BT00QU · Bertrand theorem

two_mul_eq_add_self

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Left multiplication by two is explicit doubling.

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.

Statement with defined notation

forall n. 2 * n = n + n

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n. 2 * n = n + n

Proof neighborhood

Direct theorem prerequisites

BT0006 mul_comm BT000A mul_one

Direct theorem dependents

BT00R4 square_six_shift_identity BT00SW bertrand_h_six_step_transport_from_total BT00SX bertrand_j_six_step_transport_from_total BT00TU central_binom_succ_recurrence BT00VK central_binom_strong_upper_step BT00VL central_binom_recurrence_double_bundle BT00VP central_binom_odd_middle_le_four_pow BT00VR double_half_predecessor_data BT00VS odd_positive_prefix_predecessor_bound BT00VV primorial_le_four_pow_bounded BT00X4 bertrand_floor_power_product_le_h_from_total BT00X8 bertrand_main_inequality_nat BT0126 bertrand_upper_endpoint_factorization

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

7 script commands · 6 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.

Named ingredients (2)
mul commmul one
01Fix variables and assumptionsL1–1

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

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

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

  1. L2
    trans n * 2
03Use earlier factsL3–3

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

  1. L3
    apply mul_comm
04Calculate and transport equalitiesL4–4

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

  1. L4
    rewrite PA6
05Use earlier factsL5–5

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

  1. L5
    specialize mul_one n
06Calculate and transport equalitiesL6–7

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

  1. L6
    rewrite mul_one
  2. L7
    refl

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro n
  2. 0002trans n * 2
  3. 0003apply mul_comm
  4. 0004rewrite PA6
  5. 0005specialize mul_one n
  6. 0006rewrite mul_one
  7. 0007refl