PF0012 · theorem body

pythagorean_triple_legs_not_both_odd

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

Statement with defined notation

∀ a. ∀ b. ∀ c. Pythagorean(a,b,c) → Odd(a) → ¬Odd(b)

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

CF0011 Pythagorean PD0010 Odd

In local proof propositions

none
Exact expanded first-order statement
forall a b c. ((a) * (a) + (b) * (b) = (c) * (c)) -> (exists i. a = 2 * i + 1) -> (exists j. b = 2 * j + 1) -> false

Proof neighborhood

Direct theorem prerequisites

PF0010 pythagorean_odd_square_pair_two_mod_four PF0011 pythagorean_two_mod_four_not_square

Direct theorem dependents

PF0014 pythagorean_primitive_legs_opposite_parity

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

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.

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 (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 defined 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