TS0013 · theorem body

negative_one_scaled_square_identity

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

Distributivity identifies the scaled negative-one polynomial with the linear square plus its coordinate square.

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 r x. (r * r + 1) * (x * x) = (r * x) * (r * x) + x * x

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 r x. (r * r + 1) * (x * x) = (r * x) * (r * x) + x * x

Proof neighborhood

Direct theorem prerequisites

add_mul · Stable closed one_mul · Stable closed mul_assoc · Stable closed mul_comm · Stable closed

Direct theorem dependents

TS0014 negative_one_scaled_square_congruent_zero

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

29 script commands · 15 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 assumptionsL1–2

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

  1. L1
    intro r
  2. L2
    intro x
02Use earlier factsL3–5

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

  1. L3
    specialize add_mul (r * r)
  2. L4
    specialize add_mul 1
  3. L5
    specialize add_mul (x * x)
03Calculate and transport equalitiesL6–6

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

  1. L6
    rewrite add_mul
04Use earlier factsL7–7

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

  1. L7
    specialize one_mul (x * x)
05Calculate and transport equalitiesL8–10

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

  1. L8
    rewrite one_mul
  2. L9
    congr
  3. L10
    trans r * (r * (x * x))
06Use earlier factsL11–11

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

  1. L11
    apply mul_assoc
07Calculate and transport equalitiesL12–15

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

  1. L12
    trans r * ((r * x) * x)
  2. L13
    congr
  3. L14
    refl
  4. L15
    symm
08Use earlier factsL16–16

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

  1. L16
    apply mul_assoc
09Calculate and transport equalitiesL17–20

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

  1. L17
    trans r * ((x * r) * x)
  2. L18
    congr
  3. L19
    refl
  4. L20
    congr
10Use earlier factsL21–21

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

  1. L21
    apply mul_comm
11Calculate and transport equalitiesL22–25

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

  1. L22
    refl
  2. L23
    trans r * (x * (r * x))
  3. L24
    congr
  4. L25
    refl
12Use earlier factsL26–26

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

  1. L26
    apply mul_assoc
13Calculate and transport equalitiesL27–27

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

  1. L27
    symm
14Use earlier factsL28–28

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

  1. L28
    apply mul_assoc
15Calculate and transport equalitiesL29–29

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

  1. L29
    refl

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro r
  2. 0002intro x
  3. 0003specialize add_mul (r * r)
  4. 0004specialize add_mul 1
  5. 0005specialize add_mul (x * x)
  6. 0006rewrite add_mul
  7. 0007specialize one_mul (x * x)
  8. 0008rewrite one_mul
  9. 0009congr
  10. 0010trans r * (r * (x * x))
  11. 0011apply mul_assoc
  12. 0012trans r * ((r * x) * x)
  13. 0013congr
  14. 0014refl
  15. 0015symm
  16. 0016apply mul_assoc
  17. 0017trans r * ((x * r) * x)
  18. 0018congr
  19. 0019refl
  20. 0020congr
  21. 0021apply mul_comm
  22. 0022refl
  23. 0023trans r * (x * (r * x))
  24. 0024congr
  25. 0025refl
  26. 0026apply mul_assoc
  27. 0027symm
  28. 0028apply mul_assoc
  29. 0029refl