TF0008

three_mod_four_progression_nonunit

Every witnessed natural of the form 4k+3 differs from the unit one.

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 = 1

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 = 1)

Complete unchanged native tactic proof

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

Read the argument

Proof checkpoints

12 script commands · 4 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.

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 hone
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 hone
  2. L6
    rewrite PA4 at hone
04Establish hzeroL7–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA2.

  1. L7
    have hzero : 4 * x + 2 = 0
  2. L8
    apply PA2
  3. L9
    exact hone
  4. L10
    rewrite PA4 at hzero
  5. L11
    apply PA1
  6. L12
    exact hzero

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro n
  2. 0002intro hthree
  3. 0003intro hone
  4. 0004cases hthree
  5. 0005rewrite hthree_witness at hone
  6. 0006rewrite PA4 at hone
  7. 0007have hzero : 4 * x + 2 = 0
  8. 0008apply PA2
  9. 0009exact hone
  10. 0010rewrite PA4 at hzero
  11. 0011apply PA1
  12. 0012exact hzero