CD003A

finite_modular_dyson_lower_subset

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

The lower Dyson transform is an actual subset of the original second set.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall b c d e u v p t. (forall cd_output_lower. (exists fms_gap_cd_lower_bound. fms_gap_cd_lower_bound + S (cd_output_lower) = (p)) -> ((((((exists fs_h_cd_lower_result. fs_h_cd_lower_result + S (1) = S ((S (cd_output_lower)) * v)) /\ exists fs_q_cd_lower_result. u = fs_q_cd_lower_result * S ((S (cd_output_lower)) * v) + (1))) -> ((((exists fs_h_cd_lower_old. fs_h_cd_lower_old + S (1) = S ((S (cd_output_lower)) * e)) /\ exists fs_q_cd_lower_old. d = fs_q_cd_lower_old * S ((S (cd_output_lower)) * e) + (1))) /\ (exists cd_source_lower. (((exists fms_gap_cd_lower_member. fms_gap_cd_lower_member + S (cd_source_lower) = (p)) /\ (((exists fs_h_fms_cd_lower_member. fs_h_fms_cd_lower_member + S (1) = S ((S (cd_source_lower)) * c)) /\ exists fs_q_fms_cd_lower_member. b = fs_q_fms_cd_lower_member * S ((S (cd_source_lower)) * c) + (1))))) /\ (exists fms_u_cd_lower_mod fms_v_cd_lower_mod. (cd_output_lower+t) + (p) * fms_u_cd_lower_mod = (cd_source_lower) + (p) * fms_v_cd_lower_mod)))) /\ (((((exists fs_h_cd_lower_old. fs_h_cd_lower_old + S (1) = S ((S (cd_output_lower)) * e)) /\ exists fs_q_cd_lower_old. d = fs_q_cd_lower_old * S ((S (cd_output_lower)) * e) + (1))) /\ (exists cd_source_lower. (((exists fms_gap_cd_lower_member. fms_gap_cd_lower_member + S (cd_source_lower) = (p)) /\ (((exists fs_h_fms_cd_lower_member. fs_h_fms_cd_lower_member + S (1) = S ((S (cd_source_lower)) * c)) /\ exists fs_q_fms_cd_lower_member. b = fs_q_fms_cd_lower_member * S ((S (cd_source_lower)) * c) + (1))))) /\ (exists fms_u_cd_lower_mod fms_v_cd_lower_mod. (cd_output_lower+t) + (p) * fms_u_cd_lower_mod = (cd_source_lower) + (p) * fms_v_cd_lower_mod))) -> (((exists fs_h_cd_lower_result. fs_h_cd_lower_result + S (1) = S ((S (cd_output_lower)) * v)) /\ exists fs_q_cd_lower_result. u = fs_q_cd_lower_result * S ((S (cd_output_lower)) * v) + (1))))))) -> (forall fms_i_subset. (exists fms_gap_subset. fms_gap_subset + S (fms_i_subset) = (p)) -> (((exists fs_h_fms_subset_left. fs_h_fms_subset_left + S (1) = S ((S (fms_i_subset)) * v)) /\ exists fs_q_fms_subset_left. u = fs_q_fms_subset_left * S ((S (fms_i_subset)) * v) + (1))) -> (((exists fs_h_fms_subset_right. fs_h_fms_subset_right + S (1) = S ((S (fms_i_subset)) * e)) /\ exists fs_q_fms_subset_right. d = fs_q_fms_subset_right * S ((S (fms_i_subset)) * e) + (1))))

Constructive proof overview

Generated structural guide

The lower Dyson transform is an actual subset of the original second set.

The unchanged tactic script uses 0 declared prerequisites and contains 22 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

CD003E finite_modular_dyson_strict_sizes

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

22 script commands · 7 reading checkpoints · 2 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 b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro u
  6. L6
    intro v
  7. L7
    intro p
  8. L8
    intro t
  9. L9
    intro hlower
  10. L10
    intro i
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hi
  2. L12
    intro hmember
03Establish heL13–16

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

  1. L13
    have he : (BetaAt(u,v,i,1) → BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x))) ∧ (BetaAt(d,e,i,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,i + t,x)) → BetaAt(u,v,i,1))Definitions: ModularSetMemberModEqBetaAt
  2. L14
    specialize hlower i
  3. L15
    apply hlower
  4. L16
    exact hi
04Separate the logical casesL17–17

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

  1. L17
    cases he
05Establish hbothL18–20

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

  1. L18
    have hboth : (((exists fs_h_cd_lower_member_B. fs_h_cd_lower_member_B + S (1) = S ((S (i)) * e)) /\ exists fs_q_cd_lower_member_B. d = fs_q_cd_lower_member_B * S ((S (i)) * e) + (1))) /\ exists j. (((exists fms_gap_member. fms_gap_member + S (j) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (j)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (j)) * c) + (1))))) /\ (exists fms_u_mod fms_v_mod. (i+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)
  2. L19
    apply he_left
  3. L20
    exact hmember
06Separate the logical casesL21–21

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

  1. L21
    cases hboth
07Use earlier factsL22–22

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

  1. L22
    exact hboth_left

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro u
  6. 0006intro v
  7. 0007intro p
  8. 0008intro t
  9. 0009intro hlower
  10. 0010intro i
  11. 0011intro hi
  12. 0012intro hmember
  13. 0013have he : (((((exists fs_h_cd_lower_member_V. fs_h_cd_lower_member_V + S (1) = S ((S (i)) * v)) /\ exists fs_q_cd_lower_member_V. u = fs_q_cd_lower_member_V * S ((S (i)) * v) + (1))) -> ((((exists fs_h_cd_lower_member_B. fs_h_cd_lower_member_B + S (1) = S ((S (i)) * e)) /\ exists fs_q_cd_lower_member_B. d = fs_q_cd_lower_member_B * S ((S (i)) * e) + (1))) /\ exists j. (((exists fms_gap_member. fms_gap_member + S (j) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (j)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (j)) * c) + (1))))) /\ (exists fms_u_mod fms_v_mod. (i+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod))) /\ (((((exists fs_h_cd_lower_member_B. fs_h_cd_lower_member_B + S (1) = S ((S (i)) * e)) /\ exists fs_q_cd_lower_member_B. d = fs_q_cd_lower_member_B * S ((S (i)) * e) + (1))) /\ exists j. (((exists fms_gap_member. fms_gap_member + S (j) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (j)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (j)) * c) + (1))))) /\ (exists fms_u_mod fms_v_mod. (i+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)) -> (((exists fs_h_cd_lower_member_V. fs_h_cd_lower_member_V + S (1) = S ((S (i)) * v)) /\ exists fs_q_cd_lower_member_V. u = fs_q_cd_lower_member_V * S ((S (i)) * v) + (1)))))
  14. 0014specialize hlower i
  15. 0015apply hlower
  16. 0016exact hi
  17. 0017cases he
  18. 0018have hboth : (((exists fs_h_cd_lower_member_B. fs_h_cd_lower_member_B + S (1) = S ((S (i)) * e)) /\ exists fs_q_cd_lower_member_B. d = fs_q_cd_lower_member_B * S ((S (i)) * e) + (1))) /\ exists j. (((exists fms_gap_member. fms_gap_member + S (j) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (j)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (j)) * c) + (1))))) /\ (exists fms_u_mod fms_v_mod. (i+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)
  19. 0019apply he_left
  20. 0020exact hmember
  21. 0021cases hboth
  22. 0022exact hboth_left