BT0017 · Bertrand theorem

le_of_succ_le_succ

Stable checked-use theorem · independently kernel verified

Successor order reflects to the underlying naturals.

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,S 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 = S b) -> exists r. r + a = b

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

BT002U gcd_exists_up_to BT0034 gcd_balanced_bezout_exists_up_to BT003D factor_property_succ BT003K prime_divisor_exists_up_to BT0059 beta_exclusive_accumulated_product_step BT005A beta_exclusive_recode_congruence_step BT005E beta_prefix_product_trace_exists BT005K beta_product_succ_append BT0069 beta_factor_divides_product BT007V beta_repeat_succ_extend BT0085 beta_range_succ_extend BT0089 beta_prefix_sum_trace_exists BT0097 lt_three_cases BT00AA finite_lt_succ_eq_or_lt BT00JD eisenstein_initial_segment_bit_count_functional BT00PS bounded_prime_interval_search BT00Q5 bounded_power_valuation_search BT00TF beta_pascal_table_diagonal_boundary BT00TM choose_positive BT00VV primorial_le_four_pow_bounded BT00WW bertrand_hj_base_window_thirty_two_from_total BT00X1 six_block_window_decomposition_above_thirty_two BT011A prime_le_twenty_two_cases

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

10 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.

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 x
04Use earlier factsL6–6

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

  1. L6
    apply PA2
05Calculate and transport equalitiesL7–8

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

  1. L7
    trans x + S a
  2. L8
    symm
06Use earlier factsL9–10

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

  1. L9
    apply PA4
  2. L10
    exact h_witness

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005exists x
  6. 0006apply PA2
  7. 0007trans x + S a
  8. 0008symm
  9. 0009apply PA4
  10. 0010exact h_witness