PF001X · theorem body

pythagorean_primitive_odd_even_inverse

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

Every primitive Pythagorean triangle with positive legs in odd-even order has a fully witnessed Euclidean inverse parametrization.

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. ∀ b. ∀ c. PrimitivePythagorean(a,b,c) → ¬a = 0 → ¬b = 0 → Odd(a) → Even(b) → ∃ x. ∃ y. EuclidParameters(a,b,c,x,y)

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 ND0070 EuclidParameters PD0009 Even PD0010 Odd

In local proof propositions

PD0005 Coprime
Exact expanded first-order statement
forall a b c. ((((a) * (a) + (b) * (b) = (c) * (c)) /\ (forall pff_divisor_pi_source. (exists pff_left_pi_source. (a) = pff_divisor_pi_source * pff_left_pi_source) -> (exists pff_right_pi_source. (b) = pff_divisor_pi_source * pff_right_pi_source) -> pff_divisor_pi_source = 1))) -> ~(a = 0) -> ~(b = 0) -> (exists pp_odd_pi_inverse_a. (a) = 2 * pp_odd_pi_inverse_a + 1) -> (exists pp_even_pi_inverse_b. (b) = 2 * pp_even_pi_inverse_b) -> exists m n. ((~((n) = 0) /\ ((exists pi_gap_inverse_result. pi_gap_inverse_result + S (n) = (m)) /\ ((forall pff_divisor_pi_inverse_result_coprime. (exists pff_left_pi_inverse_result_coprime. (m) = pff_divisor_pi_inverse_result_coprime * pff_left_pi_inverse_result_coprime) -> (exists pff_right_pi_inverse_result_coprime. (n) = pff_divisor_pi_inverse_result_coprime * pff_right_pi_inverse_result_coprime) -> pff_divisor_pi_inverse_result_coprime = 1) /\ (((((exists pp_even_pi_inverse_result_parity_first_even. (m) = 2 * pp_even_pi_inverse_result_parity_first_even) /\ (exists pp_odd_pi_inverse_result_parity_second_odd. (n) = 2 * pp_odd_pi_inverse_result_parity_second_odd + 1)) \/ ((exists pp_odd_pi_inverse_result_parity_first_odd. (m) = 2 * pp_odd_pi_inverse_result_parity_first_odd + 1) /\ (exists pp_even_pi_inverse_result_parity_second_even. (n) = 2 * pp_even_pi_inverse_result_parity_second_even)))) /\ ((c) = (m) * (m) + (n) * (n) /\ ((m) * (m) = (n) * (n) + (a) /\ (b) = 2 * ((m) * (n)))))))))

Proof neighborhood

Direct theorem prerequisites

PF001V pythagorean_odd_even_half_factors PF0016 pythagorean_primitive_hypotenuse_odd PF001W pythagorean_half_factors_extract_parameters

Direct theorem dependents

PF001Z pythagorean_positive_primitive_inverse 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

39 script commands · 5 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 (3)
pythagorean odd even half factorspythagorean primitive hypotenuse oddpythagorean half factors extract parameters
01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro hp
  5. L5
    intro ha
  6. L6
    intro hb
  7. L7
    intro haodd
  8. L8
    intro hbeven
02Establish hhalvesL9–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean odd even half factors.

  1. L9
    have hhalves : ∃ h. ∃ t. b = 2 · h ∧ (c = a + 2 · t ∧ ((a + t) · t = h · h ∧ Coprime(a + t,t)))Definitions: Coprime(a + t,t)Original native command in the exact edition
  2. L10
    specialize pythagorean_odd_even_half_factors a
  3. L11
    specialize pythagorean_odd_even_half_factors b
  4. L12
    specialize pythagorean_odd_even_half_factors c
  5. L13
    apply pythagorean_odd_even_half_factors
  6. L14
    exact hp
  7. L15
    exact hb
  8. L16
    exact haodd
  9. L17
    exact hbeven
03Separate the logical casesL18–22

