PF001X

pythagorean_primitive_odd_even_inverse

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 39 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

PF001V pythagorean_odd_even_half_factors PF0016 pythagorean_primitive_hypotenuse_odd PF001W pythagorean_half_factors_extract_parameters

Direct dependents

PF001Z pythagorean_positive_primitive_inverse PF002K fermat_four_primitive_odd_even_descent

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

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.

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 : 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))))
  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 exact 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 : 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