FC000A

crt_prefix_gcd_congruences_drop_last

Pointwise congruence modulo decoded gcds restricts to the preceding finite prefix.

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

∀ b. ∀ c. ∀ l. ∀ n. ∀ u. ∀ v. CRTPrefixGcdCongruences(b,c,S l,n,u,v) → CRTPrefixGcdCongruences(b,c,l,n,u,v)

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

Definition DAG

CRTPrefixGcdCongruences(b,c,l,m,u,v) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall b c l n u v. (forall gfull_index_drop_source gfull_modulus_drop_source gfull_gcd_drop_source. (exists ff_lt_gcrt_gfull_drop_source_bound. ff_lt_gcrt_gfull_drop_source_bound + S gfull_index_drop_source = S l) -> (((exists ff_h_gcrt_gfull_drop_source_entry. ff_h_gcrt_gfull_drop_source_entry + S (gfull_modulus_drop_source) = S ((S (gfull_index_drop_source)) * c)) /\ exists ff_q_gcrt_gfull_drop_source_entry. b = ff_q_gcrt_gfull_drop_source_entry * S ((S (gfull_index_drop_source)) * c) + (gfull_modulus_drop_source))) -> ((((exists ec_gcd_left_gfull_drop_source_gcd. gfull_modulus_drop_source = gfull_gcd_drop_source * ec_gcd_left_gfull_drop_source_gcd) /\ (exists ec_gcd_right_gfull_drop_source_gcd. n = gfull_gcd_drop_source * ec_gcd_right_gfull_drop_source_gcd)) /\ forall ec_gcd_common_gfull_drop_source_gcd. (exists ec_gcd_common_left_gfull_drop_source_gcd. gfull_modulus_drop_source = ec_gcd_common_gfull_drop_source_gcd * ec_gcd_common_left_gfull_drop_source_gcd) -> (exists ec_gcd_common_right_gfull_drop_source_gcd. n = ec_gcd_common_gfull_drop_source_gcd * ec_gcd_common_right_gfull_drop_source_gcd) -> exists ec_gcd_greatest_gfull_drop_source_gcd. gfull_gcd_drop_source = ec_gcd_common_gfull_drop_source_gcd * ec_gcd_greatest_gfull_drop_source_gcd)) -> (exists hgcrt_mod_left_gfull_drop_source_mod hgcrt_mod_right_gfull_drop_source_mod. u + gfull_gcd_drop_source * hgcrt_mod_left_gfull_drop_source_mod = v + gfull_gcd_drop_source * hgcrt_mod_right_gfull_drop_source_mod)) -> (forall gfull_index_drop_result gfull_modulus_drop_result gfull_gcd_drop_result. (exists ff_lt_gcrt_gfull_drop_result_bound. ff_lt_gcrt_gfull_drop_result_bound + S gfull_index_drop_result = l) -> (((exists ff_h_gcrt_gfull_drop_result_entry. ff_h_gcrt_gfull_drop_result_entry + S (gfull_modulus_drop_result) = S ((S (gfull_index_drop_result)) * c)) /\ exists ff_q_gcrt_gfull_drop_result_entry. b = ff_q_gcrt_gfull_drop_result_entry * S ((S (gfull_index_drop_result)) * c) + (gfull_modulus_drop_result))) -> ((((exists ec_gcd_left_gfull_drop_result_gcd. gfull_modulus_drop_result = gfull_gcd_drop_result * ec_gcd_left_gfull_drop_result_gcd) /\ (exists ec_gcd_right_gfull_drop_result_gcd. n = gfull_gcd_drop_result * ec_gcd_right_gfull_drop_result_gcd)) /\ forall ec_gcd_common_gfull_drop_result_gcd. (exists ec_gcd_common_left_gfull_drop_result_gcd. gfull_modulus_drop_result = ec_gcd_common_gfull_drop_result_gcd * ec_gcd_common_left_gfull_drop_result_gcd) -> (exists ec_gcd_common_right_gfull_drop_result_gcd. n = ec_gcd_common_gfull_drop_result_gcd * ec_gcd_common_right_gfull_drop_result_gcd) -> exists ec_gcd_greatest_gfull_drop_result_gcd. gfull_gcd_drop_result = ec_gcd_common_gfull_drop_result_gcd * ec_gcd_greatest_gfull_drop_result_gcd)) -> (exists hgcrt_mod_left_gfull_drop_result_mod hgcrt_mod_right_gfull_drop_result_mod. u + gfull_gcd_drop_result * hgcrt_mod_left_gfull_drop_result_mod = v + gfull_gcd_drop_result * hgcrt_mod_right_gfull_drop_result_mod))

Complete tactic proof in conservative notation

All 23 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

23 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro n
  5. L5
    intro u
  6. L6
    intro v
  7. L7
    intro hp
  8. L8
    intro i
  9. L9
    intro m
  10. L10
    intro d
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro hm
  3. L13
    intro hd
03Use earlier factsL14–23

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

  1. L14
    specialize hp i
  2. L15
    specialize hp m
  3. L16
    specialize hp d
  4. L17
    apply hp
  5. L18
    specialize le_succ (S i)
  6. L19
    specialize le_succ l
  7. L20
    apply le_succ
  8. L21
    exact hi
  9. L22
    exact hm
  10. L23
    exact hd

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro n
  5. 0005intro u
  6. 0006intro v
  7. 0007intro hp
  8. 0008intro i
  9. 0009intro m
  10. 0010intro d
  11. 0011intro hi
  12. 0012intro hm
  13. 0013intro hd
  14. 0014specialize hp i
  15. 0015specialize hp m
  16. 0016specialize hp d
  17. 0017apply hp
  18. 0018specialize le_succ (S i)
  19. 0019specialize le_succ l
  20. 0020apply le_succ
  21. 0021exact hi
  22. 0022exact hm
  23. 0023exact hd