BT001C · Bertrand theorem

le_eq_or_lt

Stable checked-use theorem · independently kernel verified

A witnessed inequality is either equality or a witnessed strict 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. Le(a,b) → a = b ∨ Lt(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 + a = b) -> a = b \/ exists k. k + S a = b

Proof neighborhood

Direct theorem prerequisites

BT000Q zero_or_succ BT0000 zero_add BT0001 add_succ_left

Direct theorem dependents

BT002U gcd_exists_up_to BT0034 gcd_balanced_bezout_exists_up_to BT003D factor_property_succ BT003J proper_factor_lt 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 BT00RA floor_sqrt_total BT00SG division_remainder_successor_cases BT00VV primorial_le_four_pow_bounded BT00VY no_bertrand_central_prime_divisor_le BT00W2 no_bertrand_central_prime_divisor_ranges BT00WW bertrand_hj_base_window_thirty_two_from_total BT00XG division_double_quotient_bit BT010D floor_sqrt_three_mul_le_double BT011A prime_le_twenty_two_cases BT0127 bertrand_strict

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

21 script commands · 16 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 (3)
zero or succzero addadd 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
03Use earlier factsL5–5

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

  1. L5
    specialize zero_or_succ x
04Separate the logical casesL6–7

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

  1. L6
    cases zero_or_succ
  2. L7
    left
05Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite zero_or_succ_left at h_witness
06Use earlier factsL9–9

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

  1. L9
    specialize zero_add a
07Calculate and transport equalitiesL10–10

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

  1. L10
    rewrite zero_add at h_witness
08Use earlier factsL11–11

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

  1. L11
    exact h_witness
09Separate the logical casesL12–13

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

  1. L12
    cases zero_or_succ_right
  2. L13
    right
10Construct an explicit witnessL14–14

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

  1. L14
    exists x1
11Calculate and transport equalitiesL15–16

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

  1. L15
    trans S x1 + a
  2. L16
    trans S (x1 + a)
12Use earlier factsL17–17

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

  1. L17
    apply PA4
13Calculate and transport equalitiesL18–18

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

  1. L18
    symm
14Use earlier factsL19–19

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

  1. L19
    apply add_succ_left
15Calculate and transport equalitiesL20–20

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

  1. L20
    rewrite <- zero_or_succ_right_witness
16Use earlier factsL21–21

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

  1. L21
    exact h_witness

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005specialize zero_or_succ x
  6. 0006cases zero_or_succ
  7. 0007left
  8. 0008rewrite zero_or_succ_left at h_witness
  9. 0009specialize zero_add a
  10. 0010rewrite zero_add at h_witness
  11. 0011exact h_witness
  12. 0012cases zero_or_succ_right
  13. 0013right
  14. 0014exists x1
  15. 0015trans S x1 + a
  16. 0016trans S (x1 + a)
  17. 0017apply PA4
  18. 0018symm
  19. 0019apply add_succ_left
  20. 0020rewrite <- zero_or_succ_right_witness
  21. 0021exact h_witness