GI0003

signed_integer_floor_quotient_transport

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

Replacing a quotient pair by any equal integer preserves the exact floor equation and the same strict remainder.

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 xp xn m qp qn Qp Qn r. qp + Qn = qn + Qp -> (((xp) + (m) * (qn) = ((xn) + (m) * (qp)) + (r) /\ exists sif_gap_source. sif_gap_source + S (r) = (m))) -> (((xp) + (m) * (Qn) = ((xn) + (m) * (Qp)) + (r) /\ exists sif_gap_target. sif_gap_target + S (r) = (m)))

Constructive proof overview

Generated structural guide

Replacing a quotient pair by any equal integer preserves the exact floor equation and the same strict remainder.

The unchanged tactic script uses 5 declared prerequisites and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_left_cancel Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

GI0004 signed_integer_canonical_floor_exists

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

24 script commands · 5 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 assumptionsL1–10

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

  1. L1
    intro xp
  2. L2
    intro xn
  3. L3
    intro m
  4. L4
    intro qp
  5. L5
    intro qn
  6. L6
    intro Qp
  7. L7
    intro Qn
  8. L8
    intro r
  9. L9
    intro hbalance
  10. L10
    intro hfloor
02Separate the logical casesL11–12

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

  1. L11
    cases hfloor
  2. L12
    split
03Use earlier factsL13–16

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

  1. L13
    specialize add_left_cancel (m * qp)
  2. L14
    specialize add_left_cancel (xp + m * Qn)
  3. L15
    specialize add_left_cancel ((xn + m * Qp) + r)
  4. L16
    apply add_left_cancel
04Calculate and transport equalitiesL17–23

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

  1. L17
    trans xp + m * (qp + Qn)
  2. L18
    simp [mul_add, add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L19
    rewrite hbalance
  4. L20
    trans (xp + m * qn) + m * Qp
  5. L21
    simp [mul_add, add_assoc]
  6. L22
    rewrite hfloor_left
  7. L23
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]
05Use earlier factsL24–24

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

  1. L24
    exact hfloor_right

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro xp
  2. 0002intro xn
  3. 0003intro m
  4. 0004intro qp
  5. 0005intro qn
  6. 0006intro Qp
  7. 0007intro Qn
  8. 0008intro r
  9. 0009intro hbalance
  10. 0010intro hfloor
  11. 0011cases hfloor
  12. 0012split
  13. 0013specialize add_left_cancel (m * qp)
  14. 0014specialize add_left_cancel (xp + m * Qn)
  15. 0015specialize add_left_cancel ((xn + m * Qp) + r)
  16. 0016apply add_left_cancel
  17. 0017trans xp + m * (qp + Qn)
  18. 0018simp [mul_add, add_assoc, add_comm, four_square_add_swap_right_tail]
  19. 0019rewrite hbalance
  20. 0020trans (xp + m * qn) + m * Qp
  21. 0021simp [mul_add, add_assoc]
  22. 0022rewrite hfloor_left
  23. 0023simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  24. 0024exact hfloor_right