PF000N

pythagorean_odd_coordinate_coprime_two

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

Every witnessed odd natural is coprime to two, using the actual prime-two theorem and constructive prime nondivisibility.

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 d. (exists q. d = 2 * q + 1) -> (forall pff_divisor_odd_two_result. (exists pff_left_odd_two_result. (d) = pff_divisor_odd_two_result * pff_left_odd_two_result) -> (exists pff_right_odd_two_result. (2) = pff_divisor_odd_two_result * pff_right_odd_two_result) -> pff_divisor_odd_two_result = 1)

Constructive proof overview

Generated structural guide

Every witnessed odd natural is coprime to two, using the actual prime-two theorem and constructive prime nondivisibility.

The unchanged tactic script uses 4 declared prerequisites and contains 22 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

even_odd_exclusive_pointwise Stable theorem; checked-use authorized prime_two Stable theorem; checked-use authorized prime_not_divides_coprime Stable theorem; checked-use authorized coprime_symm Stable theorem; checked-use authorized

Direct dependents

PF000S pythagorean_primitive_euclidean_legs PF002E fermat_four_odd_double_square_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

22 script commands · 5 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro d
  2. L2
    intro hodd
02Establish hnotL3–4

Establish this local claim before using it. It is not an additional assumption.

  1. L3
    have hnot : ~(exists q. d = 2 * q)
  2. L4
    intro heven
03Separate the logical casesL5–6

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

  1. L5
    cases heven
  2. L6
    cases hodd
04Use earlier factsL7–12

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

  1. L7
    specialize even_odd_exclusive_pointwise d
  2. L8
    specialize even_odd_exclusive_pointwise x
  3. L9
    specialize even_odd_exclusive_pointwise x1
  4. L10
    apply even_odd_exclusive_pointwise
  5. L11
    exact heven_witness
  6. L12
    exact hodd_witness
05Establish htwoL13–22

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

  1. L13
    have htwo : forall pff_divisor_odd_two_local. (exists pff_left_odd_two_local. (2) = pff_divisor_odd_two_local * pff_left_odd_two_local) -> (exists pff_right_odd_two_local. (d) = pff_divisor_odd_two_local * pff_right_odd_two_local) -> pff_divisor_odd_two_local = 1
  2. L14
    specialize prime_not_divides_coprime 2
  3. L15
    specialize prime_not_divides_coprime d
  4. L16
    apply prime_not_divides_coprime
  5. L17
    exact prime_two
  6. L18
    exact hnot
  7. L19
    specialize coprime_symm 2
  8. L20
    specialize coprime_symm d
  9. L21
    apply coprime_symm
  10. L22
    exact htwo

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro d
  2. 0002intro hodd
  3. 0003have hnot : ~(exists q. d = 2 * q)
  4. 0004intro heven
  5. 0005cases heven
  6. 0006cases hodd
  7. 0007specialize even_odd_exclusive_pointwise d
  8. 0008specialize even_odd_exclusive_pointwise x
  9. 0009specialize even_odd_exclusive_pointwise x1
  10. 0010apply even_odd_exclusive_pointwise
  11. 0011exact heven_witness
  12. 0012exact hodd_witness
  13. 0013have htwo : forall pff_divisor_odd_two_local. (exists pff_left_odd_two_local. (2) = pff_divisor_odd_two_local * pff_left_odd_two_local) -> (exists pff_right_odd_two_local. (d) = pff_divisor_odd_two_local * pff_right_odd_two_local) -> pff_divisor_odd_two_local = 1
  14. 0014specialize prime_not_divides_coprime 2
  15. 0015specialize prime_not_divides_coprime d
  16. 0016apply prime_not_divides_coprime
  17. 0017exact prime_two
  18. 0018exact hnot
  19. 0019specialize coprime_symm 2
  20. 0020specialize coprime_symm d
  21. 0021apply coprime_symm
  22. 0022exact htwo