BT010G

floor_sqrt_third_quotient_gap_exists

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

The floor-root cut has an exact additive gap to the third quotient.

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 s q r. (exists bcf_lt_gap_b5rbfsltq_positive. bcf_lt_gap_b5rbfsltq_positive + S (2) = n) -> (((exists bcs_sqrt_lower_gap_b5rbfsltq_floor. bcs_sqrt_lower_gap_b5rbfsltq_floor + (s) * (s) = (n + n)) /\ exists bcs_sqrt_upper_gap_b5rbfsltq_floor. bcs_sqrt_upper_gap_b5rbfsltq_floor + S (n + n) = S (s) * S (s))) -> (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_b5rbfsltq_division_bound. bcf_lt_gap_b5rbfsltq_division_bound + S (r) = 3))) -> exists g. s + g = q

Structural proof guide

The floor-root cut has an exact additive gap to the third quotient.

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

Proof neighborhood

Direct dependencies

BT0002 add_comm BT010F floor_sqrt_le_third_quotient

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

22 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 commfloor sqrt le third quotient
01Fix variables and assumptionsL1–7

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

  1. L1
    intro n
  2. L2
    intro s
  3. L3
    intro q
  4. L4
    intro r
  5. L5
    intro hpositive
  6. L6
    intro hfloor
  7. L7
    intro hdivision
02Establish hboundL8–16

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

  1. L8
    have hbound : exists bcf_le_gap_b5rbfsltq_result. bcf_le_gap_b5rbfsltq_result + (s) = q
  2. L9
    specialize floor_sqrt_le_third_quotient n
  3. L10
    specialize floor_sqrt_le_third_quotient s
  4. L11
    specialize floor_sqrt_le_third_quotient q
  5. L12
    specialize floor_sqrt_le_third_quotient r
  6. L13
    apply floor_sqrt_le_third_quotient
  7. L14
    exact hpositive
  8. L15
    exact hfloor
  9. L16
    exact hdivision
03Separate the logical casesL17–17

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

  1. L17
    cases hbound
04Construct an explicit witnessL18–18

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

  1. L18
    exists x
05Use earlier factsL19–20

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

  1. L19
    specialize add_comm s
  2. L20
    specialize add_comm x
06Calculate and transport equalitiesL21–21

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

  1. L21
    rewrite add_comm
07Use earlier factsL22–22

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

  1. L22
    exact hbound_witness

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro n
  2. 0002intro s
  3. 0003intro q
  4. 0004intro r
  5. 0005intro hpositive
  6. 0006intro hfloor
  7. 0007intro hdivision
  8. 0008have hbound : exists bcf_le_gap_b5rbfsltq_result. bcf_le_gap_b5rbfsltq_result + (s) = q
  9. 0009specialize floor_sqrt_le_third_quotient n
  10. 0010specialize floor_sqrt_le_third_quotient s
  11. 0011specialize floor_sqrt_le_third_quotient q
  12. 0012specialize floor_sqrt_le_third_quotient r
  13. 0013apply floor_sqrt_le_third_quotient
  14. 0014exact hpositive
  15. 0015exact hfloor
  16. 0016exact hdivision
  17. 0017cases hbound
  18. 0018exists x
  19. 0019specialize add_comm s
  20. 0020specialize add_comm x
  21. 0021rewrite add_comm
  22. 0022exact hbound_witness