GC0008

generalized_binary_crt_merge_step

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

Alpha v34 checked-use · first admitted v25 · 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.

Historical partial components only: this chapter proves canonical solutions under successive-merge compatibility and in the pairwise-compatible dominating-last case. G011 is now closed in the separate Alpha-v27 generalized-crt branch for arbitrary pairwise-compatible finite lists, including noncoprime moduli. Full G011 proof · Alpha v27

Exact theorem in conservative defined notation

∀ P. ∀ n. ∀ x. ∀ a. ∀ g. IsGCD(g,P,n) → ModEq(g,x,a) → ∃ y. ModEq(P,y,x) ∧ ModEq(n,y,a)

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

Definition DAG

IsGCD(g,a,b) · 1ModEq(m,a,b) · 3

Actual proof prerequisites

generalized_binary_crt_sufficient · checked external prerequisite
Original expanded first-order 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)))

Complete unchanged native tactic proof

All 15 lines are the exact independently kernel-checked original script.

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.

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 assumptionsL1–7

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 defined 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