BT0019 · Bertrand theorem

lt_to_le

Stable checked-use theorem · independently kernel verified

A witnessed strict inequality entails the corresponding weak inequality.

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

∀ a. ∀ b. Lt(a,b) → Le(a,b)

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

PD0001 Le PD0002 Lt

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a b. (exists k. k + S a = b) -> exists r. r + a = b

Proof neighborhood

Direct theorem prerequisites

BT0001 add_succ_left

Direct theorem dependents

BT0059 beta_exclusive_accumulated_product_step BT005A beta_exclusive_recode_congruence_step BT00PR prime_strictly_above_decidable BT00R1 ceil_div_six_total BT00T6 beta_pascal_table_prefix_extend BT00TA beta_pascal_row_step_pointwise_functional BT00TB beta_pascal_table_row_pointwise_functional BT00TF beta_pascal_table_diagonal_boundary BT00TI choose_succ_succ_of_lt BT00X4 bertrand_floor_power_product_le_h_from_total BT00YI central_binom_prime_above_floor_sqrt_valuation_le_one BT00YL no_bertrand_central_nonzero_contribution_factor_ranges BT010V no_bertrand_small_contribution_choice_le_double

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

11 script commands · 7 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 (1)
add succ left
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
02Separate the logical casesL4–4

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

  1. L4
    cases h
03Construct an explicit witnessL5–5

Supply the displayed value, then prove that it has the required property.

  1. L5
    exists S x
04Calculate and transport equalitiesL6–7

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

  1. L6
    trans x + S a
  2. L7
    trans S (x + a)
05Use earlier factsL8–8

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

  1. L8
    apply add_succ_left
06Calculate and transport equalitiesL9–9

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

  1. L9
    symm
07Use earlier factsL10–11

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

  1. L10
    apply PA4
  2. L11
    exact h_witness

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005exists S x
  6. 0006trans x + S a
  7. 0007trans S (x + a)
  8. 0008apply add_succ_left
  9. 0009symm
  10. 0010apply PA4
  11. 0011exact h_witness