TS000Y · theorem body

prime_floor_affine_residue_grid_exists

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

A prime floor-square grid admits a canonical beta-coded affine residue map on all successor-square points.

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. ∀ s. ∀ r. ∀ l. l = S s · S s → Prime(p) → ∃ x. ∃ y. ∀ z. Lt(z,l) → ∃ n. ∃ m. ∃ k. ∃ i. z = S s · n + m ∧ (Lt(m,S s) ∧ (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 PD0004 Prime PD0013 BetaAt

In local proof propositions

none
Exact expanded first-order statement
forall p s r l. l = S s * S s -> ((~(p = 1) /\ forall frm_prime_left_ftrg_prime frm_prime_right_ftrg_prime. p = frm_prime_left_ftrg_prime * frm_prime_right_ftrg_prime -> frm_prime_left_ftrg_prime = 1 \/ frm_prime_right_ftrg_prime = 1)) -> exists b c. (forall ftrg_index_square_grid. (exists ftrg_gap_square_grid_index. ftrg_gap_square_grid_index + S (ftrg_index_square_grid) = (l)) -> exists ftrg_row_square_grid ftrg_column_square_grid ftrg_quotient_square_grid ftrg_remainder_square_grid. ((ftrg_index_square_grid) = (S s) * ftrg_row_square_grid + ftrg_column_square_grid /\ ((exists ftrg_gap_square_grid_column. ftrg_gap_square_grid_column + S (ftrg_column_square_grid) = (S s)) /\ ((r * ftrg_row_square_grid + ftrg_column_square_grid = (p) * ftrg_quotient_square_grid + ftrg_remainder_square_grid) /\ ((exists ftrg_gap_square_grid_residue. ftrg_gap_square_grid_residue + S (ftrg_remainder_square_grid) = (p)) /\ (((exists ff_h_ftrg_square_grid_entry. ff_h_ftrg_square_grid_entry + S (ftrg_remainder_square_grid) = S ((S (ftrg_index_square_grid)) * c)) /\ exists ff_q_ftrg_square_grid_entry. b = ff_q_ftrg_square_grid_entry * S ((S (ftrg_index_square_grid)) * c) + (ftrg_remainder_square_grid))))))))

Proof neighborhood

Direct theorem prerequisites

prime_nonzero · Stable closed succ_ne_zero · Stable closed TS000W beta_affine_residue_grid_exists

Direct theorem dependents

TS000Z prime_floor_affine_residue_grid_collision

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

18 script commands · 4 reading checkpoints · 0 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 (1)
beta affine residue grid exists
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro s
  3. L3
    intro r
  4. L4
    intro l
  5. L5
    intro hlength
  6. L6
    intro hprime
02Use earlier factsL7–11

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

  1. L7
    specialize beta_affine_residue_grid_exists p
  2. L8
    specialize beta_affine_residue_grid_exists (S s)
  3. L9
    specialize beta_affine_residue_grid_exists r
  4. L10
    specialize beta_affine_residue_grid_exists l
  5. L11
    apply beta_affine_residue_grid_exists
03Fix variables and assumptionsL12–12

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

  1. L12
    intro hpzero
04Use earlier factsL13–18

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

  1. L13
    specialize prime_nonzero p
  2. L14
    apply prime_nonzero
  3. L15
    exact hprime
  4. L16
    exact hpzero
  5. L17
    specialize succ_ne_zero s
  6. L18
    exact succ_ne_zero

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro s
  3. 0003intro r
  4. 0004intro l
  5. 0005intro hlength
  6. 0006intro hprime
  7. 0007specialize beta_affine_residue_grid_exists p
  8. 0008specialize beta_affine_residue_grid_exists (S s)
  9. 0009specialize beta_affine_residue_grid_exists r
  10. 0010specialize beta_affine_residue_grid_exists l
  11. 0011apply beta_affine_residue_grid_exists
  12. 0012intro hpzero
  13. 0013specialize prime_nonzero p
  14. 0014apply prime_nonzero
  15. 0015exact hprime
  16. 0016exact hpzero
  17. 0017specialize succ_ne_zero s
  18. 0018exact succ_ne_zero