TF000E

euclid_three_number_successor_balance

The subtraction-free Euclid number 4d+3 satisfies the exact identity (4d+3)+1=4(d+1).

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

∀ d. 4 · d + 3 + 1 = 4 · S d

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

Definition DAG

none

Actual proof prerequisites

none
Original expanded first-order statement 
forall d. (4 * d + 3) + 1 = 4 * S d

Complete unchanged native tactic proof

All 2 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

2 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 assumptionsL1–1

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

  1. L1
    intro d
02Calculate and transport equalitiesL2–2

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

  1. L2
    simp

Library-wide reading audit

Original defined command ledger · 2 lines
  1. 0001intro d
  2. 0002simp