FC0003

crt_gcd_nonzero_right

A gcd of a nonzero right input is nonzero, including a zero left 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. ¬b = 0 → IsGCD(g,a,b) → ¬g = 0

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

Definition DAG

IsGCD(g,a,b) · 1

Actual proof prerequisites

crt_gcd_nonzero_leftis_gcd_symm · checked external prerequisite
Original expanded first-order statement 
forall a b g. ~(b = 0) -> ((((exists ec_gcd_left_gfull_nonzero_right. a = g * ec_gcd_left_gfull_nonzero_right) /\ (exists ec_gcd_right_gfull_nonzero_right. b = g * ec_gcd_right_gfull_nonzero_right)) /\ forall ec_gcd_common_gfull_nonzero_right. (exists ec_gcd_common_left_gfull_nonzero_right. a = ec_gcd_common_gfull_nonzero_right * ec_gcd_common_left_gfull_nonzero_right) -> (exists ec_gcd_common_right_gfull_nonzero_right. b = ec_gcd_common_gfull_nonzero_right * ec_gcd_common_right_gfull_nonzero_right) -> exists ec_gcd_greatest_gfull_nonzero_right. g = ec_gcd_common_gfull_nonzero_right * ec_gcd_greatest_gfull_nonzero_right)) -> ~(g = 0)

Complete tactic proof in conservative notation

All 17 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

17 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.

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

Named ingredients (1)
crt gcd nonzero left
01Fix variables and assumptionsL1–6

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 hb
  5. L5
    intro hg
  6. L6
    intro hz
02Use earlier factsL7–16

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

  1. L7
    specialize crt_gcd_nonzero_left b
  2. L8
    specialize crt_gcd_nonzero_left a
  3. L9
    specialize crt_gcd_nonzero_left g
  4. L10
    apply crt_gcd_nonzero_left
  5. L11
    exact hb
  6. L12
    specialize is_gcd_symm g
  7. L13
    specialize is_gcd_symm a
  8. L14
    specialize is_gcd_symm b
  9. L15
    apply is_gcd_symm
  10. L16
    exact hg
03Use earlier factsL17–17

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

  1. L17
    exact hz

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro g
  4. 0004intro hb
  5. 0005intro hg
  6. 0006intro hz
  7. 0007specialize crt_gcd_nonzero_left b
  8. 0008specialize crt_gcd_nonzero_left a
  9. 0009specialize crt_gcd_nonzero_left g
  10. 0010apply crt_gcd_nonzero_left
  11. 0011exact hb
  12. 0012specialize is_gcd_symm g
  13. 0013specialize is_gcd_symm a
  14. 0014specialize is_gcd_symm b
  15. 0015apply is_gcd_symm
  16. 0016exact hg
  17. 0017exact hz