BT00U7 · Bertrand theorem

primorial_factor_prefix_extend

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Append one selector factor while preserving the previous prefix.

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

∀ b. ∀ c. ∀ m. (∀ x. Lt(x,m) → ∃ y. BetaAt(b,c,x,y) ∧ (Prime(S x) ∧ y = S x ∨ ¬Prime(S x) ∧ y = 1)) → ∃ x. ∃ y. ∀ z. Lt(z,S m) → ∃ n. BetaAt(x,y,z,n) ∧ (Prime(S z) ∧ n = S z ∨ ¬Prime(S z) ∧ n = 1)

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

PD0002 Lt PD0004 Prime PD0013 BetaAt

8 occurrences

In local proof propositions

PD0002 Lt PD0004 Prime PD0013 BetaAt

10 occurrences

Exact expanded native-PA statement
forall b c m. (forall bpr_index_bpfpe_before. (exists bpr_gap_bpfpe_before_bound. bpr_gap_bpfpe_before_bound + S (bpr_index_bpfpe_before) = m) -> exists bpr_value_bpfpe_before. ((((exists bpr_height_bpfpe_before_decoded. bpr_height_bpfpe_before_decoded + S (bpr_value_bpfpe_before) = S ((S (bpr_index_bpfpe_before)) * c)) /\ exists bpr_quotient_bpfpe_before_decoded. b = bpr_quotient_bpfpe_before_decoded * S ((S (bpr_index_bpfpe_before)) * c) + (bpr_value_bpfpe_before))) /\ (((((~(S (bpr_index_bpfpe_before) = 1) /\ forall bpr_left_bpfpe_before_choice_prime bpr_right_bpfpe_before_choice_prime. S (bpr_index_bpfpe_before) = bpr_left_bpfpe_before_choice_prime * bpr_right_bpfpe_before_choice_prime -> bpr_left_bpfpe_before_choice_prime = 1 \/ bpr_right_bpfpe_before_choice_prime = 1)) /\ bpr_value_bpfpe_before = S (bpr_index_bpfpe_before)) \/ (~((~(S (bpr_index_bpfpe_before) = 1) /\ forall bpr_left_bpfpe_before_choice_prime bpr_right_bpfpe_before_choice_prime. S (bpr_index_bpfpe_before) = bpr_left_bpfpe_before_choice_prime * bpr_right_bpfpe_before_choice_prime -> bpr_left_bpfpe_before_choice_prime = 1 \/ bpr_right_bpfpe_before_choice_prime = 1)) /\ bpr_value_bpfpe_before = 1))))) -> exists d e. (forall bpr_index_bpfpe_after. (exists bpr_gap_bpfpe_after_bound. bpr_gap_bpfpe_after_bound + S (bpr_index_bpfpe_after) = S m) -> exists bpr_value_bpfpe_after. ((((exists bpr_height_bpfpe_after_decoded. bpr_height_bpfpe_after_decoded + S (bpr_value_bpfpe_after) = S ((S (bpr_index_bpfpe_after)) * e)) /\ exists bpr_quotient_bpfpe_after_decoded. d = bpr_quotient_bpfpe_after_decoded * S ((S (bpr_index_bpfpe_after)) * e) + (bpr_value_bpfpe_after))) /\ (((((~(S (bpr_index_bpfpe_after) = 1) /\ forall bpr_left_bpfpe_after_choice_prime bpr_right_bpfpe_after_choice_prime. S (bpr_index_bpfpe_after) = bpr_left_bpfpe_after_choice_prime * bpr_right_bpfpe_after_choice_prime -> bpr_left_bpfpe_after_choice_prime = 1 \/ bpr_right_bpfpe_after_choice_prime = 1)) /\ bpr_value_bpfpe_after = S (bpr_index_bpfpe_after)) \/ (~((~(S (bpr_index_bpfpe_after) = 1) /\ forall bpr_left_bpfpe_after_choice_prime bpr_right_bpfpe_after_choice_prime. S (bpr_index_bpfpe_after) = bpr_left_bpfpe_after_choice_prime * bpr_right_bpfpe_after_choice_prime -> bpr_left_bpfpe_after_choice_prime = 1 \/ bpr_right_bpfpe_after_choice_prime = 1)) /\ bpr_value_bpfpe_after = 1)))))

Proof neighborhood

Direct theorem prerequisites

BT00U5 primorial_factor_choice_exists BT005D beta_prefix_extend BT00AA finite_lt_succ_eq_or_lt

Direct theorem dependents

BT00U8 primorial_factor_prefix_exists

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

42 script commands · 18 reading checkpoints · 4 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)
primorial factor choice existsbeta prefix extendfinite lt succ eq or lt
01Fix variables and assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro m
  4. L4
    intro hprefix
02Establish hchoiceL5–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply primorial factor choice exists.

