SL000H

doubling_floor_above_implies_double_above_half

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites and contains 17 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_add Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm 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

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.

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 exact 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]