GT0005

euclidean_common_divisor_iff

An exact bounded Euclidean division preserves the entire witnessed common-divisor relation in both constructive directions.

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,a,b) → EuclideanCommonDivisor(d,b,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) · 4

Actual proof prerequisites

euclidean_common_divisor_forwardeuclidean_common_divisor_backward
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_before_left. a = d * egt_factor_before_left) /\ (exists egt_factor_before_right. b = d * egt_factor_before_right))) -> (((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_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 25 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

25 script commands · 6 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 (2)
euclidean common divisor forwardeuclidean common divisor backward
01Fix variables and assumptionsL1–6

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
02Separate the logical casesL7–7

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

  1. L7
    split
03Fix variables and assumptionsL8–8

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

  1. L8
    intro hcommon
04Use earlier factsL9–16

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

  1. L9
    specialize euclidean_common_divisor_forward d
  2. L10
    specialize euclidean_common_divisor_forward a
  3. L11
    specialize euclidean_common_divisor_forward b
  4. L12
    specialize euclidean_common_divisor_forward q
  5. L13
    specialize euclidean_common_divisor_forward r
  6. L14
    apply euclidean_common_divisor_forward
  7. L15
    exact hstep
  8. L16
    exact hcommon
05Fix variables and assumptionsL17–17

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

  1. L17
    intro hcommon
06Use earlier factsL18–25

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

  1. L18
    specialize euclidean_common_divisor_backward d
  2. L19
    specialize euclidean_common_divisor_backward a
  3. L20
    specialize euclidean_common_divisor_backward b
  4. L21
    specialize euclidean_common_divisor_backward q
  5. L22
    specialize euclidean_common_divisor_backward r
  6. L23
    apply euclidean_common_divisor_backward
  7. L24
    exact hstep
  8. L25
    exact hcommon

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro d
  2. 0002intro a
  3. 0003intro b
  4. 0004intro q
  5. 0005intro r
  6. 0006intro hstep
  7. 0007split
  8. 0008intro hcommon
  9. 0009specialize euclidean_common_divisor_forward d
  10. 0010specialize euclidean_common_divisor_forward a
  11. 0011specialize euclidean_common_divisor_forward b
  12. 0012specialize euclidean_common_divisor_forward q
  13. 0013specialize euclidean_common_divisor_forward r
  14. 0014apply euclidean_common_divisor_forward
  15. 0015exact hstep
  16. 0016exact hcommon
  17. 0017intro hcommon
  18. 0018specialize euclidean_common_divisor_backward d
  19. 0019specialize euclidean_common_divisor_backward a
  20. 0020specialize euclidean_common_divisor_backward b
  21. 0021specialize euclidean_common_divisor_backward q
  22. 0022specialize euclidean_common_divisor_backward r
  23. 0023apply euclidean_common_divisor_backward
  24. 0024exact hstep
  25. 0025exact hcommon