PF000K · theorem body

pythagorean_odd_even_square_gap_odd

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

An odd first parameter and even second parameter force their witnessed square difference to be odd.

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

∀ m. ∀ n. ∀ d. m · m = n · n + d → Odd(m) → Even(n) → Odd(d)

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

PD0009 Even PD0010 Odd

In local proof propositions

PD0009 Even PD0010 Odd
Exact expanded first-order statement
forall m n d. m * m = n * n + d -> (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b) -> exists q. d = 2 * q + 1

Proof neighborhood

Direct theorem prerequisites

PF000I pythagorean_parameter_odd_square PF000H pythagorean_parameter_even_square parity_cases · Stable closed even_add_even · Stable closed even_odd_exclusive_pointwise · Stable closed

Direct theorem dependents

PF000L pythagorean_opposite_parity_square_gap_odd

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

35 script commands · 13 reading checkpoints · 3 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 parameter odd squarepythagorean parameter even square
01Fix variables and assumptionsL1–6

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro d
  4. L4
    intro hgap
  5. L5
    intro hmodd
  6. L6
    intro hneven
02Establish hmsquareL7–9

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

  1. L7
    have hmsquare : Odd(m · m)Definitions: Odd(m · m)Original native command in the exact edition
  2. L8
    apply pythagorean_parameter_odd_square
  3. L9
    exact hmodd
03Establish hnsquareL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean parameter even square.

  1. L10
    have hnsquare : Even(n · n)Definitions: Even(n · n)Original native command in the exact edition
  2. L11
    apply pythagorean_parameter_even_square
  3. L12
    exact hneven
  4. L13
    specialize parity_cases d
04Separate the logical casesL14–16

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

  1. L14
    cases parity_cases
  2. L15
    cases parity_cases_witness
  3. L16
    exfalso
05Establish hsumevenL17–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply even add even.

  1. L17
    have hsumeven : Even(n · n + d)Definitions: Even(n · n + d)Original native command in the exact edition
  2. L18
    specialize even_add_even (n * n)
  3. L19
    specialize even_add_even d
  4. L20
    apply even_add_even
  5. L21
    exact hnsquare
06Construct an explicit witnessL22–22

Supply the displayed value, then prove that it has the required property.

  1. L22
    exists x
07Use earlier factsL23–23

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

  1. L23
    exact parity_cases_witness_left
08Separate the logical casesL24–25

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

  1. L24
    cases hsumeven
  2. L25
    cases hmsquare
09Use earlier factsL26–29

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

  1. L26
    specialize even_odd_exclusive_pointwise (m * m)
  2. L27
    specialize even_odd_exclusive_pointwise x1
  3. L28
    specialize even_odd_exclusive_pointwise x2
  4. L29
    apply even_odd_exclusive_pointwise
10Calculate and transport equalitiesL30–30

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

  1. L30
    trans n * n + d
11Use earlier factsL31–33

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

  1. L31
    exact hgap
  2. L32
    exact hsumeven_witness
  3. L33
    exact hmsquare_witness
12Construct an explicit witnessL34–34

Supply the displayed value, then prove that it has the required property.

  1. L34
    exists x
13Use earlier factsL35–35

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

  1. L35
    exact parity_cases_witness_right

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hmodd
  6. 0006intro hneven
  7. 0007have hmsquare : Odd(m · m)
    Exact native replay linehave hmsquare : exists q. m * m = 2 * q + 1
  8. 0008apply pythagorean_parameter_odd_square
  9. 0009exact hmodd
  10. 0010have hnsquare : Even(n · n)
    Exact native replay linehave hnsquare : exists q. n * n = 2 * q
  11. 0011apply pythagorean_parameter_even_square
  12. 0012exact hneven
  13. 0013specialize parity_cases d
  14. 0014cases parity_cases
  15. 0015cases parity_cases_witness
  16. 0016exfalso
  17. 0017have hsumeven : Even(n · n + d)
    Exact native replay linehave hsumeven : exists q. n * n + d = 2 * q
  18. 0018specialize even_add_even (n * n)
  19. 0019specialize even_add_even d
  20. 0020apply even_add_even
  21. 0021exact hnsquare
  22. 0022exists x
  23. 0023exact parity_cases_witness_left
  24. 0024cases hsumeven
  25. 0025cases hmsquare
  26. 0026specialize even_odd_exclusive_pointwise (m * m)
  27. 0027specialize even_odd_exclusive_pointwise x1
  28. 0028specialize even_odd_exclusive_pointwise x2
  29. 0029apply even_odd_exclusive_pointwise
  30. 0030trans n * n + d
  31. 0031exact hgap
  32. 0032exact hsumeven_witness
  33. 0033exact hmsquare_witness
  34. 0034exists x
  35. 0035exact parity_cases_witness_right