PF001J · theorem body

pythagorean_odd_ordered_difference_even

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

The natural difference of ordered odd numbers has a constructive even-half witness.

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. ∀ c. Le(a,c) → Odd(a) → Odd(c) → ∃ x. c = a + 2 · x

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

PD0001 Le PD0010 Odd

In local proof propositions

CF0016 OppositeParity PD0010 Odd
Exact expanded first-order statement
forall a c. (exists gap. gap + a = c) -> (exists pp_odd_pi_difference_a. (a) = 2 * pp_odd_pi_difference_a + 1) -> (exists pp_odd_pi_difference_c. (c) = 2 * pp_odd_pi_difference_c + 1) -> exists t. c = a + 2 * t

Proof neighborhood

Direct theorem prerequisites

odd_sum_parity_cases · Alpha closed even_odd_exclusive_pointwise · Stable closed add_comm · Stable closed

Direct theorem dependents

PF001V pythagorean_odd_even_half_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

41 script commands · 16 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro c
  3. L3
    intro hbound
  4. L4
    intro ha
  5. L5
    intro hc
02Separate the logical casesL6–6

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

  1. L6
    cases hbound
03Establish hsumL7–7

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

  1. L7
04Separate the logical casesL8–8

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

  1. L8
    cases hc
05Construct an explicit witnessL9–9

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

  1. L9
    exists x1
06Calculate and transport equalitiesL10–11

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

  1. L10
    trans c
  2. L11
    trans x + a
07Use earlier factsL12–14

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

  1. L12
    apply add_comm
  2. L13
    exact hbound_witness
  3. L14
    exact hc_witness
08Establish hparityL15–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply odd sum parity cases.

  1. L15
    have hparity : OppositeParity(a,x)Definitions: OppositeParity(a,x)Original native command in the exact edition
  2. L16
    specialize odd_sum_parity_cases a
  3. L17
    specialize odd_sum_parity_cases x
  4. L18
    apply odd_sum_parity_cases
  5. L19
    exact hsum
09Separate the logical casesL20–24

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

  1. L20
    cases hparity
  2. L21
    cases hparity_left
  3. L22
    exfalso
  4. L23
    cases hparity_left_left
  5. L24
    cases ha
10Use earlier factsL25–30

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

  1. L25
    specialize even_odd_exclusive_pointwise a
  2. L26
    specialize even_odd_exclusive_pointwise x1
  3. L27
    specialize even_odd_exclusive_pointwise x2
  4. L28
    apply even_odd_exclusive_pointwise
  5. L29
    exact hparity_left_left_witness
  6. L30
    exact ha_witness
11Separate the logical casesL31–32

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

  1. L31
    cases hparity_right
  2. L32
    cases hparity_right_right
12Construct an explicit witnessL33–33

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

  1. L33
    exists x1
13Calculate and transport equalitiesL34–36

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

  1. L34
    trans a + x
  2. L35
    trans x + a
  3. L36
    symm
14Use earlier factsL37–38

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

  1. L37
    exact hbound_witness
  2. L38
    apply add_comm
15Calculate and transport equalitiesL39–40

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

  1. L39
    congr
  2. L40
    refl
16Use earlier factsL41–41

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

  1. L41
    exact hparity_right_right_witness

Library-wide reading audit

Original defined command ledger · 41 lines
  1. 0001intro a
  2. 0002intro c
  3. 0003intro hbound
  4. 0004intro ha
  5. 0005intro hc
  6. 0006cases hbound
  7. 0007have hsum : Odd(a + x)
    Exact native replay linehave hsum : exists pp_odd_pi_difference_sum. (a + x) = 2 * pp_odd_pi_difference_sum + 1
  8. 0008cases hc
  9. 0009exists x1
  10. 0010trans c
  11. 0011trans x + a
  12. 0012apply add_comm
  13. 0013exact hbound_witness
  14. 0014exact hc_witness
  15. 0015have hparity : OppositeParity(a,x)
    Exact native replay linehave hparity : (((exists pp_even_pi_difference_parity_first_even. (a) = 2 * pp_even_pi_difference_parity_first_even) /\ (exists pp_odd_pi_difference_parity_second_odd. (x) = 2 * pp_odd_pi_difference_parity_second_odd + 1)) \/ ((exists pp_odd_pi_difference_parity_first_odd. (a) = 2 * pp_odd_pi_difference_parity_first_odd + 1) /\ (exists pp_even_pi_difference_parity_second_even. (x) = 2 * pp_even_pi_difference_parity_second_even)))
  16. 0016specialize odd_sum_parity_cases a
  17. 0017specialize odd_sum_parity_cases x
  18. 0018apply odd_sum_parity_cases
  19. 0019exact hsum
  20. 0020cases hparity
  21. 0021cases hparity_left
  22. 0022exfalso
  23. 0023cases hparity_left_left
  24. 0024cases ha
  25. 0025specialize even_odd_exclusive_pointwise a
  26. 0026specialize even_odd_exclusive_pointwise x1
  27. 0027specialize even_odd_exclusive_pointwise x2
  28. 0028apply even_odd_exclusive_pointwise
  29. 0029exact hparity_left_left_witness
  30. 0030exact ha_witness
  31. 0031cases hparity_right
  32. 0032cases hparity_right_right
  33. 0033exists x1
  34. 0034trans a + x
  35. 0035trans x + a
  36. 0036symm
  37. 0037exact hbound_witness
  38. 0038apply add_comm
  39. 0039congr
  40. 0040refl
  41. 0041exact hparity_right_right_witness