CD0039

finite_modular_dyson_upper_member

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

The upper Dyson transform contains every actual member of the original first 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 a. (forall cd_output_upper. (exists fms_gap_cd_upper_bound. fms_gap_cd_upper_bound + S (cd_output_upper) = (p)) -> ((((((exists fs_h_cd_upper_result. fs_h_cd_upper_result + S (1) = S ((S (cd_output_upper)) * v)) /\ exists fs_q_cd_upper_result. u = fs_q_cd_upper_result * S ((S (cd_output_upper)) * v) + (1))) -> ((((exists fs_h_cd_upper_old. fs_h_cd_upper_old + S (1) = S ((S (cd_output_upper)) * c)) /\ exists fs_q_cd_upper_old. b = fs_q_cd_upper_old * S ((S (cd_output_upper)) * c) + (1))) \/ (exists cd_source_upper. (((exists fms_gap_cd_upper_member. fms_gap_cd_upper_member + S (cd_source_upper) = (p)) /\ (((exists fs_h_fms_cd_upper_member. fs_h_fms_cd_upper_member + S (1) = S ((S (cd_source_upper)) * e)) /\ exists fs_q_fms_cd_upper_member. d = fs_q_fms_cd_upper_member * S ((S (cd_source_upper)) * e) + (1))))) /\ (exists fms_u_cd_upper_mod fms_v_cd_upper_mod. (cd_source_upper+t) + (p) * fms_u_cd_upper_mod = (cd_output_upper) + (p) * fms_v_cd_upper_mod)))) /\ (((((exists fs_h_cd_upper_old. fs_h_cd_upper_old + S (1) = S ((S (cd_output_upper)) * c)) /\ exists fs_q_cd_upper_old. b = fs_q_cd_upper_old * S ((S (cd_output_upper)) * c) + (1))) \/ (exists cd_source_upper. (((exists fms_gap_cd_upper_member. fms_gap_cd_upper_member + S (cd_source_upper) = (p)) /\ (((exists fs_h_fms_cd_upper_member. fs_h_fms_cd_upper_member + S (1) = S ((S (cd_source_upper)) * e)) /\ exists fs_q_fms_cd_upper_member. d = fs_q_fms_cd_upper_member * S ((S (cd_source_upper)) * e) + (1))))) /\ (exists fms_u_cd_upper_mod fms_v_cd_upper_mod. (cd_source_upper+t) + (p) * fms_u_cd_upper_mod = (cd_output_upper) + (p) * fms_v_cd_upper_mod))) -> (((exists fs_h_cd_upper_result. fs_h_cd_upper_result + S (1) = S ((S (cd_output_upper)) * v)) /\ exists fs_q_cd_upper_result. u = fs_q_cd_upper_result * S ((S (cd_output_upper)) * v) + (1))))))) -> (((exists fms_gap_member. fms_gap_member + S (a) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (a)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (a)) * c) + (1))))) -> (((exists fms_gap_member. fms_gap_member + S (a) = (p)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (a)) * v)) /\ exists fs_q_fms_member. u = fs_q_fms_member * S ((S (a)) * v) + (1)))))

Constructive proof overview

Generated structural guide

The upper Dyson transform contains every actual member of the original first 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 · 9 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 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 a
  10. L10
    intro hupper
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hmember
03Separate the logical casesL12–13

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

  1. L12
    cases hmember
  2. L13
    split
04Use earlier factsL14–14

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

  1. L14
    exact hmember_left
05Establish heL15–18

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

  1. L15
    have he : (BetaAt(u,v,a,1) → BetaAt(b,c,a,1) ∨ (∃ x. ModularSetMember(d,e,p,x) ∧ ModEq(p,x + t,a))) ∧ (BetaAt(b,c,a,1) ∨ (∃ x. ModularSetMember(d,e,p,x) ∧ ModEq(p,x + t,a)) → BetaAt(u,v,a,1))Definitions: ModularSetMemberModEqBetaAt
  2. L16
    specialize hupper a
  3. L17
    apply hupper
  4. L18
    exact hmember_left
06Separate the logical casesL19–19

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

  1. L19
    cases he
07Use earlier factsL20–20

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

  1. L20
    apply he_right
08Separate the logical casesL21–21

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

  1. L21
    left
09Use earlier factsL22–22

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

  1. L22
    exact hmember_right

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 a
  10. 0010intro hupper
  11. 0011intro hmember
  12. 0012cases hmember
  13. 0013split
  14. 0014exact hmember_left
  15. 0015have he : (((((exists fs_h_cd_upper_member_U. fs_h_cd_upper_member_U + S (1) = S ((S (a)) * v)) /\ exists fs_q_cd_upper_member_U. u = fs_q_cd_upper_member_U * S ((S (a)) * v) + (1))) -> ((((exists fs_h_cd_upper_member_A. fs_h_cd_upper_member_A + S (1) = S ((S (a)) * c)) /\ exists fs_q_cd_upper_member_A. b = fs_q_cd_upper_member_A * S ((S (a)) * c) + (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)) * e)) /\ exists fs_q_fms_member. d = fs_q_fms_member * S ((S (j)) * e) + (1))))) /\ (exists fms_u_mod fms_v_mod. (j+t) + (p) * fms_u_mod = (a) + (p) * fms_v_mod))) /\ (((((exists fs_h_cd_upper_member_A. fs_h_cd_upper_member_A + S (1) = S ((S (a)) * c)) /\ exists fs_q_cd_upper_member_A. b = fs_q_cd_upper_member_A * S ((S (a)) * c) + (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)) * e)) /\ exists fs_q_fms_member. d = fs_q_fms_member * S ((S (j)) * e) + (1))))) /\ (exists fms_u_mod fms_v_mod. (j+t) + (p) * fms_u_mod = (a) + (p) * fms_v_mod)) -> (((exists fs_h_cd_upper_member_U. fs_h_cd_upper_member_U + S (1) = S ((S (a)) * v)) /\ exists fs_q_cd_upper_member_U. u = fs_q_cd_upper_member_U * S ((S (a)) * v) + (1)))))
  16. 0016specialize hupper a
  17. 0017apply hupper
  18. 0018exact hmember_left
  19. 0019cases he
  20. 0020apply he_right
  21. 0021left
  22. 0022exact hmember_right