TS000W · theorem body

beta_affine_residue_grid_exists

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

Every nonzero modulus and positive grid width admit a full beta-coded prefix of bounded affine residues.

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.

Statement with defined notation

∀ p. ∀ w. ∀ r. ∀ l. ¬p = 0 → ¬w = 0 → ∃ x. ∃ y. ∀ z. Lt(z,l) → ∃ n. ∃ m. ∃ k. ∃ i. z = w · n + m ∧ (Lt(m,w) ∧ (r · n + m = p · k + i ∧ (Lt(i,p) ∧ BetaAt(x,y,z,i))))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0013 BetaAt

In local proof propositions

PD0002 Lt PD0007 DivRem PD0013 BetaAt
Exact expanded first-order statement
forall p w r l. ~(p = 0) -> ~(w = 0) -> exists b c. (forall ftrg_index_existence. (exists ftrg_gap_existence_index. ftrg_gap_existence_index + S (ftrg_index_existence) = (l)) -> exists ftrg_row_existence ftrg_column_existence ftrg_quotient_existence ftrg_remainder_existence. ((ftrg_index_existence) = (w) * ftrg_row_existence + ftrg_column_existence /\ ((exists ftrg_gap_existence_column. ftrg_gap_existence_column + S (ftrg_column_existence) = (w)) /\ ((r * ftrg_row_existence + ftrg_column_existence = (p) * ftrg_quotient_existence + ftrg_remainder_existence) /\ ((exists ftrg_gap_existence_residue. ftrg_gap_existence_residue + S (ftrg_remainder_existence) = (p)) /\ (((exists ff_h_ftrg_existence_entry. ff_h_ftrg_existence_entry + S (ftrg_remainder_existence) = S ((S (ftrg_index_existence)) * c)) /\ exists ff_q_ftrg_existence_entry. b = ff_q_ftrg_existence_entry * S ((S (ftrg_index_existence)) * c) + (ftrg_remainder_existence))))))))

Proof neighborhood

Direct theorem prerequisites

add_eq_zero_right · Stable closed succ_ne_zero · Stable closed TS000U affine_grid_point_remainder_exists TS000V beta_affine_residue_grid_extend

Direct theorem dependents

TS000Y prime_floor_affine_residue_grid_exists

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

47 script commands · 11 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
affine grid point remainder existsbeta affine residue grid extend
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro w
  3. L3
    intro r
02Induction on lL4–6

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L4
    induction l
  2. L5
    intro hp
  3. L6
    intro hw
03Construct an explicit witnessL7–8

Supply the displayed value, then prove that it has the required property.

  1. L7
    exists 0
  2. L8
    exists 0
04Fix variables and assumptionsL9–10

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

  1. L9
    intro k
  2. L10
    intro hk
05Separate the logical casesL11–12

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

  1. L11
    exfalso
  2. L12
    cases hk
06Establish hzeroL13–22

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

  1. L13
    have hzero : S k = 0
  2. L14
    specialize add_eq_zero_right x
  3. L15
    specialize add_eq_zero_right (S k)
  4. L16
    apply add_eq_zero_right
  5. L17
    exact hk_witness
  6. L18
    specialize succ_ne_zero k
  7. L19
    apply succ_ne_zero
  8. L20
    exact hzero
  9. L21
    intro hp
  10. L22
    intro hw
07Establish hpreviousL23–26

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

  1. L23
    have hprevious : ∃ b. ∃ c. ∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. ∃ m. x = w · y + z ∧ (Lt(z,w) ∧ (r · y + z = p · n + m ∧ (Lt(m,p) ∧ BetaAt(b,c,x,m))))Definitions: Lt(x,l)Lt(z,w)Lt(m,p)BetaAt(b,c,x,m)Original native command in the exact edition
  2. L24
    apply IH
  3. L25
    exact hp
  4. L26
    exact hw
08Separate the logical casesL27–28

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

  1. L27
    cases hprevious
  2. L28
    cases hprevious_witness
09Establish hpointL29–36

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply affine grid point remainder exists.

