PQ0013

prime_field_polynomial_subtract_from_add

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

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

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 p ab ac bb bc rb rc l. (forall pfp_index_sub_bridge_old_from_add. (exists pfa_gap_sub_bridge_old_from_addindex. pfa_gap_sub_bridge_old_from_addindex + S (pfp_index_sub_bridge_old_from_add) = (l)) -> exists pfp_left_sub_bridge_old_from_add pfp_right_sub_bridge_old_from_add pfp_value_sub_bridge_old_from_add. ((((exists ff_h_pfp_sub_bridge_old_from_addleft. ff_h_pfp_sub_bridge_old_from_addleft + S (pfp_left_sub_bridge_old_from_add) = S ((S (pfp_index_sub_bridge_old_from_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_old_from_addleft. bb = ff_q_pfp_sub_bridge_old_from_addleft * S ((S (pfp_index_sub_bridge_old_from_add)) * bc) + (pfp_left_sub_bridge_old_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_from_addright. ff_h_pfp_sub_bridge_old_from_addright + S (pfp_right_sub_bridge_old_from_add) = S ((S (pfp_index_sub_bridge_old_from_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_old_from_addright. rb = ff_q_pfp_sub_bridge_old_from_addright * S ((S (pfp_index_sub_bridge_old_from_add)) * rc) + (pfp_right_sub_bridge_old_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_from_addtarget. ff_h_pfp_sub_bridge_old_from_addtarget + S (pfp_value_sub_bridge_old_from_add) = S ((S (pfp_index_sub_bridge_old_from_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_old_from_addtarget. ab = ff_q_pfp_sub_bridge_old_from_addtarget * S ((S (pfp_index_sub_bridge_old_from_add)) * ac) + (pfp_value_sub_bridge_old_from_add))) /\ ((((exists pfa_gap_sub_bridge_old_from_addoperationleft. pfa_gap_sub_bridge_old_from_addoperationleft + S (pfp_left_sub_bridge_old_from_add) = (p)) /\ (((exists pfa_gap_sub_bridge_old_from_addoperationright. pfa_gap_sub_bridge_old_from_addoperationright + S (pfp_right_sub_bridge_old_from_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_old_from_addoperationresultbound. pfa_gap_sub_bridge_old_from_addoperationresultbound + S (pfp_value_sub_bridge_old_from_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_old_from_addoperationresultcongruence pfa_offset_right_sub_bridge_old_from_addoperationresultcongruence. ((pfp_left_sub_bridge_old_from_add) + (pfp_right_sub_bridge_old_from_add)) + (p) * pfa_offset_left_sub_bridge_old_from_addoperationresultcongruence = (pfp_value_sub_bridge_old_from_add) + (p) * pfa_offset_right_sub_bridge_old_from_addoperationresultcongruence)))))))))))))))) -> (forall pfs_index_sub_bridge_new_from_add. (exists pfa_gap_sub_bridge_new_from_addindex. pfa_gap_sub_bridge_new_from_addindex + S (pfs_index_sub_bridge_new_from_add) = (l)) -> exists pfs_left_sub_bridge_new_from_add pfs_right_sub_bridge_new_from_add pfs_result_sub_bridge_new_from_add. ((((exists ff_h_pfp_sub_bridge_new_from_addleft. ff_h_pfp_sub_bridge_new_from_addleft + S (pfs_left_sub_bridge_new_from_add) = S ((S (pfs_index_sub_bridge_new_from_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_new_from_addleft. ab = ff_q_pfp_sub_bridge_new_from_addleft * S ((S (pfs_index_sub_bridge_new_from_add)) * ac) + (pfs_left_sub_bridge_new_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_from_addright. ff_h_pfp_sub_bridge_new_from_addright + S (pfs_right_sub_bridge_new_from_add) = S ((S (pfs_index_sub_bridge_new_from_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_new_from_addright. bb = ff_q_pfp_sub_bridge_new_from_addright * S ((S (pfs_index_sub_bridge_new_from_add)) * bc) + (pfs_right_sub_bridge_new_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_from_addresult. ff_h_pfp_sub_bridge_new_from_addresult + S (pfs_result_sub_bridge_new_from_add) = S ((S (pfs_index_sub_bridge_new_from_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_new_from_addresult. rb = ff_q_pfp_sub_bridge_new_from_addresult * S ((S (pfs_index_sub_bridge_new_from_add)) * rc) + (pfs_result_sub_bridge_new_from_add))) /\ ((((exists pfa_gap_sub_bridge_new_from_addoperationleft. pfa_gap_sub_bridge_new_from_addoperationleft + S (pfs_right_sub_bridge_new_from_add) = (p)) /\ (((exists pfa_gap_sub_bridge_new_from_addoperationright. pfa_gap_sub_bridge_new_from_addoperationright + S (pfs_result_sub_bridge_new_from_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_new_from_addoperationresultbound. pfa_gap_sub_bridge_new_from_addoperationresultbound + S (pfs_left_sub_bridge_new_from_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_new_from_addoperationresultcongruence pfa_offset_right_sub_bridge_new_from_addoperationresultcongruence. ((pfs_right_sub_bridge_new_from_add) + (pfs_result_sub_bridge_new_from_add)) + (p) * pfa_offset_left_sub_bridge_new_from_addoperationresultcongruence = (pfs_left_sub_bridge_new_from_add) + (p) * pfa_offset_right_sub_bridge_new_from_addoperationresultcongruence))))))))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 0 declared prerequisites and contains 31 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

