GT0004

euclidean_common_divisor_backward

Every constructive common divisor of (b,r) is already a common divisor of the previous Euclidean pair (a,b).

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) → EuclideanCommonDivisor(d,b,r) → EuclideanCommonDivisor(d,a,b)

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

Definition DAG

EuclideanDivision(a,b,q,r) · 1EuclideanCommonDivisor(d,a,b) · 2

Actual proof prerequisites

euclidean_divisor_dividend_transport
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_after_left. b = d * egt_factor_after_left) /\ (exists egt_factor_after_right. r = d * egt_factor_after_right))) -> (((exists egt_factor_before_left. a = d * egt_factor_before_left) /\ (exists egt_factor_before_right. b = d * egt_factor_before_right)))

Complete unchanged native tactic proof

All 19 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

19 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.

Named ingredients (1)
euclidean divisor dividend transport
01Fix variables and assumptionsL1–7

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 hcommon
02Separate the logical casesL8–9

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

  1. L8
    cases hcommon
  2. L9
    split
03Use earlier factsL10–19

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

  1. L10
    specialize euclidean_divisor_dividend_transport d
  2. L11
    specialize euclidean_divisor_dividend_transport a
  3. L12
    specialize euclidean_divisor_dividend_transport b
  4. L13
    specialize euclidean_divisor_dividend_transport q
  5. L14
    specialize euclidean_divisor_dividend_transport r
  6. L15
    apply euclidean_divisor_dividend_transport
  7. L16
    exact hstep
  8. L17
    exact hcommon_left
  9. L18
    exact hcommon_right
  10. L19
    exact hcommon_left

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro d
  2. 0002intro a
  3. 0003intro b
  4. 0004intro q
  5. 0005intro r
  6. 0006intro hstep
  7. 0007intro hcommon
  8. 0008cases hcommon
  9. 0009split
  10. 0010specialize euclidean_divisor_dividend_transport d
  11. 0011specialize euclidean_divisor_dividend_transport a
  12. 0012specialize euclidean_divisor_dividend_transport b
  13. 0013specialize euclidean_divisor_dividend_transport q
  14. 0014specialize euclidean_divisor_dividend_transport r
  15. 0015apply euclidean_divisor_dividend_transport
  16. 0016exact hstep
  17. 0017exact hcommon_left
  18. 0018exact hcommon_right
  19. 0019exact hcommon_left