GT0001

euclidean_divisor_remainder_transport

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

Every witnessed common divisor of dividend and divisor also divides the exact Euclidean 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 d a b q r. ((a = b * q + r /\ (exists ff_lt_ec_egt_step_division. ff_lt_ec_egt_step_division + S r = b))) -> (exists egt_factor_da. a = d * egt_factor_da) -> (exists egt_factor_db. b = d * egt_factor_db) -> (exists egt_factor_dr. r = d * egt_factor_dr)

Constructive proof overview

Generated structural guide

Every witnessed common divisor of dividend and divisor also divides the exact Euclidean remainder.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

divides_remainder Stable theorem; checked-use authorized

Direct dependents

GT0003 euclidean_common_divisor_forward

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

18 script commands · 3 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 assumptionsL1–8

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

  1. L1
    intro d
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro q
  5. L5
    intro r
  6. L6
    intro hstep
  7. L7
    intro ha
  8. L8
    intro hb
02Separate the logical casesL9–9

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

  1. L9
    cases hstep
03Use earlier factsL10–18

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

  1. L10
    specialize divides_remainder d
  2. L11
    specialize divides_remainder a
  3. L12
    specialize divides_remainder b
  4. L13
    specialize divides_remainder q
  5. L14
    specialize divides_remainder r
  6. L15
    apply divides_remainder
  7. L16
    exact ha
  8. L17
    exact hb
  9. L18
    exact hstep_left

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro d
  2. 0002intro a
  3. 0003intro b
  4. 0004intro q
  5. 0005intro r
  6. 0006intro hstep
  7. 0007intro ha
  8. 0008intro hb
  9. 0009cases hstep
  10. 0010specialize divides_remainder d
  11. 0011specialize divides_remainder a
  12. 0012specialize divides_remainder b
  13. 0013specialize divides_remainder q
  14. 0014specialize divides_remainder r
  15. 0015apply divides_remainder
  16. 0016exact ha
  17. 0017exact hb
  18. 0018exact hstep_left