FC0001

crt_gcd_zero_right_value

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

Alpha v34 checked-use · first admitted v27 · 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.

All finite lists are included, even the empty list and zero moduli. A positive LCM gives x<M; at zero LCM congruence is exact equality and normalization deliberately does not require the impossible x<0.

Exact theorem in conservative defined notation

∀ a. ∀ g. IsGCD(g,a,0) → g = a

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

Definition DAG

IsGCD(g,a,b) · 1

Actual proof prerequisites

canonical_gcd_zero_right_iff · checked external prerequisite
Original expanded first-order 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

Complete tactic proof in conservative notation

All 8 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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