GC0004

crt_pairwise_compatible_prefix_drop_last

Pairwise gcd compatibility of any finite list restricts to its predecessor prefix.

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

∀ r. ∀ s. ∀ b. ∀ c. ∀ l. CRTPairwiseCompatiblePrefix(r,s,b,c,S l) → CRTPairwiseCompatiblePrefix(r,s,b,c,l)

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

Definition DAG

CRTPairwiseCompatiblePrefix(r,s,b,c,l) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall r s b c l. (forall gcomp_left_index_pairs_drop_source gcomp_right_index_pairs_drop_source gcomp_left_residue_pairs_drop_source gcomp_right_residue_pairs_drop_source gcomp_left_modulus_pairs_drop_source gcomp_right_modulus_pairs_drop_source gcomp_pair_gcd_pairs_drop_source. (exists ff_lt_gcrt_pairs_drop_source_left_bound. ff_lt_gcrt_pairs_drop_source_left_bound + S gcomp_left_index_pairs_drop_source = S l) -> (exists ff_lt_gcrt_pairs_drop_source_right_bound. ff_lt_gcrt_pairs_drop_source_right_bound + S gcomp_right_index_pairs_drop_source = S l) -> (((exists ff_h_gcrt_pairs_drop_source_left_residue. ff_h_gcrt_pairs_drop_source_left_residue + S (gcomp_left_residue_pairs_drop_source) = S ((S (gcomp_left_index_pairs_drop_source)) * s)) /\ exists ff_q_gcrt_pairs_drop_source_left_residue. r = ff_q_gcrt_pairs_drop_source_left_residue * S ((S (gcomp_left_index_pairs_drop_source)) * s) + (gcomp_left_residue_pairs_drop_source))) -> (((exists ff_h_gcrt_pairs_drop_source_right_residue. ff_h_gcrt_pairs_drop_source_right_residue + S (gcomp_right_residue_pairs_drop_source) = S ((S (gcomp_right_index_pairs_drop_source)) * s)) /\ exists ff_q_gcrt_pairs_drop_source_right_residue. r = ff_q_gcrt_pairs_drop_source_right_residue * S ((S (gcomp_right_index_pairs_drop_source)) * s) + (gcomp_right_residue_pairs_drop_source))) -> (((exists ff_h_gcrt_pairs_drop_source_left_modulus. ff_h_gcrt_pairs_drop_source_left_modulus + S (gcomp_left_modulus_pairs_drop_source) = S ((S (gcomp_left_index_pairs_drop_source)) * c)) /\ exists ff_q_gcrt_pairs_drop_source_left_modulus. b = ff_q_gcrt_pairs_drop_source_left_modulus * S ((S (gcomp_left_index_pairs_drop_source)) * c) + (gcomp_left_modulus_pairs_drop_source))) -> (((exists ff_h_gcrt_pairs_drop_source_right_modulus. ff_h_gcrt_pairs_drop_source_right_modulus + S (gcomp_right_modulus_pairs_drop_source) = S ((S (gcomp_right_index_pairs_drop_source)) * c)) /\ exists ff_q_gcrt_pairs_drop_source_right_modulus. b = ff_q_gcrt_pairs_drop_source_right_modulus * S ((S (gcomp_right_index_pairs_drop_source)) * c) + (gcomp_right_modulus_pairs_drop_source))) -> ((((exists hag_left_factor_gcomp_pairs_drop_source_gcd. gcomp_left_modulus_pairs_drop_source = gcomp_pair_gcd_pairs_drop_source * hag_left_factor_gcomp_pairs_drop_source_gcd) /\ (exists hag_right_factor_gcomp_pairs_drop_source_gcd. gcomp_right_modulus_pairs_drop_source = gcomp_pair_gcd_pairs_drop_source * hag_right_factor_gcomp_pairs_drop_source_gcd)) /\ forall hag_divisor_gcomp_pairs_drop_source_gcd. (exists hag_common_left_gcomp_pairs_drop_source_gcd. gcomp_left_modulus_pairs_drop_source = hag_divisor_gcomp_pairs_drop_source_gcd * hag_common_left_gcomp_pairs_drop_source_gcd) -> (exists hag_common_right_gcomp_pairs_drop_source_gcd. gcomp_right_modulus_pairs_drop_source = hag_divisor_gcomp_pairs_drop_source_gcd * hag_common_right_gcomp_pairs_drop_source_gcd) -> exists hag_greatest_factor_gcomp_pairs_drop_source_gcd. gcomp_pair_gcd_pairs_drop_source = hag_divisor_gcomp_pairs_drop_source_gcd * hag_greatest_factor_gcomp_pairs_drop_source_gcd)) -> (exists hgcrt_mod_left_gcrt_pairs_drop_source_result hgcrt_mod_right_gcrt_pairs_drop_source_result. gcomp_left_residue_pairs_drop_source + gcomp_pair_gcd_pairs_drop_source * hgcrt_mod_left_gcrt_pairs_drop_source_result = gcomp_right_residue_pairs_drop_source + gcomp_pair_gcd_pairs_drop_source * hgcrt_mod_right_gcrt_pairs_drop_source_result)) -> (forall gcomp_left_index_pairs_drop_result gcomp_right_index_pairs_drop_result gcomp_left_residue_pairs_drop_result gcomp_right_residue_pairs_drop_result gcomp_left_modulus_pairs_drop_result gcomp_right_modulus_pairs_drop_result gcomp_pair_gcd_pairs_drop_result. (exists ff_lt_gcrt_pairs_drop_result_left_bound. ff_lt_gcrt_pairs_drop_result_left_bound + S gcomp_left_index_pairs_drop_result = l) -> (exists ff_lt_gcrt_pairs_drop_result_right_bound. ff_lt_gcrt_pairs_drop_result_right_bound + S gcomp_right_index_pairs_drop_result = l) -> (((exists ff_h_gcrt_pairs_drop_result_left_residue. ff_h_gcrt_pairs_drop_result_left_residue + S (gcomp_left_residue_pairs_drop_result) = S ((S (gcomp_left_index_pairs_drop_result)) * s)) /\ exists ff_q_gcrt_pairs_drop_result_left_residue. r = ff_q_gcrt_pairs_drop_result_left_residue * S ((S (gcomp_left_index_pairs_drop_result)) * s) + (gcomp_left_residue_pairs_drop_result))) -> (((exists ff_h_gcrt_pairs_drop_result_right_residue. ff_h_gcrt_pairs_drop_result_right_residue + S (gcomp_right_residue_pairs_drop_result) = S ((S (gcomp_right_index_pairs_drop_result)) * s)) /\ exists ff_q_gcrt_pairs_drop_result_right_residue. r = ff_q_gcrt_pairs_drop_result_right_residue * S ((S (gcomp_right_index_pairs_drop_result)) * s) + (gcomp_right_residue_pairs_drop_result))) -> (((exists ff_h_gcrt_pairs_drop_result_left_modulus. ff_h_gcrt_pairs_drop_result_left_modulus + S (gcomp_left_modulus_pairs_drop_result) = S ((S (gcomp_left_index_pairs_drop_result)) * c)) /\ exists ff_q_gcrt_pairs_drop_result_left_modulus. b = ff_q_gcrt_pairs_drop_result_left_modulus * S ((S (gcomp_left_index_pairs_drop_result)) * c) + (gcomp_left_modulus_pairs_drop_result))) -> (((exists ff_h_gcrt_pairs_drop_result_right_modulus. ff_h_gcrt_pairs_drop_result_right_modulus + S (gcomp_right_modulus_pairs_drop_result) = S ((S (gcomp_right_index_pairs_drop_result)) * c)) /\ exists ff_q_gcrt_pairs_drop_result_right_modulus. b = ff_q_gcrt_pairs_drop_result_right_modulus * S ((S (gcomp_right_index_pairs_drop_result)) * c) + (gcomp_right_modulus_pairs_drop_result))) -> ((((exists hag_left_factor_gcomp_pairs_drop_result_gcd. gcomp_left_modulus_pairs_drop_result = gcomp_pair_gcd_pairs_drop_result * hag_left_factor_gcomp_pairs_drop_result_gcd) /\ (exists hag_right_factor_gcomp_pairs_drop_result_gcd. gcomp_right_modulus_pairs_drop_result = gcomp_pair_gcd_pairs_drop_result * hag_right_factor_gcomp_pairs_drop_result_gcd)) /\ forall hag_divisor_gcomp_pairs_drop_result_gcd. (exists hag_common_left_gcomp_pairs_drop_result_gcd. gcomp_left_modulus_pairs_drop_result = hag_divisor_gcomp_pairs_drop_result_gcd * hag_common_left_gcomp_pairs_drop_result_gcd) -> (exists hag_common_right_gcomp_pairs_drop_result_gcd. gcomp_right_modulus_pairs_drop_result = hag_divisor_gcomp_pairs_drop_result_gcd * hag_common_right_gcomp_pairs_drop_result_gcd) -> exists hag_greatest_factor_gcomp_pairs_drop_result_gcd. gcomp_pair_gcd_pairs_drop_result = hag_divisor_gcomp_pairs_drop_result_gcd * hag_greatest_factor_gcomp_pairs_drop_result_gcd)) -> (exists hgcrt_mod_left_gcrt_pairs_drop_result_result hgcrt_mod_right_gcrt_pairs_drop_result_result. gcomp_left_residue_pairs_drop_result + gcomp_pair_gcd_pairs_drop_result * hgcrt_mod_left_gcrt_pairs_drop_result_result = gcomp_right_residue_pairs_drop_result + gcomp_pair_gcd_pairs_drop_result * hgcrt_mod_right_gcrt_pairs_drop_result_result))

