PF000P · theorem body

pythagorean_square_gap_coprime_first_parameter

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

A square-difference leg is coprime to the first 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,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

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_first_result. (exists pff_left_gap_first_result. (d) = pff_divisor_gap_first_result * pff_left_gap_first_result) -> (exists pff_right_gap_first_result. (m) = pff_divisor_gap_first_result * pff_right_gap_first_result) -> pff_divisor_gap_first_result = 1)

Proof neighborhood

Direct theorem prerequisites

PF000O pythagorean_parameter_divisor_divides_square divides_remainder · Stable closed mul_one · Stable closed add_comm · Stable closed coprime_mul_right · Stable closed

Direct theorem dependents

PF000R pythagorean_square_gap_coprime_parameter_product

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

38 script commands · 7 reading checkpoints · 3 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 hm
02Establish hmsquareL9–13

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 hmsquare : Dvd(divisor,m · m)Definitions: Dvd(divisor,m · m)Original native command in the exact edition
  2. L10
    specialize pythagorean_parameter_divisor_divides_square divisor
  3. L11
    specialize pythagorean_parameter_divisor_divides_square m
  4. L12
    apply pythagorean_parameter_divisor_divides_square
  5. L13
    exact hm
03Establish hnsquareL14–23

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

  1. L14
    have hnsquare : Dvd(divisor,n · n)Definitions: Dvd(divisor,n · n)Original native command in the exact edition
  2. L15
    specialize divides_remainder divisor
  3. L16
    specialize divides_remainder (m * m)
  4. L17
    specialize divides_remainder d
  5. L18
    specialize divides_remainder 1
  6. L19
    specialize divides_remainder (n * n)
  7. L20
    apply divides_remainder
  8. L21
    exact hmsquare
  9. L22
    exact hd
  10. L23
    specialize mul_one d
04Calculate and transport equalitiesL24–25

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

  1. L24
    rewrite mul_one
  2. L25
    trans n * n + d
05Use earlier factsL26–27

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

  1. L26
    exact hgap
  2. L27
    apply add_comm
06Establish hsquarecopL28–37

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

  1. L28
    have hsquarecop : Coprime(m,n · n)Definitions: Coprime(m,n · n)Original native command in the exact edition
  2. L29
    specialize coprime_mul_right m
  3. L30
    specialize coprime_mul_right n
  4. L31
    specialize coprime_mul_right n
  5. L32
    apply coprime_mul_right
  6. L33
    exact hcoprime
  7. L34
    exact hcoprime
  8. L35
    specialize hsquarecop divisor
  9. L36
    apply hsquarecop
  10. L37
    exact hm
07Use earlier factsL38–38

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

  1. L38
    exact hnsquare

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hcoprime
  6. 0006intro divisor
  7. 0007intro hd
  8. 0008intro hm
  9. 0009have hmsquare : Dvd(divisor,m · m)
    Exact native replay linehave hmsquare : exists q. m * m = divisor * q
  10. 0010specialize pythagorean_parameter_divisor_divides_square divisor
  11. 0011specialize pythagorean_parameter_divisor_divides_square m
  12. 0012apply pythagorean_parameter_divisor_divides_square
  13. 0013exact hm
  14. 0014have hnsquare : Dvd(divisor,n · n)
    Exact native replay linehave hnsquare : exists q. n * n = divisor * q
  15. 0015specialize divides_remainder divisor
  16. 0016specialize divides_remainder (m * m)
  17. 0017specialize divides_remainder d
  18. 0018specialize divides_remainder 1
  19. 0019specialize divides_remainder (n * n)
  20. 0020apply divides_remainder
  21. 0021exact hmsquare
  22. 0022exact hd
  23. 0023specialize mul_one d
  24. 0024rewrite mul_one
  25. 0025trans n * n + d
  26. 0026exact hgap
  27. 0027apply add_comm
  28. 0028have hsquarecop : Coprime(m,n · n)
    Exact native replay linehave hsquarecop : forall pff_divisor_first_square_local. (exists pff_left_first_square_local. (m) = pff_divisor_first_square_local * pff_left_first_square_local) -> (exists pff_right_first_square_local. (n * n) = pff_divisor_first_square_local * pff_right_first_square_local) -> pff_divisor_first_square_local = 1
  29. 0029specialize coprime_mul_right m
  30. 0030specialize coprime_mul_right n
  31. 0031specialize coprime_mul_right n
  32. 0032apply coprime_mul_right
  33. 0033exact hcoprime
  34. 0034exact hcoprime
  35. 0035specialize hsquarecop divisor
  36. 0036apply hsquarecop
  37. 0037exact hm
  38. 0038exact hnsquare