GC0008

generalized_binary_crt_merge_step

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

Exact gcd compatibility merges two arbitrary natural moduli, including zero, without any coprimality assumption.

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 P n x a g. ((((exists hag_left_factor_gcomp_binary_merge_gcd. P = g * hag_left_factor_gcomp_binary_merge_gcd) /\ (exists hag_right_factor_gcomp_binary_merge_gcd. n = g * hag_right_factor_gcomp_binary_merge_gcd)) /\ forall hag_divisor_gcomp_binary_merge_gcd. (exists hag_common_left_gcomp_binary_merge_gcd. P = hag_divisor_gcomp_binary_merge_gcd * hag_common_left_gcomp_binary_merge_gcd) -> (exists hag_common_right_gcomp_binary_merge_gcd. n = hag_divisor_gcomp_binary_merge_gcd * hag_common_right_gcomp_binary_merge_gcd) -> exists hag_greatest_factor_gcomp_binary_merge_gcd. g = hag_divisor_gcomp_binary_merge_gcd * hag_greatest_factor_gcomp_binary_merge_gcd)) -> (exists hgcrt_mod_left_gcrt_gcomp_binary_merge_compatible hgcrt_mod_right_gcrt_gcomp_binary_merge_compatible. x + g * hgcrt_mod_left_gcrt_gcomp_binary_merge_compatible = a + g * hgcrt_mod_right_gcrt_gcomp_binary_merge_compatible) -> exists z. (((exists hgcrt_mod_left_gcrt_gcomp_binary_merge_left hgcrt_mod_right_gcrt_gcomp_binary_merge_left. z + P * hgcrt_mod_left_gcrt_gcomp_binary_merge_left = x + P * hgcrt_mod_right_gcrt_gcomp_binary_merge_left) /\ (exists hgcrt_mod_left_gcrt_gcomp_binary_merge_right hgcrt_mod_right_gcrt_gcomp_binary_merge_right. z + n * hgcrt_mod_left_gcrt_gcomp_binary_merge_right = a + n * hgcrt_mod_right_gcrt_gcomp_binary_merge_right)))

Constructive proof overview

Generated structural guide

Exact gcd compatibility merges two arbitrary natural moduli, including zero, without any coprimality assumption.

The unchanged tactic script uses 1 declared prerequisite and contains 15 exact native proof lines.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

generalized_binary_crt_sufficient Stable theorem; checked-use authorized

Direct dependents

GC0009 crt_merge_compatible_prefix_solution_exists

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

15 script commands · 2 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 assumptionsL1–7

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

  1. L1
    intro P
  2. L2
    intro n
  3. L3
    intro x
  4. L4
    intro a
  5. L5
    intro g
  6. L6
    intro hg
  7. L7
    intro hcompat
02Use earlier factsL8–15

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

  1. L8
    specialize generalized_binary_crt_sufficient g
  2. L9
    specialize generalized_binary_crt_sufficient P
  3. L10
    specialize generalized_binary_crt_sufficient n
  4. L11
    specialize generalized_binary_crt_sufficient x
  5. L12
    specialize generalized_binary_crt_sufficient a
  6. L13
    apply generalized_binary_crt_sufficient
  7. L14
    exact hg
  8. L15
    exact hcompat

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro P
  2. 0002intro n
  3. 0003intro x
  4. 0004intro a
  5. 0005intro g
  6. 0006intro hg
  7. 0007intro hcompat
  8. 0008specialize generalized_binary_crt_sufficient g
  9. 0009specialize generalized_binary_crt_sufficient P
  10. 0010specialize generalized_binary_crt_sufficient n
  11. 0011specialize generalized_binary_crt_sufficient x
  12. 0012specialize generalized_binary_crt_sufficient a
  13. 0013apply generalized_binary_crt_sufficient
  14. 0014exact hg
  15. 0015exact hcompat

Separate complete second-wave branches: Full G011 proof · Alpha v27.