BT001I · Bertrand theorem

lt_not_le

Stable checked-use theorem · independently kernel verified

A strict inequality excludes the reverse 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(b,a)

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 k. k + b = a)

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

BT001J le_not_lt BT00JD eisenstein_initial_segment_bit_count_functional BT00PR prime_strictly_above_decidable BT00QH power_valuation_successor_not_divides BT00RC floor_sqrt_monotone BT00SI valuation_threshold_bit_decides_power_divides BT00TC choose_functional BT00TD choose_out_of_range_zero BT00TE choose_zero BT00TI choose_succ_succ_of_lt BT00U4 four_pow_lt_mul_central_binom BT00VC choose_prime_divides_between BT00XG division_double_quotient_bit BT00Y6 central_binom_prime_square_tail_exponent_not_two_le BT00YA prime_square_tail_of_two_three_range BT010B floor_sqrt_two_le_of_two_lt BT010E division_quotient_lower_of_scaled_le BT010V no_bertrand_small_contribution_choice_le_double BT010W no_bertrand_middle_contribution_choice_le_selector BT010X no_bertrand_high_contribution_choice_eq_one BT0115 bertrand_eventually_closed_upper BT0117 factor_pair_has_small_member_below_square BT011A prime_le_twenty_two_cases BT011B nonzero_remainder_not_multiple BT011F prime_five BT011G prime_seven BT011H prime_thirteen BT011I prime_twenty_three BT011J prime_forty_three BT011K prime_eighty_three BT011L prime_one_hundred_sixty_three BT011M prime_three_hundred_seventeen BT0126 bertrand_upper_endpoint_factorization

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

34 script commands · 11 reading checkpoints · 1 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.

01Establish hbadL1–2

Establish this local claim before using it. It is not an additional assumption.

  1. L1
    have hbad : forall z u v. ~(z = u + (v + S z))
  2. L2
    intro z
02Induction on zL3–12

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

  1. L3
    induction z
  2. L4
    intro u
  3. L5
    intro v
  4. L6
    intro hz
  5. L7
    apply PA1
  6. L8
    symm
  7. L9
    rewrite PA4 at hz
  8. L10
    rewrite PA4 at hz
  9. L11
    exact hz
  10. L12
    intro u
03Fix variables and assumptionsL13–14

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

  1. L13
    intro v
  2. L14
    intro hz
04Use earlier factsL15–18

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

  1. L15
    specialize IH u
  2. L16
    specialize IH v
  3. L17
    apply IH
  4. L18
    apply PA2
05Calculate and transport equalitiesL19–20

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

  1. L19
    rewrite PA4 at hz
  2. L20
    rewrite PA4 at hz
06Use earlier factsL21–21

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

  1. L21
    exact hz
07Fix variables and assumptionsL22–25

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

  1. L22
    intro a
  2. L23
    intro b
  3. L24
    intro hab
  4. L25
    intro hba
08Separate the logical casesL26–27

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

  1. L26
    cases hab
  2. L27
    cases hba
09Use earlier factsL28–31

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

  1. L28
    specialize hbad a
  2. L29
    specialize hbad x1
  3. L30
    specialize hbad x
  4. L31
    apply hbad
10Calculate and transport equalitiesL32–33

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

  1. L32
    symm
  2. L33
    rewrite <- hab_witness at hba_witness
11Use earlier factsL34–34

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

  1. L34
    exact hba_witness

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001have hbad : forall z u v. ~(z = u + (v + S z))
  2. 0002intro z
  3. 0003induction z
  4. 0004intro u
  5. 0005intro v
  6. 0006intro hz
  7. 0007apply PA1
  8. 0008symm
  9. 0009rewrite PA4 at hz
  10. 0010rewrite PA4 at hz
  11. 0011exact hz
  12. 0012intro u
  13. 0013intro v
  14. 0014intro hz
  15. 0015specialize IH u
  16. 0016specialize IH v
  17. 0017apply IH
  18. 0018apply PA2
  19. 0019rewrite PA4 at hz
  20. 0020rewrite PA4 at hz
  21. 0021exact hz
  22. 0022intro a
  23. 0023intro b
  24. 0024intro hab
  25. 0025intro hba
  26. 0026cases hab
  27. 0027cases hba
  28. 0028specialize hbad a
  29. 0029specialize hbad x1
  30. 0030specialize hbad x
  31. 0031apply hbad
  32. 0032symm
  33. 0033rewrite <- hab_witness at hba_witness
  34. 0034exact hba_witness