SL000H · theorem body

doubling_floor_above_implies_double_above_half

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

A source strictly above the odd-half floor doubles beyond the half.

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

PD0002 Lt

In local proof propositions

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

Proof neighborhood

Direct theorem prerequisites

mul_add · Stable closed add_assoc · Stable closed add_comm · Stable closed add_succ_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

17 script commands · 7 reading checkpoints · 0 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 h
  2. L2
    intro k
  3. L3
    intro x
  4. L4
    intro hhalf
  5. L5
    intro habove
02Separate the logical casesL6–7

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

  1. L6
    cases habove
  2. L7
    cases hhalf
03Construct an explicit witnessL8–8

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

  1. L8
    exists 2 * x1 + 1
04Calculate and transport equalitiesL9–12

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

  1. L9
    rewrite hhalf_left
  2. L10
    rewrite <- habove_witness
  3. L11
    simp [mul_add, add_assoc, add_comm]
  4. L12
    congr
05Use earlier factsL13–13

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

  1. L13
    apply add_succ_left
06Construct an explicit witnessL14–14

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

  1. L14
    exists 2 * x1
07Calculate and transport equalitiesL15–17

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

  1. L15
    rewrite hhalf_right
  2. L16
    rewrite <- habove_witness
  3. L17
    simp [mul_add, add_assoc, add_comm]

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro h
  2. 0002intro k
  3. 0003intro x
  4. 0004intro hhalf
  5. 0005intro habove
  6. 0006cases habove
  7. 0007cases hhalf
  8. 0008exists 2 * x1 + 1
  9. 0009rewrite hhalf_left
  10. 0010rewrite <- habove_witness
  11. 0011simp [mul_add, add_assoc, add_comm]
  12. 0012congr
  13. 0013apply add_succ_left
  14. 0014exists 2 * x1
  15. 0015rewrite hhalf_right
  16. 0016rewrite <- habove_witness
  17. 0017simp [mul_add, add_assoc, add_comm]