GT0007

euclidean_gcd_step_backward

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

A relational gcd of the previous exact Euclidean pair is a relational gcd of the next pair.

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 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))

Constructive proof overview

Generated structural guide

A relational gcd of the previous exact Euclidean pair is a relational gcd of the next pair.

The unchanged tactic script uses 1 declared prerequisite and contains 16 exact native proof lines.

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

Proof neighborhood

Direct dependencies

is_gcd_euclid_backward Stable theorem; checked-use authorized

Direct dependents

GT0008 euclidean_gcd_step_iff

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

16 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.

01Fix variables and assumptionsL1–7

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
  7. L7
    intro hg
02Separate the logical casesL8–8

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

  1. L8
    cases hstep
03Use earlier factsL9–16

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

  1. L9
    specialize is_gcd_euclid_backward g
  2. L10
    specialize is_gcd_euclid_backward a
  3. L11
    specialize is_gcd_euclid_backward b
  4. L12
    specialize is_gcd_euclid_backward q
  5. L13
    specialize is_gcd_euclid_backward r
  6. L14
    apply is_gcd_euclid_backward
  7. L15
    exact hstep_left
  8. L16
    exact hg

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro g
  2. 0002intro a
  3. 0003intro b
  4. 0004intro q
  5. 0005intro r
  6. 0006intro hstep
  7. 0007intro hg
  8. 0008cases hstep
  9. 0009specialize is_gcd_euclid_backward g
  10. 0010specialize is_gcd_euclid_backward a
  11. 0011specialize is_gcd_euclid_backward b
  12. 0012specialize is_gcd_euclid_backward q
  13. 0013specialize is_gcd_euclid_backward r
  14. 0014apply is_gcd_euclid_backward
  15. 0015exact hstep_left
  16. 0016exact hg