BT000K · Bertrand theorem

le_total

Stable checked-use theorem · independently kernel verified

Every pair of natural numbers is comparable in the defined order.

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

∀ n. ∀ m. Le(n,m) ∨ Le(m,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

PD0001 Le

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n m. n <= m \/ m <= n

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

BT001U division_remainder_unique BT00VY no_bertrand_central_prime_divisor_le BT00W2 no_bertrand_central_prime_divisor_ranges BT0117 factor_pair_has_small_member_below_square

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

23 script commands · 17 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.

01Induction on nL1–2

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L1
    induction n
  2. L2
    intro m
02Separate the logical casesL3–3

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

  1. L3
    left
03Construct an explicit witnessL4–4

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

  1. L4
    exists m
04Calculate and transport equalitiesL5–5

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

  1. L5
    simp
05Induction on mL6–6

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L6
    induction m
06Separate the logical casesL7–7

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

  1. L7
    right
07Construct an explicit witnessL8–8

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

  1. L8
    exists (S n)
08Calculate and transport equalitiesL9–9

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

  1. L9
    simp
09Use earlier factsL10–10

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

  1. L10
    specialize IH m
10Separate the logical casesL11–13

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

  1. L11
    cases IH
  2. L12
    cases IH_left
  3. L13
    left
11Construct an explicit witnessL14–14

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

  1. L14
    exists x
12Calculate and transport equalitiesL15–16

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

  1. L15
    rewrite PA4
  2. L16
    congr
13Use earlier factsL17–17

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

  1. L17
    exact IH_left_witness
14Separate the logical casesL18–19

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

  1. L18
    cases IH_right
  2. L19
    right
15Construct an explicit witnessL20–20

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

  1. L20
    exists x
16Calculate and transport equalitiesL21–22

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

  1. L21
    rewrite PA4
  2. L22
    congr
17Use earlier factsL23–23

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

  1. L23
    exact IH_right_witness

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001induction n
  2. 0002intro m
  3. 0003left
  4. 0004exists m
  5. 0005simp
  6. 0006induction m
  7. 0007right
  8. 0008exists (S n)
  9. 0009simp
  10. 0010specialize IH m
  11. 0011cases IH
  12. 0012cases IH_left
  13. 0013left
  14. 0014exists x
  15. 0015rewrite PA4
  16. 0016congr
  17. 0017exact IH_left_witness
  18. 0018cases IH_right
  19. 0019right
  20. 0020exists x
  21. 0021rewrite PA4
  22. 0022congr
  23. 0023exact IH_right_witness