BX0009

binary_modular_multiply_congruence

Multiplication preserves both witnessed modular factors.

Alpha v34 checked-use · first admitted v21 · 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.

G102 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: every arbitrary exponent has actual canonical beta-coded digits, a complete modular execution, and exact counted bound operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ m. ∀ x. ∀ r. ∀ y. ∀ s. ModEq(m,x,r) → ModEq(m,y,s) → ModEq(m,x · y,r · s)

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

Definition DAG

ModEq(m,a,b) · 3

Actual proof prerequisites

mod_eq_mul · checked external prerequisite
Original expanded first-order statement 
forall m x r y s. (exists ff_left_binary_multiply_left ff_right_binary_multiply_left. (x) + m * ff_left_binary_multiply_left = (r) + m * ff_right_binary_multiply_left) -> (exists ff_left_binary_multiply_right ff_right_binary_multiply_right. (y) + m * ff_left_binary_multiply_right = (s) + m * ff_right_binary_multiply_right) -> (exists ff_left_binary_multiply_result ff_right_binary_multiply_result. (x * y) + m * ff_left_binary_multiply_result = (r * s) + m * ff_right_binary_multiply_result)

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 m
  2. L2
    intro x
  3. L3
    intro r
  4. L4
    intro y
  5. L5
    intro s
  6. L6
    intro hleft
  7. L7
    intro hright
02Use earlier factsL8–15

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

  1. L8
    specialize mod_eq_mul m
  2. L9
    specialize mod_eq_mul x
  3. L10
    specialize mod_eq_mul r
  4. L11
    specialize mod_eq_mul y
  5. L12
    specialize mod_eq_mul s
  6. L13
    apply mod_eq_mul
  7. L14
    exact hleft
  8. L15
    exact hright

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro m
  2. 0002intro x
  3. 0003intro r
  4. 0004intro y
  5. 0005intro s
  6. 0006intro hleft
  7. 0007intro hright
  8. 0008specialize mod_eq_mul m
  9. 0009specialize mod_eq_mul x
  10. 0010specialize mod_eq_mul r
  11. 0011specialize mod_eq_mul y
  12. 0012specialize mod_eq_mul s
  13. 0013apply mod_eq_mul
  14. 0014exact hleft
  15. 0015exact hright