CD003C

finite_modular_dyson_lower_boundary_nonmember

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

An actual translation-boundary direction is absent from the lower Dyson transform.

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 h r. (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))))))) -> ((((exists fms_gap_cd_boundary_source. fms_gap_cd_boundary_source + S (t) = (p)) /\ (((exists fs_h_fms_cd_boundary_source. fs_h_fms_cd_boundary_source + S (1) = S ((S (t)) * c)) /\ exists fs_q_fms_cd_boundary_source. b = fs_q_fms_cd_boundary_source * S ((S (t)) * c) + (1))))) /\ ((exists fms_gap_cd_boundary_target. fms_gap_cd_boundary_target + S (r) = (p)) /\ ((exists fms_u_cd_boundary_shift fms_v_cd_boundary_shift. (t+h) + (p) * fms_u_cd_boundary_shift = (r) + (p) * fms_v_cd_boundary_shift) /\ ~(((exists fs_h_cd_boundary_outside. fs_h_cd_boundary_outside + S (1) = S ((S (r)) * c)) /\ exists fs_q_cd_boundary_outside. b = fs_q_cd_boundary_outside * S ((S (r)) * c) + (1)))))) -> (exists fms_gap_lt. fms_gap_lt + S (h) = (p)) -> ~(((exists fs_h_cd_boundary_missing. fs_h_cd_boundary_missing + S (1) = S ((S (h)) * v)) /\ exists fs_q_cd_boundary_missing. u = fs_q_cd_boundary_missing * S ((S (h)) * v) + (1)))

Constructive proof overview

Generated structural guide

An actual translation-boundary direction is absent from the lower Dyson transform.

The unchanged tactic script uses 4 declared prerequisites and contains 56 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mod_eq_bounded_unique Stable theorem; checked-use authorized mod_eq_trans Stable theorem; checked-use authorized mod_eq_symm Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

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

56 script commands · 10 reading checkpoints · 4 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 h
  10. L10
    intro r
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hlower
  2. L12
    intro hboundary
  3. L13
    intro hh
  4. L14
    intro hmember
03Separate the logical casesL15–17

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

  1. L15
    cases hboundary
  2. L16
    cases hboundary_right
  3. L17
    cases hboundary_right_right
04Establish heL18–21

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

  1. L18
    have he : (BetaAt(u,v,h,1) → BetaAt(d,e,h,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,h + t,x))) ∧ (BetaAt(d,e,h,1) ∧ (∃ x. ModularSetMember(b,c,p,x) ∧ ModEq(p,h + t,x)) → BetaAt(u,v,h,1))Definitions: ModularSetMemberModEqBetaAt
  2. L19
    specialize hlower h
  3. L20
    apply hlower
  4. L21
    exact hh
05Separate the logical casesL22–22

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

  1. L22
    cases he
06Establish hbothL23–25

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

  1. L23
    have hboth : (((exists fs_h_cd_lower_boundary_B. fs_h_cd_lower_boundary_B + S (1) = S ((S (h)) * e)) /\ exists fs_q_cd_lower_boundary_B. d = fs_q_cd_lower_boundary_B * S ((S (h)) * 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. (h+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)
  2. L24
    apply he_left
  3. L25
    exact hmember
07Separate the logical casesL26–29

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

  1. L26
    cases hboth
  2. L27
    cases hboth_right
  3. L28
    cases hboth_right_witness
  4. L29
    cases hboth_right_witness_left
08Establish heqL30–39

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq bounded unique.

  1. L30
    have heq : x=r
  2. L31
    specialize mod_eq_bounded_unique p
  3. L32
    specialize mod_eq_bounded_unique x
  4. L33
    specialize mod_eq_bounded_unique r
  5. L34
    apply mod_eq_bounded_unique
  6. L35
    exact hboth_right_witness_left_left
  7. L36
    exact hboundary_right_left
  8. L37
    specialize mod_eq_trans p
  9. L38
    specialize mod_eq_trans x
  10. L39
    specialize mod_eq_trans h+t
09Use earlier factsL40–46

