PF001N · theorem body

pythagorean_half_factors_coprime

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

Every common divisor of the two half-factors divides the original odd leg and hypotenuse, so the factors 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

∀ a. ∀ c. ∀ t. Coprime(a,c) → c = a + 2 · t → Coprime(a + t,t)

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
Exact expanded first-order statement
forall a c t. (forall pff_divisor_pi_half_source. (exists pff_left_pi_half_source. (a) = pff_divisor_pi_half_source * pff_left_pi_half_source) -> (exists pff_right_pi_half_source. (c) = pff_divisor_pi_half_source * pff_right_pi_half_source) -> pff_divisor_pi_half_source = 1) -> c = a + 2 * t -> (forall pff_divisor_pi_half_result. (exists pff_left_pi_half_result. (a + t) = pff_divisor_pi_half_result * pff_left_pi_half_result) -> (exists pff_right_pi_half_result. (t) = pff_divisor_pi_half_result * pff_right_pi_half_result) -> pff_divisor_pi_half_result = 1)

Proof neighborhood

Direct theorem prerequisites

divides_remainder · Stable closed multiple_add · Stable closed PF001L pythagorean_half_hypotenuse_reassociation mul_one · Stable closed add_comm · Stable closed

Direct theorem dependents

PF001V pythagorean_odd_even_half_factors

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 · 8 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 half hypotenuse reassociation
01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro c
  3. L3
    intro t
  4. L4
    intro hcop
  5. L5
    intro hgap
  6. L6
    intro divisor
  7. L7
    intro hu
  8. L8
    intro hv
02Establish haL9–18

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

  1. L9
  2. L10
    specialize divides_remainder divisor
  3. L11
    specialize divides_remainder (a + t)
  4. L12
    specialize divides_remainder t
  5. L13
    specialize divides_remainder 1
  6. L14
    specialize divides_remainder a
  7. L15
    apply divides_remainder
  8. L16
    exact hu
  9. L17
    exact hv
  10. L18
    trans t + a
03Use earlier factsL19–19

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

  1. L19
    apply add_comm
04Calculate and transport equalitiesL20–21

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

  1. L20
    congr
  2. L21
    symm
05Use earlier factsL22–22

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

  1. L22
    apply mul_one
06Calculate and transport equalitiesL23–23

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

  1. L23
    refl
07Establish hcL24–33

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

  1. L24
  2. L25
    rewrite hgap
  3. L26
    specialize pythagorean_half_hypotenuse_reassociation a
  4. L27
    specialize pythagorean_half_hypotenuse_reassociation t
  5. L28
    rewrite pythagorean_half_hypotenuse_reassociation
  6. L29
    specialize multiple_add divisor
  7. L30
    specialize multiple_add (a + t)
  8. L31
    specialize multiple_add t
  9. L32
    apply multiple_add
  10. L33
    exact hu
08Use earlier factsL34–38

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

  1. L34
    exact hv
  2. L35
    specialize hcop divisor
  3. L36
    apply hcop
  4. L37
    exact ha
  5. L38
    exact hc

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro a
  2. 0002intro c
  3. 0003intro t
  4. 0004intro hcop
  5. 0005intro hgap
  6. 0006intro divisor
  7. 0007intro hu
  8. 0008intro hv
  9. 0009have ha : Dvd(divisor,a)
    Exact native replay linehave ha : exists q. a = divisor * q
  10. 0010specialize divides_remainder divisor
  11. 0011specialize divides_remainder (a + t)
  12. 0012specialize divides_remainder t
  13. 0013specialize divides_remainder 1
  14. 0014specialize divides_remainder a
  15. 0015apply divides_remainder
  16. 0016exact hu
  17. 0017exact hv
  18. 0018trans t + a
  19. 0019apply add_comm
  20. 0020congr
  21. 0021symm
  22. 0022apply mul_one
  23. 0023refl
  24. 0024have hc : Dvd(divisor,c)
    Exact native replay linehave hc : exists q. c = divisor * q
  25. 0025rewrite hgap
  26. 0026specialize pythagorean_half_hypotenuse_reassociation a
  27. 0027specialize pythagorean_half_hypotenuse_reassociation t
  28. 0028rewrite pythagorean_half_hypotenuse_reassociation
  29. 0029specialize multiple_add divisor
  30. 0030specialize multiple_add (a + t)
  31. 0031specialize multiple_add t
  32. 0032apply multiple_add
  33. 0033exact hu
  34. 0034exact hv
  35. 0035specialize hcop divisor
  36. 0036apply hcop
  37. 0037exact ha
  38. 0038exact hc