  1. L29
    have hpoint : ∃ ftrg_row_induction_point. ∃ ftrg_column_induction_point. ∃ ftrg_quotient_induction_point. ∃ ftrg_remainder_induction_point. l = w · ftrg_row_induction_point + ftrg_column_induction_point ∧ (Lt(ftrg_column_induction_point,w) ∧ DivRem(r · ftrg_row_induction_point + ftrg_column_induction_point,p,ftrg_quotient_induction_point,ftrg_remainder_induction_point))Definitions: Lt(ftrg_column_induction_point,w)DivRem(r · ftrg_row_induction_point + ftrg_column_induction_point,p,ftrg_quotient_induction_point,ftrg_remainder_induction_point)Original native command in the exact edition
  2. L30
    specialize affine_grid_point_remainder_exists p
  3. L31
    specialize affine_grid_point_remainder_exists w
  4. L32
    specialize affine_grid_point_remainder_exists r
  5. L33
    specialize affine_grid_point_remainder_exists l
  6. L34
    apply affine_grid_point_remainder_exists
  7. L35
    exact hp
  8. L36
    exact hw
10Establish hnextL37–46

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta affine residue grid extend.

  1. L37
    have hnext : ∃ b. ∃ c. ∀ x. Lt(x,S l) → ∃ y. ∃ z. ∃ n. ∃ m. x = w · y + z ∧ (Lt(z,w) ∧ (r · y + z = p · n + m ∧ (Lt(m,p) ∧ BetaAt(b,c,x,m))))Definitions: Lt(x,S l)Lt(z,w)Lt(m,p)BetaAt(b,c,x,m)Original native command in the exact edition
  2. L38
    specialize beta_affine_residue_grid_extend p
  3. L39
    specialize beta_affine_residue_grid_extend w
  4. L40
    specialize beta_affine_residue_grid_extend r
  5. L41
    specialize beta_affine_residue_grid_extend x
  6. L42
    specialize beta_affine_residue_grid_extend x1
  7. L43
    specialize beta_affine_residue_grid_extend l
  8. L44
    apply beta_affine_residue_grid_extend
  9. L45
    exact hprevious_witness_witness
  10. L46
    exact hpoint
11Use earlier factsL47–47