PQ0014 prime_field_polynomial_subtract_self_zero PQ0015 prime_field_polynomial_subtract_zero_right PQ0016 prime_field_polynomial_subtract_zero_left PQ0019 prime_field_polynomial_subtract_add_cancel

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

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.

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 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 : ∃ b. ∃ r. ∃ a. BetaAt(bb,bc,i,b) ∧ (BetaAt(rb,rc,i,r) ∧ (BetaAt(ab,ac,i,a) ∧ FpAdd(p,b,r,a)))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 x2
  2. L23
    exists x
  3. L24
    exists x1
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_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 factsL30–31

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

  1. L30
    exact hv_witness_witness_witness_right_left
  2. L31
    exact hv_witness_witness_witness_right_right_right

Library-wide reading audit

Original exact 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 : exists b r a. (((((exists ff_h_pfp_sub_bridge_from_addb. ff_h_pfp_sub_bridge_from_addb + S (b) = S ((S (i)) * bc)) /\ exists ff_q_pfp_sub_bridge_from_addb. bb = ff_q_pfp_sub_bridge_from_addb * S ((S (i)) * bc) + (b))) /\ (((((exists ff_h_pfp_sub_bridge_from_addr. ff_h_pfp_sub_bridge_from_addr + S (r) = S ((S (i)) * rc)) /\ exists ff_q_pfp_sub_bridge_from_addr. rb = ff_q_pfp_sub_bridge_from_addr * S ((S (i)) * rc) + (r))) /\ (((((exists ff_h_pfp_sub_bridge_from_adda. ff_h_pfp_sub_bridge_from_adda + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_sub_bridge_from_adda. ab = ff_q_pfp_sub_bridge_from_adda * S ((S (i)) * ac) + (a))) /\ ((((exists pfa_gap_sub_bridge_from_addvalueleft. pfa_gap_sub_bridge_from_addvalueleft + S (b) = (p)) /\ (((exists pfa_gap_sub_bridge_from_addvalueright. pfa_gap_sub_bridge_from_addvalueright + S (r) = (p)) /\ ((((exists pfa_gap_sub_bridge_from_addvalueresultbound. pfa_gap_sub_bridge_from_addvalueresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_sub_bridge_from_addvalueresultcongruence pfa_offset_right_sub_bridge_from_addvalueresultcongruence. ((b) + (r)) + (p) * pfa_offset_left_sub_bridge_from_addvalueresultcongruence = (a) + (p) * pfa_offset_right_sub_bridge_from_addvalueresultcongruence))))))))))))))))
  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 x2
  23. 0023exists x
  24. 0024exists x1
  25. 0025split
  26. 0026exact hv_witness_witness_witness_right_right_left
  27. 0027split
  28. 0028exact hv_witness_witness_witness_left
  29. 0029split
  30. 0030exact hv_witness_witness_witness_right_left
  31. 0031exact hv_witness_witness_witness_right_right_right