BT010I

third_quotient_double_gap_exists

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

The third quotient has an exact additive gap to the doubled input.

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 PA statement

forall n q r. (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_b5rbdqld_division_bound. bcf_lt_gap_b5rbdqld_division_bound + S (r) = 3))) -> exists h. q + h = n + n

Structural proof guide

The third quotient has an exact additive gap to the doubled input.

Direct prerequisites: add_comm, division_quotient_le_dividend. The authored body proceeds by case analysis (1), intermediate claims (1), equality transport (1).

Proof neighborhood

Direct dependencies

BT0002 add_comm BT010H division_quotient_le_dividend

Direct dependents

BT010J floor_third_double_gap_package

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

16 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.

Named ingredients (2)
add commdivision quotient le dividend
01Fix variables and assumptionsL1–4

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

  1. L1
    intro n
  2. L2
    intro q
  3. L3
    intro r
  4. L4
    intro hdivision
02Establish hboundL5–10

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

  1. L5
    have hbound : exists bcf_le_gap_b5rbdqld_result. bcf_le_gap_b5rbdqld_result + (q) = n + n
  2. L6
    specialize division_quotient_le_dividend n
  3. L7
    specialize division_quotient_le_dividend q
  4. L8
    specialize division_quotient_le_dividend r
  5. L9
    apply division_quotient_le_dividend
  6. L10
    exact hdivision
03Separate the logical casesL11–11

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

  1. L11
    cases hbound
04Construct an explicit witnessL12–12

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

  1. L12
    exists x
05Use earlier factsL13–14

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

  1. L13
    specialize add_comm q
  2. L14
    specialize add_comm x
06Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite add_comm
07Use earlier factsL16–16

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

  1. L16
    exact hbound_witness

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro n
  2. 0002intro q
  3. 0003intro r
  4. 0004intro hdivision
  5. 0005have hbound : exists bcf_le_gap_b5rbdqld_result. bcf_le_gap_b5rbdqld_result + (q) = n + n
  6. 0006specialize division_quotient_le_dividend n
  7. 0007specialize division_quotient_le_dividend q
  8. 0008specialize division_quotient_le_dividend r
  9. 0009apply division_quotient_le_dividend
  10. 0010exact hdivision
  11. 0011cases hbound
  12. 0012exists x
  13. 0013specialize add_comm q
  14. 0014specialize add_comm x
  15. 0015rewrite add_comm
  16. 0016exact hbound_witness