PF0001 · theorem body

pythagorean_euclidean_identity

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

Euclid's subtraction-free Pythagorean identity follows from the witnessed square difference and the checked Brahmagupta identity.

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

∀ m. ∀ n. ∀ d. m · m = n · n + d → Pythagorean(d,2 · (m · n),m · m + n · n)

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

CF0011 Pythagorean

In local proof propositions

none
Exact expanded first-order statement
forall m n d. m * m = n * n + d -> ((d) * (d) + (2 * (m * n)) * (2 * (m * n)) = (m * m + n * n) * (m * m + n * n))

Proof neighborhood

Direct theorem prerequisites

brahmagupta_fibonacci_two_square_identity · Alpha closed add_comm · Stable closed PF0000 pythagorean_double_product

Direct theorem dependents

PF0002 pythagorean_euclidean_constructor PF0004 pythagorean_euclidean_swapped_constructor PF000T pythagorean_primitive_euclidean_constructor

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

24 script commands · 6 reading checkpoints · 2 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)
pythagorean double product
01Fix variables and assumptionsL1–4

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro d
  4. L4
    intro hgap
02Establish hidentityL5–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply brahmagupta fibonacci two square identity.

  1. L5
    have hidentity : (m * m + n * n) * (n * n + m * m) = (m * n + n * m) * (m * n + n * m) + d * d
  2. L6
    specialize brahmagupta_fibonacci_two_square_identity m
  3. L7
    specialize brahmagupta_fibonacci_two_square_identity n
  4. L8
    specialize brahmagupta_fibonacci_two_square_identity n
  5. L9
    specialize brahmagupta_fibonacci_two_square_identity m
  6. L10
    specialize brahmagupta_fibonacci_two_square_identity d
  7. L11
    apply brahmagupta_fibonacci_two_square_identity
03Separate the logical casesL12–12

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

  1. L12
    left
04Use earlier factsL13–13

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

  1. L13
    exact hgap
05Establish hsumL14–23

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

  1. L14
    have hsum : n * n + m * m = m * m + n * n
  2. L15
    apply add_comm
  3. L16
    rewrite hsum at hidentity
  4. L17
    specialize pythagorean_double_product m
  5. L18
    specialize pythagorean_double_product n
  6. L19
    rewrite pythagorean_double_product at hidentity
  7. L20
    rewrite pythagorean_double_product at hidentity
  8. L21
    trans (2 * (m * n)) * (2 * (m * n)) + d * d
  9. L22
    apply add_comm
  10. L23
    symm
06Use earlier factsL24–24

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

  1. L24
    exact hidentity

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005have hidentity : (m * m + n * n) * (n * n + m * m) = (m * n + n * m) * (m * n + n * m) + d * d
  6. 0006specialize brahmagupta_fibonacci_two_square_identity m
  7. 0007specialize brahmagupta_fibonacci_two_square_identity n
  8. 0008specialize brahmagupta_fibonacci_two_square_identity n
  9. 0009specialize brahmagupta_fibonacci_two_square_identity m
  10. 0010specialize brahmagupta_fibonacci_two_square_identity d
  11. 0011apply brahmagupta_fibonacci_two_square_identity
  12. 0012left
  13. 0013exact hgap
  14. 0014have hsum : n * n + m * m = m * m + n * n
  15. 0015apply add_comm
  16. 0016rewrite hsum at hidentity
  17. 0017specialize pythagorean_double_product m
  18. 0018specialize pythagorean_double_product n
  19. 0019rewrite pythagorean_double_product at hidentity
  20. 0020rewrite pythagorean_double_product at hidentity
  21. 0021trans (2 * (m * n)) * (2 * (m * n)) + d * d
  22. 0022apply add_comm
  23. 0023symm
  24. 0024exact hidentity