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

  1. L18
    cases hhalves
  2. L19
    cases hhalves_witness
  3. L20
    cases hhalves_witness_witness
  4. L21
    cases hhalves_witness_witness_right
  5. L22
    cases hhalves_witness_witness_right_right
04Use earlier factsL23–32

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

  1. L23
    specialize pythagorean_half_factors_extract_parameters a
  2. L24
    specialize pythagorean_half_factors_extract_parameters b
  3. L25
    specialize pythagorean_half_factors_extract_parameters c
  4. L26
    specialize pythagorean_half_factors_extract_parameters x
  5. L27
    specialize pythagorean_half_factors_extract_parameters x1
  6. L28
    apply pythagorean_half_factors_extract_parameters
  7. L29
    exact ha
  8. L30
    exact hb
  9. L31
    specialize pythagorean_primitive_hypotenuse_odd a
  10. L32
    specialize pythagorean_primitive_hypotenuse_odd b
05Use earlier factsL33–39

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

  1. L33
    specialize pythagorean_primitive_hypotenuse_odd c
  2. L34
    apply pythagorean_primitive_hypotenuse_odd
  3. L35
    exact hp
  4. L36
    exact hhalves_witness_witness_left
  5. L37
    exact hhalves_witness_witness_right_left
  6. L38
    exact hhalves_witness_witness_right_right_left
  7. L39
    exact hhalves_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 39 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro hp
  5. 0005intro ha
  6. 0006intro hb
  7. 0007intro haodd
  8. 0008intro hbeven
  9. 0009have hhalves : ∃ h. ∃ t. b = 2 · h ∧ (c = a + 2 · t ∧ ((a + t) · t = h · h ∧ Coprime(a + t,t)))
    Exact native replay linehave hhalves : exists h t. (b = 2 * h /\ (c = a + 2 * t /\ ((a + t) * t = h * h /\ (forall pff_divisor_pi_inverse_halves. (exists pff_left_pi_inverse_halves. (a + t) = pff_divisor_pi_inverse_halves * pff_left_pi_inverse_halves) -> (exists pff_right_pi_inverse_halves. (t) = pff_divisor_pi_inverse_halves * pff_right_pi_inverse_halves) -> pff_divisor_pi_inverse_halves = 1))))
  10. 0010specialize pythagorean_odd_even_half_factors a
  11. 0011specialize pythagorean_odd_even_half_factors b
  12. 0012specialize pythagorean_odd_even_half_factors c
  13. 0013apply pythagorean_odd_even_half_factors
  14. 0014exact hp
  15. 0015exact hb
  16. 0016exact haodd
  17. 0017exact hbeven
  18. 0018cases hhalves
  19. 0019cases hhalves_witness
  20. 0020cases hhalves_witness_witness
  21. 0021cases hhalves_witness_witness_right
  22. 0022cases hhalves_witness_witness_right_right
  23. 0023specialize pythagorean_half_factors_extract_parameters a
  24. 0024specialize pythagorean_half_factors_extract_parameters b
  25. 0025specialize pythagorean_half_factors_extract_parameters c
  26. 0026specialize pythagorean_half_factors_extract_parameters x
  27. 0027specialize pythagorean_half_factors_extract_parameters x1
  28. 0028apply pythagorean_half_factors_extract_parameters
  29. 0029exact ha
  30. 0030exact hb
  31. 0031specialize pythagorean_primitive_hypotenuse_odd a
  32. 0032specialize pythagorean_primitive_hypotenuse_odd b
  33. 0033specialize pythagorean_primitive_hypotenuse_odd c
  34. 0034apply pythagorean_primitive_hypotenuse_odd
  35. 0035exact hp
  36. 0036exact hhalves_witness_witness_left
  37. 0037exact hhalves_witness_witness_right_left
  38. 0038exact hhalves_witness_witness_right_right_left
  39. 0039exact hhalves_witness_witness_right_right_right