GT0005

euclidean_common_divisor_iff

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 25 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GT0003 euclidean_common_divisor_forward GT0004 euclidean_common_divisor_backward

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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