CG0002

linear_congruence_solution_class_iff_reduced_modulus

Relative to any actual solution, every natural solution is exactly its class modulo the actual gcd cofactor.

Alpha v34 checked-use · first admitted v34 · 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.

Each statement retains its explicit modulus, coprimality and divisibility assumptions. These twelve arithmetic laws do not assert all order, primitive-root, Carmichael, exponential or simultaneous-polynomial congruence goals are finished.

Exact theorem in conservative defined notation

∀ a. ∀ m. ∀ g. ∀ A. ∀ M. ∀ b. ∀ r. ∀ x. ¬m = 0 → IsGCD(g,a,m) → a = g · A → m = g · M → ModEq(m,a · r,b) → (ModEq(m,a · x,b) → ModEq(M,x,r)) ∧ (ModEq(M,x,r) → ModEq(m,a · x,b))

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

Definition DAG

IsGCD(g,a,b) · 1ModEq(m,a,b) · 9

Actual proof prerequisites

mod_eq_cancel_gcd_cofactormod_eq_trans · checked external prerequisitemod_eq_symm · checked external prerequisite
Original expanded first-order statement 
forall a m g A M b r x. (~(m=0)) -> ((((exists hag_left_factor_lcc_class. a = g * hag_left_factor_lcc_class) /\ (exists hag_right_factor_lcc_class. m = g * hag_right_factor_lcc_class)) /\ forall hag_divisor_lcc_class. (exists hag_common_left_lcc_class. a = hag_divisor_lcc_class * hag_common_left_lcc_class) -> (exists hag_common_right_lcc_class. m = hag_divisor_lcc_class * hag_common_right_lcc_class) -> exists hag_greatest_factor_lcc_class. g = hag_divisor_lcc_class * hag_greatest_factor_lcc_class)) -> (a=g*A) -> (m=g*M) -> (exists hgcrt_mod_left_lcc_class_reference hgcrt_mod_right_lcc_class_reference. (a * r) + m * hgcrt_mod_left_lcc_class_reference = b + m * hgcrt_mod_right_lcc_class_reference) -> ((((exists hgcrt_mod_left_lcc_class_solution hgcrt_mod_right_lcc_class_solution. (a * x) + m * hgcrt_mod_left_lcc_class_solution = b + m * hgcrt_mod_right_lcc_class_solution) -> (exists hgcrt_mod_left_lcc_class_reduced hgcrt_mod_right_lcc_class_reduced. x + M * hgcrt_mod_left_lcc_class_reduced = r + M * hgcrt_mod_right_lcc_class_reduced)) /\ (((exists hgcrt_mod_left_lcc_class_reduced hgcrt_mod_right_lcc_class_reduced. x + M * hgcrt_mod_left_lcc_class_reduced = r + M * hgcrt_mod_right_lcc_class_reduced) -> (exists hgcrt_mod_left_lcc_class_solution hgcrt_mod_right_lcc_class_solution. (a * x) + m * hgcrt_mod_left_lcc_class_solution = b + m * hgcrt_mod_right_lcc_class_solution)))))

Complete tactic proof in conservative notation

All 50 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

50 script commands · 10 reading checkpoints · 1 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.

Named ingredients (1)
mod eq cancel gcd cofactor
01Fix variables and assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro g
  4. L4
    intro A
  5. L5
    intro M
  6. L6
    intro b
  7. L7
    intro r
  8. L8
    intro x
  9. L9
    intro hm0
  10. L10
    intro hg
02Fix variables and assumptionsL11–13

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

  1. L11
    intro ha
  2. L12
    intro hm
  3. L13
    intro hr
03Establish hcL14–23

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq cancel gcd cofactor.