Complete unchanged native tactic proof

All 41 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

41 script commands · 5 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 r
  2. L2
    intro s
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro l
  6. L6
    intro hpairs
  7. L7
    intro i
  8. L8
    intro j
  9. L9
    intro a
  10. L10
    intro d
02Fix variables and assumptionsL11–20

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

  1. L11
    intro m
  2. L12
    intro n
  3. L13
    intro g
  4. L14
    intro hi
  5. L15
    intro hj
  6. L16
    intro ha
  7. L17
    intro hd
  8. L18
    intro hm
  9. L19
    intro hn
  10. L20
    intro hg
03Use earlier factsL21–30

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

  1. L21
    specialize hpairs i
  2. L22
    specialize hpairs j
  3. L23
    specialize hpairs a
  4. L24
    specialize hpairs d
  5. L25
    specialize hpairs m
  6. L26
    specialize hpairs n
  7. L27
    specialize hpairs g
  8. L28
    apply hpairs
  9. L29
    specialize le_succ (S i)
  10. L30
    specialize le_succ l
04Use earlier factsL31–40

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

  1. L31
    apply le_succ
  2. L32
    exact hi
  3. L33
    specialize le_succ (S j)
  4. L34
    specialize le_succ l
  5. L35
    apply le_succ
  6. L36
    exact hj
  7. L37
    exact ha
  8. L38
    exact hd
  9. L39
    exact hm
  10. L40
    exact hn
05Use earlier factsL41–41

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

  1. L41
    exact hg

Library-wide reading audit

Original defined command ledger · 41 lines
  1. 0001intro r
  2. 0002intro s
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro hpairs
  7. 0007intro i
  8. 0008intro j
  9. 0009intro a
  10. 0010intro d
  11. 0011intro m
  12. 0012intro n
  13. 0013intro g
  14. 0014intro hi
  15. 0015intro hj
  16. 0016intro ha
  17. 0017intro hd
  18. 0018intro hm
  19. 0019intro hn
  20. 0020intro hg
  21. 0021specialize hpairs i
  22. 0022specialize hpairs j
  23. 0023specialize hpairs a
  24. 0024specialize hpairs d
  25. 0025specialize hpairs m
  26. 0026specialize hpairs n
  27. 0027specialize hpairs g
  28. 0028apply hpairs
  29. 0029specialize le_succ (S i)
  30. 0030specialize le_succ l
  31. 0031apply le_succ
  32. 0032exact hi
  33. 0033specialize le_succ (S j)
  34. 0034specialize le_succ l
  35. 0035apply le_succ
  36. 0036exact hj
  37. 0037exact ha
  38. 0038exact hd
  39. 0039exact hm
  40. 0040exact hn
  41. 0041exact hg