witness_right_right_right