PF000Q · theorem body

pythagorean_square_gap_coprime_second_parameter

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

A square-difference leg is coprime to the second Euclidean parameter whenever the parameters themselves are coprime.

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 → Coprime(m,n) → Coprime(d,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

PD0005 Coprime

In local proof propositions

PD0003 Dvd PD0005 Coprime
Exact expanded first-order statement
forall m n d. m * m = n * n + d -> (forall pff_divisor_primitive_parameters. (exists pff_left_primitive_parameters. (m) = pff_divisor_primitive_parameters * pff_left_primitive_parameters) -> (exists pff_right_primitive_parameters. (n) = pff_divisor_primitive_parameters * pff_right_primitive_parameters) -> pff_divisor_primitive_parameters = 1) -> (forall pff_divisor_gap_second_result. (exists pff_left_gap_second_result. (d) = pff_divisor_gap_second_result * pff_left_gap_second_result) -> (exists pff_right_gap_second_result. (n) = pff_divisor_gap_second_result * pff_right_gap_second_result) -> pff_divisor_gap_second_result = 1)

Proof neighborhood

Direct theorem prerequisites

PF000O pythagorean_parameter_divisor_divides_square multiple_add · Stable closed coprime_symm · Stable closed coprime_mul_right · Stable closed

Direct theorem dependents

PF000R pythagorean_square_gap_coprime_parameter_product PF002G fermat_four_nested_primitive_triangle

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

37 script commands · 6 reading checkpoints · 4 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 parameter divisor divides square
01Fix variables and assumptionsL1–8

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
  5. L5
    intro hcoprime
  6. L6
    intro divisor
  7. L7
    intro hd
  8. L8
    intro hn
02Establish hnsquareL9–13

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

  1. L9
    have hnsquare : Dvd(divisor,n · n)Definitions: Dvd(divisor,n · n)Original native command in the exact edition
  2. L10
    specialize pythagorean_parameter_divisor_divides_square divisor
  3. L11
    specialize pythagorean_parameter_divisor_divides_square n
  4. L12
    apply pythagorean_parameter_divisor_divides_square
  5. L13
    exact hn
03Establish hmsquareL14–21

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

  1. L14
    have hmsquare : Dvd(divisor,m · m)Definitions: Dvd(divisor,m · m)Original native command in the exact edition
  2. L15
    rewrite hgap
  3. L16
    specialize multiple_add divisor
  4. L17
    specialize multiple_add (n * n)
  5. L18
    specialize multiple_add d
  6. L19
    apply multiple_add
  7. L20
    exact hnsquare
  8. L21
    exact hd
04Establish hreverseL22–26

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

  1. L22
    have hreverse : Coprime(n,m)Definitions: Coprime(n,m)Original native command in the exact edition
  2. L23
    specialize coprime_symm m
  3. L24
    specialize coprime_symm n
  4. L25
    apply coprime_symm
  5. L26
    exact hcoprime
05Establish hsquarecopL27–36

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply coprime mul right.

  1. L27
    have hsquarecop : Coprime(n,m · m)Definitions: Coprime(n,m · m)Original native command in the exact edition
  2. L28
    specialize coprime_mul_right n
  3. L29
    specialize coprime_mul_right m
  4. L30
    specialize coprime_mul_right m
  5. L31
    apply coprime_mul_right
  6. L32
    exact hreverse
  7. L33
    exact hreverse
  8. L34
    specialize hsquarecop divisor
  9. L35
    apply hsquarecop
  10. L36
    exact hn
06Use earlier factsL37–37

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

  1. L37
    exact hmsquare

Library-wide reading audit

Original defined command ledger · 37 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hcoprime
  6. 0006intro divisor
  7. 0007intro hd
  8. 0008intro hn
  9. 0009have hnsquare : Dvd(divisor,n · n)
    Exact native replay linehave hnsquare : exists q. n * n = divisor * q
  10. 0010specialize pythagorean_parameter_divisor_divides_square divisor
  11. 0011specialize pythagorean_parameter_divisor_divides_square n
  12. 0012apply pythagorean_parameter_divisor_divides_square
  13. 0013exact hn
  14. 0014have hmsquare : Dvd(divisor,m · m)
    Exact native replay linehave hmsquare : exists q. m * m = divisor * q
  15. 0015rewrite hgap
  16. 0016specialize multiple_add divisor
  17. 0017specialize multiple_add (n * n)
  18. 0018specialize multiple_add d
  19. 0019apply multiple_add
  20. 0020exact hnsquare
  21. 0021exact hd
  22. 0022have hreverse : Coprime(n,m)
    Exact native replay linehave hreverse : forall pff_divisor_second_reverse_local. (exists pff_left_second_reverse_local. (n) = pff_divisor_second_reverse_local * pff_left_second_reverse_local) -> (exists pff_right_second_reverse_local. (m) = pff_divisor_second_reverse_local * pff_right_second_reverse_local) -> pff_divisor_second_reverse_local = 1
  23. 0023specialize coprime_symm m
  24. 0024specialize coprime_symm n
  25. 0025apply coprime_symm
  26. 0026exact hcoprime
  27. 0027have hsquarecop : Coprime(n,m · m)
    Exact native replay linehave hsquarecop : forall pff_divisor_second_square_local. (exists pff_left_second_square_local. (n) = pff_divisor_second_square_local * pff_left_second_square_local) -> (exists pff_right_second_square_local. (m * m) = pff_divisor_second_square_local * pff_right_second_square_local) -> pff_divisor_second_square_local = 1
  28. 0028specialize coprime_mul_right n
  29. 0029specialize coprime_mul_right m
  30. 0030specialize coprime_mul_right m
  31. 0031apply coprime_mul_right
  32. 0032exact hreverse
  33. 0033exact hreverse
  34. 0034specialize hsquarecop divisor
  35. 0035apply hsquarecop
  36. 0036exact hn
  37. 0037exact hmsquare