BT0018 · Bertrand theorem

le_succ

Stable checked-use theorem · independently kernel verified

A weak inequality remains true after raising its upper bound by one.

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. Le(a,b) → Le(a,S 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

2 occurrences

In local proof propositions

none

0 occurrences

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

Proof neighborhood

Direct theorem prerequisites

BT0001 add_succ_left

Direct theorem dependents

BT003E factor_search_up_to BT0054 base_le_beta_modulus BT005G beta_product_functional BT005J beta_product_succ_decompose BT0083 pow_successor_decompose BT008B beta_sum_trace_functional BT008F beta_sum_succ_decompose BT008J all_bits_prefix_succ BT0092 factorial_succ_decompose BT00DH beta_product_pointwise_coprime BT00JC beta_all_one_bit_count_exact BT00JD eisenstein_initial_segment_bit_count_functional BT00K5 beta_sum_pointwise_add BT00PS bounded_prime_interval_search BT00Q5 bounded_power_valuation_search BT00RA floor_sqrt_total BT00ST legendre_sum_zero_extended_prefix BT00TI choose_succ_succ_of_lt BT00TJ choose_succ_succ BT00UD primorial_succ_decompose BT00UQ beta_product_prefix_suffix_split BT00VB factorial_prime_le_of_divides BT00VD beta_pairwise_coprime_product_divides_common_multiple BT00X9 beta_product_pointwise_le BT00XQ power_quotient_prefix_sum_extend_zero BT00XX double_quotient_carry_prefix_exists BT00Y0 double_quotient_carry_prefix_restrict BT00Y1 bit_count_positive_last_one BT010B floor_sqrt_two_le_of_two_lt BT010U beta_product_all_one_exact

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

9 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 equalitiesL6–6

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

  1. L6
    trans S (x + a)
05Use earlier factsL7–7

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

  1. L7
    apply add_succ_left
06Calculate and transport equalitiesL8–8

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

  1. L8
    congr
07Use earlier factsL9–9

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

  1. L9
    exact h_witness

Library-wide reading audit

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