FS004P · theorem body

four_square_signed_cases_norm_quotient_zero_congruence

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

An exact centered norm quotient supplies constructive balanced zero congruence for its complete four-square norm.

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

∀ k. ∀ r. ∀ a. ∀ b. ∀ c. ∀ d. k · r = a · a + b · b + c · c + d · d → ModEq(k,a · a + b · b + c · c + d · d,0)

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

PD0008 ModEq

In local proof propositions

none
Exact expanded first-order statement
forall k r a b c d. k * r = a * a + b * b + c * c + d * d -> (exists ftcn_left_case_norm_zero ftcn_right_case_norm_zero. (a * a + b * b + c * c + d * d) + (k) * ftcn_left_case_norm_zero = (0) + (k) * ftcn_right_case_norm_zero)

Proof neighborhood

Direct theorem prerequisites

multiple_implies_balanced_zero_congruence · Alpha closed

Direct theorem dependents

FS004Q four_square_signed_orientation_mask_00 FS004R four_square_signed_orientation_mask_01 FS004S four_square_signed_orientation_mask_02 FS004T four_square_signed_orientation_mask_03 FS004U four_square_signed_orientation_mask_04 FS004V four_square_signed_orientation_mask_05 FS004W four_square_signed_orientation_mask_06 FS004X four_square_signed_orientation_mask_07 FS004Y four_square_signed_orientation_mask_08 FS004Z four_square_signed_orientation_mask_09 FS0050 four_square_signed_orientation_mask_10 FS0051 four_square_signed_orientation_mask_11 FS0052 four_square_signed_orientation_mask_12 FS0053 four_square_signed_orientation_mask_13 FS0054 four_square_signed_orientation_mask_14 FS0055 four_square_signed_orientation_mask_15

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

13 script commands · 5 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro k
  2. L2
    intro r
  3. L3
    intro a
  4. L4
    intro b
  5. L5
    intro c
  6. L6
    intro d
  7. L7
    intro hnorm
02Use earlier factsL8–10

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

  1. L8
    specialize multiple_implies_balanced_zero_congruence k
  2. L9
    specialize multiple_implies_balanced_zero_congruence (a * a + b * b + c * c + d * d)
  3. L10
    apply multiple_implies_balanced_zero_congruence
03Construct an explicit witnessL11–11

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

  1. L11
    exists r
04Calculate and transport equalitiesL12–12

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L12
    symm
05Use earlier factsL13–13

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

  1. L13
    exact hnorm

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro k
  2. 0002intro r
  3. 0003intro a
  4. 0004intro b
  5. 0005intro c
  6. 0006intro d
  7. 0007intro hnorm
  8. 0008specialize multiple_implies_balanced_zero_congruence k
  9. 0009specialize multiple_implies_balanced_zero_congruence (a * a + b * b + c * c + d * d)
  10. 0010apply multiple_implies_balanced_zero_congruence
  11. 0011exists r
  12. 0012symm
  13. 0013exact hnorm