PF000N · theorem body

pythagorean_odd_coordinate_coprime_two

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ d. Odd(d) → Coprime(d,2)

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 PD0010 Odd

In local proof propositions

PD0005 Coprime PD0009 Even
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

even_odd_exclusive_pointwise · Stable closed prime_two · Stable closed prime_not_divides_coprime · Stable closed coprime_symm · Stable closed

Direct theorem dependents

PF000S pythagorean_primitive_euclidean_legs PF002E fermat_four_odd_double_square_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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 : ¬Even(d)Definitions: Even(d)Original native command in the exact edition
  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
  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 defined command ledger · 22 lines
  1. 0001intro d
  2. 0002intro hodd
  3. 0003have hnot : ¬Even(d)
    Exact native replay linehave 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 : Coprime(2,d)
    Exact native replay linehave 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