FC0001

crt_gcd_zero_right_value

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

A relational gcd with zero on the right equals its other input.

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 a g. ((((exists ec_gcd_left_gfull_zero_right. a = g * ec_gcd_left_gfull_zero_right) /\ (exists ec_gcd_right_gfull_zero_right. 0 = g * ec_gcd_right_gfull_zero_right)) /\ forall ec_gcd_common_gfull_zero_right. (exists ec_gcd_common_left_gfull_zero_right. a = ec_gcd_common_gfull_zero_right * ec_gcd_common_left_gfull_zero_right) -> (exists ec_gcd_common_right_gfull_zero_right. 0 = ec_gcd_common_gfull_zero_right * ec_gcd_common_right_gfull_zero_right) -> exists ec_gcd_greatest_gfull_zero_right. g = ec_gcd_common_gfull_zero_right * ec_gcd_greatest_gfull_zero_right)) -> g = a

Constructive proof overview

Generated structural guide

A relational gcd with zero on the right equals its other input.

The unchanged tactic script uses 1 declared prerequisite and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

canonical_gcd_zero_right_iff Alpha theorem; checked-use authorized

Direct dependents

FC0008 crt_gcd_lcm_distributes_zero_comparison

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

8 script commands · 4 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro g
  3. L3
    intro hg
02Use earlier factsL4–5

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

  1. L4
    specialize canonical_gcd_zero_right_iff a
  2. L5
    specialize canonical_gcd_zero_right_iff g
03Separate the logical casesL6–6

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

  1. L6
    cases canonical_gcd_zero_right_iff
04Use earlier factsL7–8

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

  1. L7
    apply canonical_gcd_zero_right_iff_left
  2. L8
    exact hg

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro a
  2. 0002intro g
  3. 0003intro hg
  4. 0004specialize canonical_gcd_zero_right_iff a
  5. 0005specialize canonical_gcd_zero_right_iff g
  6. 0006cases canonical_gcd_zero_right_iff
  7. 0007apply canonical_gcd_zero_right_iff_left
  8. 0008exact hg