PF002G · theorem body

fermat_four_nested_primitive_triangle

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

The square odd leg in the first parametrization exposes a second primitive Pythagorean triangle on the unsquared base.

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. ∀ m. ∀ n. Coprime(m,n) → m · m = n · n + a · a → PrimitivePythagorean(a,n,m)

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

CF0013 PrimitivePythagorean PD0005 Coprime

In local proof propositions

PD0005 Coprime
Exact expanded first-order statement
forall a m n. (forall pff_divisor_ffd_nested_parameters. (exists pff_left_ffd_nested_parameters. (m) = pff_divisor_ffd_nested_parameters * pff_left_ffd_nested_parameters) -> (exists pff_right_ffd_nested_parameters. (n) = pff_divisor_ffd_nested_parameters * pff_right_ffd_nested_parameters) -> pff_divisor_ffd_nested_parameters = 1) -> m * m = n * n + a * a -> ((((a) * (a) + (n) * (n) = (m) * (m)) /\ (forall pff_divisor_ffd_nested_result. (exists pff_left_ffd_nested_result. (a) = pff_divisor_ffd_nested_result * pff_left_ffd_nested_result) -> (exists pff_right_ffd_nested_result. (n) = pff_divisor_ffd_nested_result * pff_right_ffd_nested_result) -> pff_divisor_ffd_nested_result = 1)))

Proof neighborhood

Direct theorem prerequisites

PF000Q pythagorean_square_gap_coprime_second_parameter crt_coprime_divisor_pair · Alpha closed mul_one · Stable closed add_comm · Stable closed

Direct theorem dependents

PF002K fermat_four_primitive_odd_even_descent

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

28 script commands · 12 reading checkpoints · 1 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 square gap coprime second parameter
01Fix variables and assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro n
  4. L4
    intro hcoprime
  5. L5
    intro hgap
02Establish hsquareL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean square gap coprime second parameter.

  1. L6
    have hsquare : Coprime(a · a,n)Definitions: Coprime(a · a,n)Original native command in the exact edition
  2. L7
    specialize pythagorean_square_gap_coprime_second_parameter (m)
  3. L8
    specialize pythagorean_square_gap_coprime_second_parameter (n)
  4. L9
    specialize pythagorean_square_gap_coprime_second_parameter (a * a)
  5. L10
    apply pythagorean_square_gap_coprime_second_parameter
  6. L11
    exact hgap
  7. L12
    exact hcoprime
03Separate the logical casesL13–13

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

  1. L13
    split
04Calculate and transport equalitiesL14–14

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

  1. L14
    trans n * n + a * a
05Use earlier factsL15–15

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

  1. L15
    apply add_comm
06Calculate and transport equalitiesL16–16

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

  1. L16
    symm
07Use earlier factsL17–23

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

  1. L17
    exact hgap
  2. L18
    specialize crt_coprime_divisor_pair (a * a)
  3. L19
    specialize crt_coprime_divisor_pair (n)
  4. L20
    specialize crt_coprime_divisor_pair (a)
  5. L21
    specialize crt_coprime_divisor_pair (n)
  6. L22
    apply crt_coprime_divisor_pair
  7. L23
    exact hsquare
08Construct an explicit witnessL24–24

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

  1. L24
    exists a
09Calculate and transport equalitiesL25–25

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

  1. L25
    refl
10Construct an explicit witnessL26–26

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

  1. L26
    exists 1
11Calculate and transport equalitiesL27–27

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

  1. L27
    symm
12Use earlier factsL28–28

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

  1. L28
    apply mul_one

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro n
  4. 0004intro hcoprime
  5. 0005intro hgap
  6. 0006have hsquare : Coprime(a · a,n)
    Exact native replay linehave hsquare : forall pff_divisor_ffd_nested_square. (exists pff_left_ffd_nested_square. (a * a) = pff_divisor_ffd_nested_square * pff_left_ffd_nested_square) -> (exists pff_right_ffd_nested_square. (n) = pff_divisor_ffd_nested_square * pff_right_ffd_nested_square) -> pff_divisor_ffd_nested_square = 1
  7. 0007specialize pythagorean_square_gap_coprime_second_parameter (m)
  8. 0008specialize pythagorean_square_gap_coprime_second_parameter (n)
  9. 0009specialize pythagorean_square_gap_coprime_second_parameter (a * a)
  10. 0010apply pythagorean_square_gap_coprime_second_parameter
  11. 0011exact hgap
  12. 0012exact hcoprime
  13. 0013split
  14. 0014trans n * n + a * a
  15. 0015apply add_comm
  16. 0016symm
  17. 0017exact hgap
  18. 0018specialize crt_coprime_divisor_pair (a * a)
  19. 0019specialize crt_coprime_divisor_pair (n)
  20. 0020specialize crt_coprime_divisor_pair (a)
  21. 0021specialize crt_coprime_divisor_pair (n)
  22. 0022apply crt_coprime_divisor_pair
  23. 0023exact hsquare
  24. 0024exists a
  25. 0025refl
  26. 0026exists 1
  27. 0027symm
  28. 0028apply mul_one