FS0047 · theorem body

four_square_ordered_square_difference_factor

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

The ordered difference of two natural squares factors subtraction-free as their gap times their sum.

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

forall x y d. y = x + d -> x * x + d * (y + x) = y * y

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

none

In local proof propositions

none
Exact expanded first-order statement
forall x y d. y = x + d -> x * x + d * (y + x) = y * y

Proof neighborhood

Direct theorem prerequisites

mul_add · Stable closed add_mul · Stable closed mul_comm · Stable closed add_assoc · Stable closed add_comm · Stable closed

Direct theorem dependents

FS0048 four_square_ordered_square_congruence_factors

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 · 7 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 assumptionsL1–4

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro d
  4. L4
    intro horder
02Calculate and transport equalitiesL5–12

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

  1. L5
    rewrite horder
  2. L6
    rewrite horder
  3. L7
    rewrite horder
  4. L8
    simp [mul_add, add_mul, mul_comm, add_assoc, add_comm]
  5. L9
    congr
  6. L10
    refl
  7. L11
    trans (d * d + d * x) + x * x
  8. L12
    symm
03Use earlier factsL13–13

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

  1. L13
    apply add_assoc
04Calculate and transport equalitiesL14–15

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

  1. L14
    trans (d * x + d * d) + x * x
  2. L15
    congr
05Use earlier factsL16–16

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

  1. L16
    apply add_comm
06Calculate and transport equalitiesL17–17

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

  1. L17
    refl
07Use earlier factsL18–18

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

  1. L18
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro d
  4. 0004intro horder
  5. 0005rewrite horder
  6. 0006rewrite horder
  7. 0007rewrite horder
  8. 0008simp [mul_add, add_mul, mul_comm, add_assoc, add_comm]
  9. 0009congr
  10. 0010refl
  11. 0011trans (d * d + d * x) + x * x
  12. 0012symm
  13. 0013apply add_assoc
  14. 0014trans (d * x + d * d) + x * x
  15. 0015congr
  16. 0016apply add_comm
  17. 0017refl
  18. 0018apply add_assoc