PF0012

pythagorean_triple_legs_not_both_odd

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

The two legs of any Pythagorean triple cannot both be odd, independently of primitiveness.

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)) -> (exists i. a = 2 * i + 1) -> (exists j. b = 2 * j + 1) -> false

Constructive proof overview

Generated structural guide

The two legs of any Pythagorean triple cannot both be odd, independently of primitiveness.

The unchanged tactic script uses 2 declared prerequisites and contains 20 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

PF0010 pythagorean_odd_square_pair_two_mod_four PF0011 pythagorean_two_mod_four_not_square

Direct dependents

PF0014 pythagorean_primitive_legs_opposite_parity

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

20 script commands · 6 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 (2)
pythagorean odd square pair two mod fourpythagorean two mod four not square
01Fix variables and assumptionsL1–6

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 htriple
  5. L5
    intro ha
  6. L6
    intro hb
02Establish htwoL7–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean odd square pair two mod four.

  1. L7
    have htwo : exists q. a * a + b * b = 4 * q + 2
  2. L8
    specialize pythagorean_odd_square_pair_two_mod_four a
  3. L9
    specialize pythagorean_odd_square_pair_two_mod_four b
  4. L10
    apply pythagorean_odd_square_pair_two_mod_four
  5. L11
    exact ha
  6. L12
    exact hb
03Separate the logical casesL13–13

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

  1. L13
    cases htwo
04Use earlier factsL14–16

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

  1. L14
    specialize pythagorean_two_mod_four_not_square c
  2. L15
    specialize pythagorean_two_mod_four_not_square x
  3. L16
    apply pythagorean_two_mod_four_not_square
05Calculate and transport equalitiesL17–18

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

  1. L17
    trans a * a + b * b
  2. L18
    symm
06Use earlier factsL19–20

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

  1. L19
    exact htriple
  2. L20
    exact htwo_witness

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro htriple
  5. 0005intro ha
  6. 0006intro hb
  7. 0007have htwo : exists q. a * a + b * b = 4 * q + 2
  8. 0008specialize pythagorean_odd_square_pair_two_mod_four a
  9. 0009specialize pythagorean_odd_square_pair_two_mod_four b
  10. 0010apply pythagorean_odd_square_pair_two_mod_four
  11. 0011exact ha
  12. 0012exact hb
  13. 0013cases htwo
  14. 0014specialize pythagorean_two_mod_four_not_square c
  15. 0015specialize pythagorean_two_mod_four_not_square x
  16. 0016apply pythagorean_two_mod_four_not_square
  17. 0017trans a * a + b * b
  18. 0018symm
  19. 0019exact htriple
  20. 0020exact htwo_witness