GT0002

euclidean_divisor_dividend_transport

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

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

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.

G101 was OPEN at this family's Alpha-v22 first admission: its complete anchored trace and actual terminal gcd were proved but its logarithmic bound was not. G101 is now CLOSED in Alpha v23, including the exact first-order bound steps≤2*BitLen(b)+1.

Exact theorem in conservative defined notation

∀ d. ∀ a. ∀ b. ∀ q. ∀ r. EuclideanDivision(a,b,q,r) → Dvd(d,b) → Dvd(d,r) → Dvd(d,a)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

EuclideanDivision(a,b,q,r) · 1Dvd(d,n) · 3

Actual proof prerequisites

divides_linear_step · checked external prerequisite
Original expanded first-order 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)

Complete unchanged native tactic proof

All 17 lines are the exact independently kernel-checked original script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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