FC0002

crt_gcd_nonzero_left

A gcd of a nonzero left input is nonzero, without restricting the right 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. ∀ b. ∀ g. ¬a = 0 → IsGCD(g,a,b) → ¬g = 0

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

Definition DAG

Dvd(d,n) · 1IsGCD(g,a,b) · 1

Actual proof prerequisites

is_gcd_dvd_left · checked external prerequisitefactor_nonzero_left · checked external prerequisite
Original expanded first-order statement 
forall a b g. ~(a = 0) -> ((((exists ec_gcd_left_gfull_nonzero. a = g * ec_gcd_left_gfull_nonzero) /\ (exists ec_gcd_right_gfull_nonzero. b = g * ec_gcd_right_gfull_nonzero)) /\ forall ec_gcd_common_gfull_nonzero. (exists ec_gcd_common_left_gfull_nonzero. a = ec_gcd_common_gfull_nonzero * ec_gcd_common_left_gfull_nonzero) -> (exists ec_gcd_common_right_gfull_nonzero. b = ec_gcd_common_gfull_nonzero * ec_gcd_common_right_gfull_nonzero) -> exists ec_gcd_greatest_gfull_nonzero. g = ec_gcd_common_gfull_nonzero * ec_gcd_greatest_gfull_nonzero)) -> ~(g = 0)

Complete tactic proof in conservative notation

All 20 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

20 script commands · 5 reading checkpoints · 1 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro g
  4. L4
    intro ha
  5. L5
    intro hg
02Establish hdivL6–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply is gcd dvd left.

  1. L6
  2. L7
    specialize is_gcd_dvd_left g
  3. L8
    specialize is_gcd_dvd_left a
  4. L9
    specialize is_gcd_dvd_left b
  5. L10
    apply is_gcd_dvd_left
  6. L11
    exact hg
03Separate the logical casesL12–12

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

  1. L12
    cases hdiv
04Fix variables and assumptionsL13–13

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

  1. L13
    intro hz
05Use earlier factsL14–20

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

  1. L14
    specialize factor_nonzero_left a
  2. L15
    specialize factor_nonzero_left g
  3. L16
    specialize factor_nonzero_left x
  4. L17
    apply factor_nonzero_left
  5. L18
    exact ha
  6. L19
    exact hdiv_witness
  7. L20
    exact hz

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro g
  4. 0004intro ha
  5. 0005intro hg
  6. 0006have hdiv : Dvd(g,a)
  7. 0007specialize is_gcd_dvd_left g
  8. 0008specialize is_gcd_dvd_left a
  9. 0009specialize is_gcd_dvd_left b
  10. 0010apply is_gcd_dvd_left
  11. 0011exact hg
  12. 0012cases hdiv
  13. 0013intro hz
  14. 0014specialize factor_nonzero_left a
  15. 0015specialize factor_nonzero_left g
  16. 0016specialize factor_nonzero_left x
  17. 0017apply factor_nonzero_left
  18. 0018exact ha
  19. 0019exact hdiv_witness
  20. 0020exact hz