  1. L5
    have hchoice : ∃ x. Prime(S m) ∧ x = S m ∨ ¬Prime(S m) ∧ x = 1Definitions: Prime(S m)Original native command in the exact edition
  2. L6
    apply primorial_factor_choice_exists
03Separate the logical casesL7–7

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

  1. L7
    cases hchoice
04Establish hextL8–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta prefix extend.

  1. L8
    have hext : ∃ d. ∃ e. BetaAt(d,e,m,x) ∧ (∀ y. ∀ z. Lt(y,m) → BetaAt(b,c,y,z) → BetaAt(d,e,y,z))Definitions: BetaAt(d,e,m,x)Lt(y,m)BetaAt(b,c,y,z)BetaAt(d,e,y,z)Original native command in the exact edition
  2. L9
    apply beta_prefix_extend
05Separate the logical casesL10–12

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

  1. L10
    cases hext
  2. L11
    cases hext_witness
  3. L12
    cases hext_witness_witness
06Construct an explicit witnessL13–14

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

  1. L13
    exists x1
  2. L14
    exists x2
07Fix variables and assumptionsL15–16

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

  1. L15
    intro i
  2. L16
    intro hi
08Establish hsplitL17–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite lt succ eq or lt.

  1. L17
    have hsplit : i = m ∨ Lt(i,m)Definitions: Lt(i,m)Original native command in the exact edition
  2. L18
    apply finite_lt_succ_eq_or_lt
  3. L19
    exact hi
09Separate the logical casesL20–20

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

  1. L20
    cases hsplit
10Calculate and transport equalitiesL21–27

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

  1. L21
    rewrite hsplit_left
  2. L22
    rewrite hsplit_left
  3. L23
    rewrite hsplit_left
  4. L24
    rewrite hsplit_left
  5. L25
    rewrite hsplit_left
  6. L26
    rewrite hsplit_left
  7. L27
    rewrite hsplit_left
11Construct an explicit witnessL28–28

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

  1. L28
    exists x
12Separate the logical casesL29–29

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

  1. L29
    split
13Use earlier factsL30–31

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

  1. L30
    exact hext_witness_witness_left
  2. L31
    exact hchoice_witness
14Establish holdL32–34

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprefix.

  1. L32
    have hold : ∃ a. BetaAt(b,c,i,a) ∧ (Prime(S i) ∧ a = S i ∨ ¬Prime(S i) ∧ a = 1)Definitions: BetaAt(b,c,i,a)Prime(S i)Original native command in the exact edition
  2. L33
    apply hprefix
  3. L34
    exact hsplit_right
15Separate the logical casesL35–36

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

  1. L35
    cases hold
  2. L36
    cases hold_witness
16Construct an explicit witnessL37–37

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

  1. L37
    exists x3
17Separate the logical casesL38–38

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

  1. L38
    split
18Use earlier factsL39–42

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

  1. L39
    apply hext_witness_witness_right
  2. L40
    exact hsplit_right
  3. L41
    exact hold_witness_left
  4. L42
    exact hold_witness_right