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

  1. L47
    exact hnext

Library-wide reading audit

Original defined command ledger · 47 lines
  1. 0001intro p
  2. 0002intro w
  3. 0003intro r
  4. 0004induction l
  5. 0005intro hp
  6. 0006intro hw
  7. 0007exists 0
  8. 0008exists 0
  9. 0009intro k
  10. 0010intro hk
  11. 0011exfalso
  12. 0012cases hk
  13. 0013have hzero : S k = 0
  14. 0014specialize add_eq_zero_right x
  15. 0015specialize add_eq_zero_right (S k)
  16. 0016apply add_eq_zero_right
  17. 0017exact hk_witness
  18. 0018specialize succ_ne_zero k
  19. 0019apply succ_ne_zero
  20. 0020exact hzero
  21. 0021intro hp
  22. 0022intro hw
  23. 0023have hprevious : ∃ b. ∃ c. ∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. ∃ m. x = w · y + z ∧ (Lt(z,w) ∧ (r · y + z = p · n + m ∧ (Lt(m,p) ∧ BetaAt(b,c,x,m))))
    Exact native replay linehave hprevious : exists b c. (forall ftrg_index_induction_previous. (exists ftrg_gap_induction_previous_index. ftrg_gap_induction_previous_index + S (ftrg_index_induction_previous) = (l)) -> exists ftrg_row_induction_previous ftrg_column_induction_previous ftrg_quotient_induction_previous ftrg_remainder_induction_previous. ((ftrg_index_induction_previous) = (w) * ftrg_row_induction_previous + ftrg_column_induction_previous /\ ((exists ftrg_gap_induction_previous_column. ftrg_gap_induction_previous_column + S (ftrg_column_induction_previous) = (w)) /\ ((r * ftrg_row_induction_previous + ftrg_column_induction_previous = (p) * ftrg_quotient_induction_previous + ftrg_remainder_induction_previous) /\ ((exists ftrg_gap_induction_previous_residue. ftrg_gap_induction_previous_residue + S (ftrg_remainder_induction_previous) = (p)) /\ (((exists ff_h_ftrg_induction_previous_entry. ff_h_ftrg_induction_previous_entry + S (ftrg_remainder_induction_previous) = S ((S (ftrg_index_induction_previous)) * c)) /\ exists ff_q_ftrg_induction_previous_entry. b = ff_q_ftrg_induction_previous_entry * S ((S (ftrg_index_induction_previous)) * c) + (ftrg_remainder_induction_previous))))))))
  24. 0024apply IH
  25. 0025exact hp
  26. 0026exact hw
  27. 0027cases hprevious
  28. 0028cases hprevious_witness
  29. 0029have hpoint : ∃ ftrg_row_induction_point. ∃ ftrg_column_induction_point. ∃ ftrg_quotient_induction_point. ∃ ftrg_remainder_induction_point. l = w · ftrg_row_induction_point + ftrg_column_induction_point ∧ (Lt(ftrg_column_induction_point,w) ∧ DivRem(r · ftrg_row_induction_point + ftrg_column_induction_point,p,ftrg_quotient_induction_point,ftrg_remainder_induction_point))
    Exact native replay linehave hpoint : exists ftrg_row_induction_point ftrg_column_induction_point ftrg_quotient_induction_point ftrg_remainder_induction_point. ((l) = (w) * ftrg_row_induction_point + ftrg_column_induction_point /\ ((exists ftrg_gap_induction_point_column. ftrg_gap_induction_point_column + S (ftrg_column_induction_point) = (w)) /\ ((r * ftrg_row_induction_point + ftrg_column_induction_point = (p) * ftrg_quotient_induction_point + ftrg_remainder_induction_point) /\ (exists ftrg_gap_induction_point_residue. ftrg_gap_induction_point_residue + S (ftrg_remainder_induction_point) = (p)))))
  30. 0030specialize affine_grid_point_remainder_exists p
  31. 0031specialize affine_grid_point_remainder_exists w
  32. 0032specialize affine_grid_point_remainder_exists r
  33. 0033specialize affine_grid_point_remainder_exists l
  34. 0034apply affine_grid_point_remainder_exists
  35. 0035exact hp
  36. 0036exact hw
  37. 0037have hnext : ∃ b. ∃ c. ∀ x. Lt(x,S l) → ∃ y. ∃ z. ∃ n. ∃ m. x = w · y + z ∧ (Lt(z,w) ∧ (r · y + z = p · n + m ∧ (Lt(m,p) ∧ BetaAt(b,c,x,m))))
    Exact native replay linehave hnext : exists b c. (forall ftrg_index_induction_next. (exists ftrg_gap_induction_next_index. ftrg_gap_induction_next_index + S (ftrg_index_induction_next) = (S l)) -> exists ftrg_row_induction_next ftrg_column_induction_next ftrg_quotient_induction_next ftrg_remainder_induction_next. ((ftrg_index_induction_next) = (w) * ftrg_row_induction_next + ftrg_column_induction_next /\ ((exists ftrg_gap_induction_next_column. ftrg_gap_induction_next_column + S (ftrg_column_induction_next) = (w)) /\ ((r * ftrg_row_induction_next + ftrg_column_induction_next = (p) * ftrg_quotient_induction_next + ftrg_remainder_induction_next) /\ ((exists ftrg_gap_induction_next_residue. ftrg_gap_induction_next_residue + S (ftrg_remainder_induction_next) = (p)) /\ (((exists ff_h_ftrg_induction_next_entry. ff_h_ftrg_induction_next_entry + S (ftrg_remainder_induction_next) = S ((S (ftrg_index_induction_next)) * c)) /\ exists ff_q_ftrg_induction_next_entry. b = ff_q_ftrg_induction_next_entry * S ((S (ftrg_index_induction_next)) * c) + (ftrg_remainder_induction_next))))))))
  38. 0038specialize beta_affine_residue_grid_extend p
  39. 0039specialize beta_affine_residue_grid_extend w
  40. 0040specialize beta_affine_residue_grid_extend r
  41. 0041specialize beta_affine_residue_grid_extend x
  42. 0042specialize beta_affine_residue_grid_extend x1
  43. 0043specialize beta_affine_residue_grid_extend l
  44. 0044apply beta_affine_residue_grid_extend
  45. 0045exact hprevious_witness_witness
  46. 0046exact hpoint
  47. 0047exact hnext