GT0008

euclidean_gcd_step_iff

The full greatest-common-divisor specification is constructively equivalent before and after each exact Euclidean division.

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

∀ g. ∀ a. ∀ b. ∀ q. ∀ r. EuclideanDivision(a,b,q,r) → (IsGCD(g,a,b) → IsGCD(g,b,r)) ∧ (IsGCD(g,b,r) → IsGCD(g,a,b))

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

Definition DAG

EuclideanDivision(a,b,q,r) · 1IsGCD(g,a,b) · 4

Actual proof prerequisites

euclidean_gcd_step_backwardeuclidean_gcd_step_forward
Original expanded first-order statement 
forall g 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 hag_left_factor_egt_before. a = g * hag_left_factor_egt_before) /\ (exists hag_right_factor_egt_before. b = g * hag_right_factor_egt_before)) /\ forall hag_divisor_egt_before. (exists hag_common_left_egt_before. a = hag_divisor_egt_before * hag_common_left_egt_before) -> (exists hag_common_right_egt_before. b = hag_divisor_egt_before * hag_common_right_egt_before) -> exists hag_greatest_factor_egt_before. g = hag_divisor_egt_before * hag_greatest_factor_egt_before)) -> ((((exists hag_left_factor_egt_after. b = g * hag_left_factor_egt_after) /\ (exists hag_right_factor_egt_after. r = g * hag_right_factor_egt_after)) /\ forall hag_divisor_egt_after. (exists hag_common_left_egt_after. b = hag_divisor_egt_after * hag_common_left_egt_after) -> (exists hag_common_right_egt_after. r = hag_divisor_egt_after * hag_common_right_egt_after) -> exists hag_greatest_factor_egt_after. g = hag_divisor_egt_after * hag_greatest_factor_egt_after))) /\ (((((exists hag_left_factor_egt_after. b = g * hag_left_factor_egt_after) /\ (exists hag_right_factor_egt_after. r = g * hag_right_factor_egt_after)) /\ forall hag_divisor_egt_after. (exists hag_common_left_egt_after. b = hag_divisor_egt_after * hag_common_left_egt_after) -> (exists hag_common_right_egt_after. r = hag_divisor_egt_after * hag_common_right_egt_after) -> exists hag_greatest_factor_egt_after. g = hag_divisor_egt_after * hag_greatest_factor_egt_after)) -> ((((exists hag_left_factor_egt_before. a = g * hag_left_factor_egt_before) /\ (exists hag_right_factor_egt_before. b = g * hag_right_factor_egt_before)) /\ forall hag_divisor_egt_before. (exists hag_common_left_egt_before. a = hag_divisor_egt_before * hag_common_left_egt_before) -> (exists hag_common_right_egt_before. b = hag_divisor_egt_before * hag_common_right_egt_before) -> exists hag_greatest_factor_egt_before. g = hag_divisor_egt_before * hag_greatest_factor_egt_before))))

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 gcd step backwardeuclidean gcd step forward
01Fix variables and assumptionsL1–6

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

  1. L1
    intro g
  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 hg
04Use earlier factsL9–16

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

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

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

  1. L17
    intro hg
06Use earlier factsL18–25

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

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

Library-wide reading audit

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