Library-wide reading audit

Original defined command ledger · 42 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro m
  4. 0004intro hprefix
  5. 0005have hchoice : ∃ x. Prime(S m) ∧ x = S m ∨ ¬Prime(S m) ∧ x = 1
    Exact native replay linehave hchoice : exists x. (((((~(S (m) = 1) /\ forall bpr_left_bpfpe_last_choice_prime bpr_right_bpfpe_last_choice_prime. S (m) = bpr_left_bpfpe_last_choice_prime * bpr_right_bpfpe_last_choice_prime -> bpr_left_bpfpe_last_choice_prime = 1 \/ bpr_right_bpfpe_last_choice_prime = 1)) /\ x = S (m)) \/ (~((~(S (m) = 1) /\ forall bpr_left_bpfpe_last_choice_prime bpr_right_bpfpe_last_choice_prime. S (m) = bpr_left_bpfpe_last_choice_prime * bpr_right_bpfpe_last_choice_prime -> bpr_left_bpfpe_last_choice_prime = 1 \/ bpr_right_bpfpe_last_choice_prime = 1)) /\ x = 1)))
  6. 0006apply primorial_factor_choice_exists
  7. 0007cases hchoice
  8. 0008have hext : ∃ d. ∃ e. BetaAt(d,e,m,x) ∧ (∀ y. ∀ z. Lt(y,m) → BetaAt(b,c,y,z) → BetaAt(d,e,y,z))
    Exact native replay linehave hext : exists d e. ((((exists bpr_height_bpfpe_append. bpr_height_bpfpe_append + S (x) = S ((S (m)) * e)) /\ exists bpr_quotient_bpfpe_append. d = bpr_quotient_bpfpe_append * S ((S (m)) * e) + (x))) /\ forall i a. (exists bpr_gap_bpfpe_old_bound. bpr_gap_bpfpe_old_bound + S (i) = m) -> (((exists bpr_height_bpfpe_old. bpr_height_bpfpe_old + S (a) = S ((S (i)) * c)) /\ exists bpr_quotient_bpfpe_old. b = bpr_quotient_bpfpe_old * S ((S (i)) * c) + (a))) -> (((exists bpr_height_bpfpe_new. bpr_height_bpfpe_new + S (a) = S ((S (i)) * e)) /\ exists bpr_quotient_bpfpe_new. d = bpr_quotient_bpfpe_new * S ((S (i)) * e) + (a))))
  9. 0009apply beta_prefix_extend
  10. 0010cases hext
  11. 0011cases hext_witness
  12. 0012cases hext_witness_witness
  13. 0013exists x1
  14. 0014exists x2
  15. 0015intro i
  16. 0016intro hi
  17. 0017have hsplit : i = m ∨ Lt(i,m)
    Exact native replay linehave hsplit : i = m \/ exists gap. gap + S i = m
  18. 0018apply finite_lt_succ_eq_or_lt
  19. 0019exact hi
  20. 0020cases hsplit
  21. 0021rewrite hsplit_left
  22. 0022rewrite hsplit_left
  23. 0023rewrite hsplit_left
  24. 0024rewrite hsplit_left
  25. 0025rewrite hsplit_left
  26. 0026rewrite hsplit_left
  27. 0027rewrite hsplit_left
  28. 0028exists x
  29. 0029split
  30. 0030exact hext_witness_witness_left
  31. 0031exact hchoice_witness
  32. 0032have hold : ∃ a. BetaAt(b,c,i,a) ∧ (Prime(S i) ∧ a = S i ∨ ¬Prime(S i) ∧ a = 1)
    Exact native replay linehave hold : exists a. ((((exists bpr_height_bpfpe_hold_decoded. bpr_height_bpfpe_hold_decoded + S (a) = S ((S (i)) * c)) /\ exists bpr_quotient_bpfpe_hold_decoded. b = bpr_quotient_bpfpe_hold_decoded * S ((S (i)) * c) + (a))) /\ (((((~(S (i) = 1) /\ forall bpr_left_bpfpe_hold_choice_prime bpr_right_bpfpe_hold_choice_prime. S (i) = bpr_left_bpfpe_hold_choice_prime * bpr_right_bpfpe_hold_choice_prime -> bpr_left_bpfpe_hold_choice_prime = 1 \/ bpr_right_bpfpe_hold_choice_prime = 1)) /\ a = S (i)) \/ (~((~(S (i) = 1) /\ forall bpr_left_bpfpe_hold_choice_prime bpr_right_bpfpe_hold_choice_prime. S (i) = bpr_left_bpfpe_hold_choice_prime * bpr_right_bpfpe_hold_choice_prime -> bpr_left_bpfpe_hold_choice_prime = 1 \/ bpr_right_bpfpe_hold_choice_prime = 1)) /\ a = 1))))
  33. 0033apply hprefix
  34. 0034exact hsplit_right
  35. 0035cases hold
  36. 0036cases hold_witness
  37. 0037exists x3
  38. 0038split
  39. 0039apply hext_witness_witness_right
  40. 0040exact hsplit_right
  41. 0041exact hold_witness_left
  42. 0042exact hold_witness_right