FS003R · theorem body

four_square_parity_pair_mod_two_sum

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

The sum of two coordinate squares is congruent modulo two to their ordinary sum.

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

∀ a. ∀ b. ModEq(2,a · a + b · b,a + b)

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 a b. (exists fsps_u_pair fsps_v_pair. (a * a + b * b) + 2 * fsps_u_pair = (a + b) + 2 * fsps_v_pair)

Proof neighborhood

Direct theorem prerequisites

FS003Q four_square_parity_square_mod_two_self mod_eq_add · Stable closed

Direct theorem dependents

FS003S four_square_parity_triple_mod_two_sum

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

10 script commands · 2 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)
four square parity square mod two self
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–10

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

  1. L3
    specialize mod_eq_add 2
  2. L4
    specialize mod_eq_add (a * a)
  3. L5
    specialize mod_eq_add a
  4. L6
    specialize mod_eq_add (b * b)
  5. L7
    specialize mod_eq_add b
  6. L8
    apply mod_eq_add
  7. L9
    apply four_square_parity_square_mod_two_self
  8. L10
    apply four_square_parity_square_mod_two_self

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003specialize mod_eq_add 2
  4. 0004specialize mod_eq_add (a * a)
  5. 0005specialize mod_eq_add a
  6. 0006specialize mod_eq_add (b * b)
  7. 0007specialize mod_eq_add b
  8. 0008apply mod_eq_add
  9. 0009apply four_square_parity_square_mod_two_self
  10. 0010apply four_square_parity_square_mod_two_self