SL000G

doubling_floor_below_implies_double_at_most_half

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

A source at most the odd-half floor has doubled value at most 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.

Exact expanded first-order arithmetic statement

forall h k x. (((h = 2 * k) \/ (h = 2 * k + 1))) -> (exists qst_le_floor. qst_le_floor + (x) = (k)) -> (exists qst_le_doubled. qst_le_doubled + (2 * x) = (h))

Constructive proof overview

Generated structural guide

A source at most the odd-half floor has doubled value at most the half.

The unchanged tactic script uses 2 declared prerequisites and contains 21 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

mul_le_mul_left Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized

Direct dependents

SL000K doubling_gauss_initial_segment_complement

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

21 script commands · 10 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.

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 hfloor
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 : exists gap. gap + 2 * x = 2 * k
  2. L7
    specialize mul_le_mul_left x
  3. L8
    specialize mul_le_mul_left k
  4. L9
    specialize mul_le_mul_left 2
  5. L10
    apply mul_le_mul_left
  6. L11
    exact hfloor
03Separate the logical casesL12–12

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

  1. L12
    cases hhalf
04Calculate and transport equalitiesL13–13

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

  1. L13
    rewrite hhalf_left
05Use earlier factsL14–14

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

  1. L14
    exact hscaled
06Separate the logical casesL15–15

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

  1. L15
    cases hscaled
07Construct an explicit witnessL16–16

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

  1. L16
    exists S x1
08Calculate 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 (x1 + 2 * x)
09Use earlier factsL18–18

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

  1. L18
    apply add_succ_left
10Calculate and transport equalitiesL19–21

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

  1. L19
    rewrite hscaled_witness
  2. L20
    rewrite hhalf_right
  3. L21
    simp

Library-wide reading audit

Original exact command ledger · 21 lines
  1. 0001intro h
  2. 0002intro k
  3. 0003intro x
  4. 0004intro hhalf
  5. 0005intro hfloor
  6. 0006have hscaled : exists gap. gap + 2 * x = 2 * k
  7. 0007specialize mul_le_mul_left x
  8. 0008specialize mul_le_mul_left k
  9. 0009specialize mul_le_mul_left 2
  10. 0010apply mul_le_mul_left
  11. 0011exact hfloor
  12. 0012cases hhalf
  13. 0013rewrite hhalf_left
  14. 0014exact hscaled
  15. 0015cases hscaled
  16. 0016exists S x1
  17. 0017trans S (x1 + 2 * x)
  18. 0018apply add_succ_left
  19. 0019rewrite hscaled_witness
  20. 0020rewrite hhalf_right
  21. 0021simp