GT0001

euclidean_divisor_remainder_transport

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

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,a) → Dvd(d,b) → Dvd(d,r)

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_remainder · 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_da. a = d * egt_factor_da) -> (exists egt_factor_db. b = d * egt_factor_db) -> (exists egt_factor_dr. r = d * egt_factor_dr)

Complete unchanged native tactic proof

All 18 lines are the exact independently kernel-checked original script.

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.

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 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 defined 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