GT0002

euclidean_divisor_dividend_transport

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

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

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_db. b = d * egt_factor_db) -> (exists egt_factor_dr. r = d * egt_factor_dr) -> (exists egt_factor_da. a = d * egt_factor_da)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 17 exact native proof lines.

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

Proof neighborhood

Direct dependencies

divides_linear_step Stable theorem; checked-use authorized

Direct dependents

GT0004 euclidean_common_divisor_backward

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 · 4 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 hb
  8. L8
    intro hr
02Separate the logical casesL9–9

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

  1. L9
    cases hstep
03Calculate and transport equalitiesL10–10

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

  1. L10
    rewrite hstep_left
04Use earlier factsL11–17

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

  1. L11
    specialize divides_linear_step d
  2. L12
    specialize divides_linear_step b
  3. L13
    specialize divides_linear_step q
  4. L14
    specialize divides_linear_step r
  5. L15
    apply divides_linear_step
  6. L16
    exact hb
  7. L17
    exact hr

Library-wide reading audit

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