SL000L · theorem body

doubling_half_decomposition_lower_bound

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

A natural half in either doubled decomposition is at most the full length.

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

∀ h. ∀ k. h = 2 · k ∨ h = 2 · k + 1 → Le(k,h)

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

In local proof propositions

none
Exact expanded first-order statement
forall h k. (((h = 2 * k) \/ (h = 2 * k + 1))) -> exists gap. gap + k = h

Proof neighborhood

Direct theorem prerequisites

mul_comm · Stable closed zero_add · Stable closed add_succ_left · Stable closed

Direct theorem dependents

SL000M doubling_gauss_count_shape_from_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

25 script commands · 15 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 assumptionsL1–3

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

  1. L1
    intro h
  2. L2
    intro k
  3. L3
    intro hshape
02Establish hdoubleL4–9

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

  1. L4
    have hdouble : k + k = 2 * k
  2. L5
    trans k * 2
  3. L6
    simp [zero_add]
  4. L7
    specialize mul_comm k
  5. L8
    specialize mul_comm 2
  6. L9
    apply mul_comm
03Separate the logical casesL10–10

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

  1. L10
    cases hshape
04Construct an explicit witnessL11–11

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

  1. L11
    exists k
05Calculate and transport equalitiesL12–12

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

  1. L12
    trans 2 * k
06Use earlier factsL13–13

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

  1. L13
    exact hdouble
07Calculate and transport equalitiesL14–14

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

  1. L14
    symm
08Use earlier factsL15–15

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

  1. L15
    exact hshape_left
09Construct an explicit witnessL16–16

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

  1. L16
    exists S k
10Calculate and transport equalitiesL17–17

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

  1. L17
    trans S (k + k)
11Use earlier factsL18–18

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

  1. L18
    apply add_succ_left
12Calculate and transport equalitiesL19–20

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

  1. L19
    trans S (2 * k)
  2. L20
    congr
13Use earlier factsL21–21

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

  1. L21
    exact hdouble
14Calculate and transport equalitiesL22–24

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

  1. L22
    trans 2 * k + 1
  2. L23
    simp
  3. L24
    symm
15Use earlier factsL25–25

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

  1. L25
    exact hshape_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro h
  2. 0002intro k
  3. 0003intro hshape
  4. 0004have hdouble : k + k = 2 * k
  5. 0005trans k * 2
  6. 0006simp [zero_add]
  7. 0007specialize mul_comm k
  8. 0008specialize mul_comm 2
  9. 0009apply mul_comm
  10. 0010cases hshape
  11. 0011exists k
  12. 0012trans 2 * k
  13. 0013exact hdouble
  14. 0014symm
  15. 0015exact hshape_left
  16. 0016exists S k
  17. 0017trans S (k + k)
  18. 0018apply add_succ_left
  19. 0019trans S (2 * k)
  20. 0020congr
  21. 0021exact hdouble
  22. 0022trans 2 * k + 1
  23. 0023simp
  24. 0024symm
  25. 0025exact hshape_right