FS004C · theorem body

four_square_square_residue_prefix_bounded

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

Every canonical square-residue prefix is pointwise bounded by its modulus.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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. ∀ b. ∀ c. ∀ l. (∀ x. Lt(x,l) → ∃ y. ∃ z. x · x = p · y + z ∧ (Lt(z,p) ∧ BetaAt(b,c,x,z))) → ∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,p)

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 PD0013 BetaAt
Exact expanded first-order statement
forall p b c l. (forall fsri_index_bounded_source. (exists fsri_gap_bounded_source_index. fsri_gap_bounded_source_index + S (fsri_index_bounded_source) = (l)) -> exists fsri_quotient_bounded_source fsri_residue_bounded_source. (fsri_index_bounded_source * fsri_index_bounded_source = (p) * fsri_quotient_bounded_source + fsri_residue_bounded_source /\ ((exists fsri_gap_bounded_source_residue. fsri_gap_bounded_source_residue + S (fsri_residue_bounded_source) = (p)) /\ (((exists fsri_height_bounded_source_entry. fsri_height_bounded_source_entry + S (fsri_residue_bounded_source) = S ((S (fsri_index_bounded_source)) * (c))) /\ exists fsri_quotient_bounded_source_entry. (b) = fsri_quotient_bounded_source_entry * S ((S (fsri_index_bounded_source)) * (c)) + (fsri_residue_bounded_source)))))) -> (forall fom_index_fsri_bounded_result. (exists fom_gap_fsri_bounded_result_index_bound. fom_gap_fsri_bounded_result_index_bound + S (fom_index_fsri_bounded_result) = l) -> exists fom_value_fsri_bounded_result. ((((exists fom_beta_height_fsri_bounded_result_entry. fom_beta_height_fsri_bounded_result_entry + S (fom_value_fsri_bounded_result) = S ((S (fom_index_fsri_bounded_result)) * c)) /\ exists fom_beta_quotient_fsri_bounded_result_entry. b = fom_beta_quotient_fsri_bounded_result_entry * S ((S (fom_index_fsri_bounded_result)) * c) + (fom_value_fsri_bounded_result))) /\ (exists fom_gap_fsri_bounded_result_value_bound. fom_gap_fsri_bounded_result_value_bound + S (fom_value_fsri_bounded_result) = p)))

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

FS000K four_square_odd_prime_half_coordinate_seed FS004K four_square_odd_prime_modular_seed

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

19 script commands · 7 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.

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

01Fix variables and assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro hprefix
  6. L6
    intro i
  7. L7
    intro hi
02Use earlier factsL8–8

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

  1. L8
    specialize hprefix i
03Establish hpointL9–11

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

  1. L9
    have hpoint : ∃ q. ∃ r. i · i = p · q + r ∧ (Lt(r,p) ∧ BetaAt(b,c,i,r))Definitions: Lt(r,p)BetaAt(b,c,i,r)Original native command in the exact edition
  2. L10
    apply hprefix
  3. L11
    exact hi
04Separate the logical casesL12–15

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

  1. L12
    cases hpoint
  2. L13
    cases hpoint_witness
  3. L14
    cases hpoint_witness_witness
  4. L15
    cases hpoint_witness_witness_right
05Construct an explicit witnessL16–16

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

  1. L16
    exists x1
06Separate the logical casesL17–17

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

  1. L17
    split
07Use earlier factsL18–19

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

  1. L18
    exact hpoint_witness_witness_right_right
  2. L19
    exact hpoint_witness_witness_right_left

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hprefix
  6. 0006intro i
  7. 0007intro hi
  8. 0008specialize hprefix i
  9. 0009have hpoint : ∃ q. ∃ r. i · i = p · q + r ∧ (Lt(r,p) ∧ BetaAt(b,c,i,r))
    Exact native replay linehave hpoint : exists q r. (i * i = p * q + r /\ ((exists fsri_gap_bounded_point. fsri_gap_bounded_point + S (r) = (p)) /\ (((exists fsri_height_bounded_entry. fsri_height_bounded_entry + S (r) = S ((S (i)) * (c))) /\ exists fsri_quotient_bounded_entry. (b) = fsri_quotient_bounded_entry * S ((S (i)) * (c)) + (r)))))
  10. 0010apply hprefix
  11. 0011exact hi
  12. 0012cases hpoint
  13. 0013cases hpoint_witness
  14. 0014cases hpoint_witness_witness
  15. 0015cases hpoint_witness_witness_right
  16. 0016exists x1
  17. 0017split
  18. 0018exact hpoint_witness_witness_right_right
  19. 0019exact hpoint_witness_witness_right_left