PF000W

pythagorean_primitive_hypotenuse_coprime_first_leg

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

In every primitive Pythagorean triple, the first leg and hypotenuse have no nonunit common divisor.

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.

Exact expanded first-order arithmetic statement

forall x y z. ((((x) * (x) + (y) * (y) = (z) * (z)) /\ (forall pff_divisor_pairwise_source. (exists pff_left_pairwise_source. (x) = pff_divisor_pairwise_source * pff_left_pairwise_source) -> (exists pff_right_pairwise_source. (y) = pff_divisor_pairwise_source * pff_right_pairwise_source) -> pff_divisor_pairwise_source = 1))) -> (forall pff_divisor_first_hypotenuse_result. (exists pff_left_first_hypotenuse_result. (x) = pff_divisor_first_hypotenuse_result * pff_left_first_hypotenuse_result) -> (exists pff_right_first_hypotenuse_result. (z) = pff_divisor_first_hypotenuse_result * pff_right_first_hypotenuse_result) -> pff_divisor_first_hypotenuse_result = 1)

Constructive proof overview

Generated structural guide

In every primitive Pythagorean triple, the first leg and hypotenuse have no nonunit common divisor.

The unchanged tactic script uses 4 declared prerequisites and contains 42 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

PF000O pythagorean_parameter_divisor_divides_square divides_remainder Stable theorem; checked-use authorized mul_one Stable theorem; checked-use authorized coprime_mul_right Stable theorem; checked-use authorized

Direct dependents

PF000X pythagorean_primitive_hypotenuse_coprime_second_leg PF000Y pythagorean_primitive_pairwise_coprime PF001V pythagorean_odd_even_half_factors

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

42 script commands · 10 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.

Named ingredients (1)
pythagorean parameter divisor divides square
01Fix variables and assumptionsL1–4

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
  4. L4
    intro hprimitive
02Separate the logical casesL5–5

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

  1. L5
    cases hprimitive
03Fix variables and assumptionsL6–8

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

  1. L6
    intro divisor
  2. L7
    intro hx
  3. L8
    intro hz
04Establish hxsquareL9–13

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 hxsquare : exists q. x * x = divisor * q
  2. L10
    specialize pythagorean_parameter_divisor_divides_square divisor
  3. L11
    specialize pythagorean_parameter_divisor_divides_square x
  4. L12
    apply pythagorean_parameter_divisor_divides_square
  5. L13
    exact hx
05Establish hzsquareL14–18

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

  1. L14
    have hzsquare : exists q. z * z = divisor * q
  2. L15
    specialize pythagorean_parameter_divisor_divides_square divisor
  3. L16
    specialize pythagorean_parameter_divisor_divides_square z
  4. L17
    apply pythagorean_parameter_divisor_divides_square
  5. L18
    exact hz
06Establish hysquareL19–28

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

  1. L19
    have hysquare : exists q. y * y = divisor * q
  2. L20
    specialize divides_remainder divisor
  3. L21
    specialize divides_remainder (z * z)
  4. L22
    specialize divides_remainder (x * x)
  5. L23
    specialize divides_remainder 1
  6. L24
    specialize divides_remainder (y * y)
  7. L25
    apply divides_remainder
  8. L26
    exact hzsquare
  9. L27
    exact hxsquare
  10. L28
    specialize mul_one (x * x)
07Calculate and transport equalitiesL29–30

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

  1. L29
    rewrite mul_one
  2. L30
    symm
08Use earlier factsL31–31

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

  1. L31
    exact hprimitive_left
09Establish hsquarecopL32–41

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

  1. L32
    have hsquarecop : forall pff_divisor_hypotenuse_first_square. (exists pff_left_hypotenuse_first_square. (x) = pff_divisor_hypotenuse_first_square * pff_left_hypotenuse_first_square) -> (exists pff_right_hypotenuse_first_square. (y * y) = pff_divisor_hypotenuse_first_square * pff_right_hypotenuse_first_square) -> pff_divisor_hypotenuse_first_square = 1
  2. L33
    specialize coprime_mul_right x
  3. L34
    specialize coprime_mul_right y
  4. L35
    specialize coprime_mul_right y
  5. L36
    apply coprime_mul_right
  6. L37
    exact hprimitive_right
  7. L38
    exact hprimitive_right
  8. L39
    specialize hsquarecop divisor
  9. L40
    apply hsquarecop
  10. L41
    exact hx
10Use earlier factsL42–42

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

  1. L42
    exact hysquare

Library-wide reading audit

Original exact command ledger · 42 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro hprimitive
  5. 0005cases hprimitive
  6. 0006intro divisor
  7. 0007intro hx
  8. 0008intro hz
  9. 0009have hxsquare : exists q. x * x = divisor * q
  10. 0010specialize pythagorean_parameter_divisor_divides_square divisor
  11. 0011specialize pythagorean_parameter_divisor_divides_square x
  12. 0012apply pythagorean_parameter_divisor_divides_square
  13. 0013exact hx
  14. 0014have hzsquare : exists q. z * z = divisor * q
  15. 0015specialize pythagorean_parameter_divisor_divides_square divisor
  16. 0016specialize pythagorean_parameter_divisor_divides_square z
  17. 0017apply pythagorean_parameter_divisor_divides_square
  18. 0018exact hz
  19. 0019have hysquare : exists q. y * y = divisor * q
  20. 0020specialize divides_remainder divisor
  21. 0021specialize divides_remainder (z * z)
  22. 0022specialize divides_remainder (x * x)
  23. 0023specialize divides_remainder 1
  24. 0024specialize divides_remainder (y * y)
  25. 0025apply divides_remainder
  26. 0026exact hzsquare
  27. 0027exact hxsquare
  28. 0028specialize mul_one (x * x)
  29. 0029rewrite mul_one
  30. 0030symm
  31. 0031exact hprimitive_left
  32. 0032have hsquarecop : forall pff_divisor_hypotenuse_first_square. (exists pff_left_hypotenuse_first_square. (x) = pff_divisor_hypotenuse_first_square * pff_left_hypotenuse_first_square) -> (exists pff_right_hypotenuse_first_square. (y * y) = pff_divisor_hypotenuse_first_square * pff_right_hypotenuse_first_square) -> pff_divisor_hypotenuse_first_square = 1
  33. 0033specialize coprime_mul_right x
  34. 0034specialize coprime_mul_right y
  35. 0035specialize coprime_mul_right y
  36. 0036apply coprime_mul_right
  37. 0037exact hprimitive_right
  38. 0038exact hprimitive_right
  39. 0039specialize hsquarecop divisor
  40. 0040apply hsquarecop
  41. 0041exact hx
  42. 0042exact hysquare