SL000I · theorem body

doubling_half_range_below_odd_modulus

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

Doubling a value in the canonical half range never wraps modulo the odd modulus.

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

∀ p. ∀ h. ∀ x. p = 2 · h + 1 → Le(x,h) → Lt(2 · x,p)

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 PD0002 Lt

In local proof propositions

PD0001 Le
Exact expanded first-order statement
forall p h x. p = 2 * h + 1 -> (exists qst_le_source. qst_le_source + (x) = (h)) -> (exists qst_lt_canonical. qst_lt_canonical + S (2 * x) = (p))

Proof neighborhood

Direct theorem prerequisites

mul_le_mul_left · Stable closed

Direct theorem dependents

SL000K doubling_gauss_initial_segment_complement

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

18 script commands · 7 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro x
  4. L4
    intro hpodd
  5. L5
    intro hxbelow
02Establish hscaledL6–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L6
    have hscaled : Le(2 · x,2 · h)Definitions: Le(2 · x,2 · h)Original native command in the exact edition
  2. L7
    specialize mul_le_mul_left x
  3. L8
    specialize mul_le_mul_left h
  4. L9
    specialize mul_le_mul_left 2
  5. L10
    apply mul_le_mul_left
  6. L11
    exact hxbelow
03Separate the logical casesL12–12

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

  1. L12
    cases hscaled
04Construct an explicit witnessL13–13

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

  1. L13
    exists x1
05Calculate and transport equalitiesL14–14

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

  1. L14
    trans S (x1 + 2 * x)
06Use earlier factsL15–15

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

  1. L15
    apply PA4
07Calculate and transport equalitiesL16–18

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

  1. L16
    rewrite hscaled_witness
  2. L17
    rewrite hpodd
  3. L18
    simp

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro x
  4. 0004intro hpodd
  5. 0005intro hxbelow
  6. 0006have hscaled : Le(2 · x,2 · h)
    Exact native replay linehave hscaled : exists gap. gap + 2 * x = 2 * h
  7. 0007specialize mul_le_mul_left x
  8. 0008specialize mul_le_mul_left h
  9. 0009specialize mul_le_mul_left 2
  10. 0010apply mul_le_mul_left
  11. 0011exact hxbelow
  12. 0012cases hscaled
  13. 0013exists x1
  14. 0014trans S (x1 + 2 * x)
  15. 0015apply PA4
  16. 0016rewrite hscaled_witness
  17. 0017rewrite hpodd
  18. 0018simp