CD0010

finite_bit_product_cases

The product of two actual zero-or-one values is again zero or one.

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.

Sets are complete characteristic-bit codes with actual finite cardinality witnesses. The proof constructs translations and the sumset; no finite-choice oracle, supplied cardinality conclusion, or unproved polynomial-method premise is used.

Exact theorem in conservative defined notation

∀ a. ∀ b. a = 0 ∨ a = 1 → b = 0 ∨ b = 1 → a · b = 0 ∨ a · b = 1

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

Definition DAG

none

Actual proof prerequisites

mul_zero_left · checked external prerequisite
Original expanded first-order statement 
forall a b. (a=0 \/ a=1) -> (b=0 \/ b=1) -> a*b=0 \/ a*b=1

Complete tactic proof in conservative notation

All 16 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

16 script commands · 7 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 a
  2. L2
    intro b
  3. L3
    intro ha
  4. L4
    intro hb
02Separate the logical casesL5–6

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

  1. L5
    cases ha
  2. L6
    left
03Calculate and transport equalitiesL7–8

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L7
    rewrite ha_left
  2. L8
    simp [mul_zero_left]
04Separate the logical casesL9–10

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

  1. L9
    cases hb
  2. L10
    left
05Calculate and transport equalitiesL11–12

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L11
    rewrite hb_left
  2. L12
    simp
06Separate the logical casesL13–13

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

  1. L13
    right
07Calculate and transport equalitiesL14–16

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L14
    rewrite ha_right
  2. L15
    rewrite hb_right
  3. L16
    simp

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro ha
  4. 0004intro hb
  5. 0005cases ha
  6. 0006left
  7. 0007rewrite ha_left
  8. 0008simp [mul_zero_left]
  9. 0009cases hb
  10. 0010left
  11. 0011rewrite hb_left
  12. 0012simp
  13. 0013right
  14. 0014rewrite ha_right
  15. 0015rewrite hb_right
  16. 0016simp