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

  1. L40
    specialize mod_eq_trans r
  2. L41
    apply mod_eq_trans
  3. L42
    specialize mod_eq_symm p
  4. L43
    specialize mod_eq_symm h+t
  5. L44
    specialize mod_eq_symm x
  6. L45
    apply mod_eq_symm
  7. L46
    exact hboth_right_witness_right
10Establish hcommL47–56

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

  1. L47
    have hcomm : h+t=t+h
  2. L48
    specialize add_comm h
  3. L49
    specialize add_comm t
  4. L50
    apply add_comm
  5. L51
    rewrite hcomm
  6. L52
    exact hboundary_right_right_left
  7. L53
    apply hboundary_right_right_right
  8. L54
    rewrite heq at hboth_right_witness_left_right
  9. L55
    rewrite heq at hboth_right_witness_left_right
  10. L56
    exact hboth_right_witness_left_right

Library-wide reading audit

Original exact command ledger · 56 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 h
  10. 0010intro r
  11. 0011intro hlower
  12. 0012intro hboundary
  13. 0013intro hh
  14. 0014intro hmember
  15. 0015cases hboundary
  16. 0016cases hboundary_right
  17. 0017cases hboundary_right_right
  18. 0018have he : (((((exists fs_h_cd_boundary_V. fs_h_cd_boundary_V + S (1) = S ((S (h)) * v)) /\ exists fs_q_cd_boundary_V. u = fs_q_cd_boundary_V * S ((S (h)) * v) + (1))) -> ((((exists fs_h_cd_lower_boundary_B. fs_h_cd_lower_boundary_B + S (1) = S ((S (h)) * e)) /\ exists fs_q_cd_lower_boundary_B. d = fs_q_cd_lower_boundary_B * S ((S (h)) * 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. (h+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod))) /\ (((((exists fs_h_cd_lower_boundary_B. fs_h_cd_lower_boundary_B + S (1) = S ((S (h)) * e)) /\ exists fs_q_cd_lower_boundary_B. d = fs_q_cd_lower_boundary_B * S ((S (h)) * 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. (h+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)) -> (((exists fs_h_cd_boundary_V. fs_h_cd_boundary_V + S (1) = S ((S (h)) * v)) /\ exists fs_q_cd_boundary_V. u = fs_q_cd_boundary_V * S ((S (h)) * v) + (1)))))
  19. 0019specialize hlower h
  20. 0020apply hlower
  21. 0021exact hh
  22. 0022cases he
  23. 0023have hboth : (((exists fs_h_cd_lower_boundary_B. fs_h_cd_lower_boundary_B + S (1) = S ((S (h)) * e)) /\ exists fs_q_cd_lower_boundary_B. d = fs_q_cd_lower_boundary_B * S ((S (h)) * 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. (h+t) + (p) * fms_u_mod = (j) + (p) * fms_v_mod)
  24. 0024apply he_left
  25. 0025exact hmember
  26. 0026cases hboth
  27. 0027cases hboth_right
  28. 0028cases hboth_right_witness
  29. 0029cases hboth_right_witness_left
  30. 0030have heq : x=r
  31. 0031specialize mod_eq_bounded_unique p
  32. 0032specialize mod_eq_bounded_unique x
  33. 0033specialize mod_eq_bounded_unique r
  34. 0034apply mod_eq_bounded_unique
  35. 0035exact hboth_right_witness_left_left
  36. 0036exact hboundary_right_left
  37. 0037specialize mod_eq_trans p
  38. 0038specialize mod_eq_trans x
  39. 0039specialize mod_eq_trans h+t
  40. 0040specialize mod_eq_trans r
  41. 0041apply mod_eq_trans
  42. 0042specialize mod_eq_symm p
  43. 0043specialize mod_eq_symm h+t
  44. 0044specialize mod_eq_symm x
  45. 0045apply mod_eq_symm
  46. 0046exact hboth_right_witness_right
  47. 0047have hcomm : h+t=t+h
  48. 0048specialize add_comm h
  49. 0049specialize add_comm t
  50. 0050apply add_comm
  51. 0051rewrite hcomm
  52. 0052exact hboundary_right_right_left
  53. 0053apply hboundary_right_right_right
  54. 0054rewrite heq at hboth_right_witness_left_right
  55. 0055rewrite heq at hboth_right_witness_left_right
  56. 0056exact hboth_right_witness_left_right