TF0009

three_mod_four_progression_not_two_square

A witnessed three-modulo-four natural cannot have a two-square representation.

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) → ¬(∃ x. ∃ y. n = x · x + y · y)

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

Definition DAG

Mod4Three(n) · 1

Actual proof prerequisites

three_mod_four_number_not_equal_represented · checked external prerequisite
Original expanded first-order statement 
forall n. (exists ff_residue_ptmf_number. (n) = 4 * ff_residue_ptmf_number + 3) -> (exists ptmf_first_represented ptmf_second_represented. (n) = ptmf_first_represented * ptmf_first_represented + ptmf_second_represented * ptmf_second_represented) -> false

Complete unchanged native tactic proof

All 9 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

9 script commands · 3 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 hrepresented
02Use earlier factsL4–8

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

  1. L4
    specialize three_mod_four_number_not_equal_represented n
  2. L5
    specialize three_mod_four_number_not_equal_represented n
  3. L6
    apply three_mod_four_number_not_equal_represented
  4. L7
    exact hthree
  5. L8
    exact hrepresented
03Calculate and transport equalitiesL9–9

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

  1. L9
    refl

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro n
  2. 0002intro hthree
  3. 0003intro hrepresented
  4. 0004specialize three_mod_four_number_not_equal_represented n
  5. 0005specialize three_mod_four_number_not_equal_represented n
  6. 0006apply three_mod_four_number_not_equal_represented
  7. 0007exact hthree
  8. 0008exact hrepresented
  9. 0009refl