FC0003

crt_gcd_nonzero_right

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

A gcd of a nonzero right input is nonzero, including a zero left 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 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)

Constructive proof overview

Generated structural guide

A gcd of a nonzero right input is nonzero, including a zero left input.

The unchanged tactic script uses 2 declared prerequisites and contains 17 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FC0002 crt_gcd_nonzero_left is_gcd_symm Stable theorem; checked-use authorized

Direct dependents

FC0006 crt_gcd_lcm_distributes_nonzero

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

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.

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 exact 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