TF0007

three_mod_four_progression_nonzero

Every witnessed natural of the form 4k+3 is nonzero.

Alpha v34 checked-use · first admitted v23 · 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.

The exact G025 progression-prime milestone is fully proved in unchanged constructive arithmetic; Mod4Three deliberately reuses its existing Quadratic Reciprocity definition PD0012. The much stronger full Dirichlet progression-prime milestone G030 remains open.

Exact theorem in conservative defined notation

∀ n. Mod4Three(n) → ¬n = 0

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

Definition DAG

Mod4Three(n) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall n. (exists ff_residue_ptmf_number. (n) = 4 * ff_residue_ptmf_number + 3) -> ~(n = 0)

Complete unchanged native tactic proof

All 8 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

8 script commands · 4 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 assumptionsL1–3

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

  1. L1
    intro n
  2. L2
    intro hthree
  3. L3
    intro hzero
02Separate the logical casesL4–4

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

  1. L4
    cases hthree
03Calculate and transport equalitiesL5–6

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

  1. L5
    rewrite hthree_witness at hzero
  2. L6
    rewrite PA4 at hzero
04Use earlier factsL7–8

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

  1. L7
    apply PA1
  2. L8
    exact hzero

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro n
  2. 0002intro hthree
  3. 0003intro hzero
  4. 0004cases hthree
  5. 0005rewrite hthree_witness at hzero
  6. 0006rewrite PA4 at hzero
  7. 0007apply PA1
  8. 0008exact hzero