  1. L14
    have hc : (ModEq(m,a · x,a · r) → ModEq(M,x,r)) ∧ (ModEq(M,x,r) → ModEq(m,a · x,a · r))Definitions: ModEq(m,a · x,a · r)ModEq(M,x,r)Original native command in the exact edition
  2. L15
    specialize mod_eq_cancel_gcd_cofactor (a)
  3. L16
    specialize mod_eq_cancel_gcd_cofactor (m)
  4. L17
    specialize mod_eq_cancel_gcd_cofactor (g)
  5. L18
    specialize mod_eq_cancel_gcd_cofactor (A)
  6. L19
    specialize mod_eq_cancel_gcd_cofactor (M)
  7. L20
    specialize mod_eq_cancel_gcd_cofactor (x)
  8. L21
    specialize mod_eq_cancel_gcd_cofactor (r)
  9. L22
    apply mod_eq_cancel_gcd_cofactor
  10. L23
    exact hm0
04Use earlier factsL24–26

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

  1. L24
    exact hg
  2. L25
    exact ha
  3. L26
    exact hm
05Separate the logical casesL27–28

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L27
    cases hc
  2. L28
    split
06Fix variables and assumptionsL29–29

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

  1. L29
    intro hx
07Use earlier factsL30–39

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

  1. L30
    apply hc_left
  2. L31
    specialize mod_eq_trans (m)
  3. L32
    specialize mod_eq_trans (a*x)
  4. L33
    specialize mod_eq_trans (b)
  5. L34
    specialize mod_eq_trans (a*r)
  6. L35
    apply mod_eq_trans
  7. L36
    exact hx
  8. L37
    specialize mod_eq_symm (m)
  9. L38
    specialize mod_eq_symm (a*r)
  10. L39
    specialize mod_eq_symm (b)
08Use earlier factsL40–41

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

  1. L40
    apply mod_eq_symm
  2. L41
    exact hr
09Fix variables and assumptionsL42–42

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

  1. L42
    intro hx
10Use earlier factsL43–50

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

  1. L43
    specialize mod_eq_trans (m)
  2. L44
    specialize mod_eq_trans (a*x)
  3. L45
    specialize mod_eq_trans (a*r)
  4. L46
    specialize mod_eq_trans (b)
  5. L47
    apply mod_eq_trans
  6. L48
    apply hc_right
  7. L49
    exact hx
  8. L50
    exact hr

Library-wide reading audit

Original defined command ledger · 50 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro g
  4. 0004intro A
  5. 0005intro M
  6. 0006intro b
  7. 0007intro r
  8. 0008intro x
  9. 0009intro hm0
  10. 0010intro hg
  11. 0011intro ha
  12. 0012intro hm
  13. 0013intro hr
  14. 0014have hc : (ModEq(m,a · x,a · r) → ModEq(M,x,r)) ∧ (ModEq(M,x,r) → ModEq(m,a · x,a · r))
  15. 0015specialize mod_eq_cancel_gcd_cofactor (a)
  16. 0016specialize mod_eq_cancel_gcd_cofactor (m)
  17. 0017specialize mod_eq_cancel_gcd_cofactor (g)
  18. 0018specialize mod_eq_cancel_gcd_cofactor (A)
  19. 0019specialize mod_eq_cancel_gcd_cofactor (M)
  20. 0020specialize mod_eq_cancel_gcd_cofactor (x)
  21. 0021specialize mod_eq_cancel_gcd_cofactor (r)
  22. 0022apply mod_eq_cancel_gcd_cofactor
  23. 0023exact hm0
  24. 0024exact hg
  25. 0025exact ha
  26. 0026exact hm
  27. 0027cases hc
  28. 0028split
  29. 0029intro hx
  30. 0030apply hc_left
  31. 0031specialize mod_eq_trans (m)
  32. 0032specialize mod_eq_trans (a*x)
  33. 0033specialize mod_eq_trans (b)
  34. 0034specialize mod_eq_trans (a*r)
  35. 0035apply mod_eq_trans
  36. 0036exact hx
  37. 0037specialize mod_eq_symm (m)
  38. 0038specialize mod_eq_symm (a*r)
  39. 0039specialize mod_eq_symm (b)
  40. 0040apply mod_eq_symm
  41. 0041exact hr
  42. 0042intro hx
  43. 0043specialize mod_eq_trans (m)
  44. 0044specialize mod_eq_trans (a*x)
  45. 0045specialize mod_eq_trans (a*r)
  46. 0046specialize mod_eq_trans (b)
  47. 0047apply mod_eq_trans
  48. 0048apply hc_right
  49. 0049exact hx
  50. 0050exact hr