BX0008

binary_modular_square_congruence

Squaring preserves witnessed balanced natural congruence.

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. ModEq(m,x,r) → ModEq(m,x · x,r · r)

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

Definition DAG

ModEq(m,a,b) · 2

Actual proof prerequisites

mod_eq_mul · checked external prerequisite
Original expanded first-order statement 
forall m x r. (exists ff_left_binary_square_source ff_right_binary_square_source. (x) + m * ff_left_binary_square_source = (r) + m * ff_right_binary_square_source) -> (exists ff_left_binary_square_result ff_right_binary_square_result. (x * x) + m * ff_left_binary_square_result = (r * r) + m * ff_right_binary_square_result)

Complete unchanged native tactic proof

All 12 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

12 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 assumptionsL1–4

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 hcongruence
02Use earlier factsL5–12

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

  1. L5
    specialize mod_eq_mul m
  2. L6
    specialize mod_eq_mul x
  3. L7
    specialize mod_eq_mul r
  4. L8
    specialize mod_eq_mul x
  5. L9
    specialize mod_eq_mul r
  6. L10
    apply mod_eq_mul
  7. L11
    exact hcongruence
  8. L12
    exact hcongruence

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro m
  2. 0002intro x
  3. 0003intro r
  4. 0004intro hcongruence
  5. 0005specialize mod_eq_mul m
  6. 0006specialize mod_eq_mul x
  7. 0007specialize mod_eq_mul r
  8. 0008specialize mod_eq_mul x
  9. 0009specialize mod_eq_mul r
  10. 0010apply mod_eq_mul
  11. 0011exact hcongruence
  12. 0012exact hcongruence