BT00UR · Bertrand theorem

primorial_interval_factor_prefix_extend

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

Append one offset selector while preserving the prior interval.

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. ∀ c. ∀ l. (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ (Prime(S (a + x)) ∧ y = S (a + x) ∨ ¬Prime(S (a + x)) ∧ y = 1)) → ∃ x. ∃ y. ∀ z. Lt(z,S l) → ∃ n. BetaAt(x,y,z,n) ∧ (Prime(S (a + z)) ∧ n = S (a + z) ∨ ¬Prime(S (a + 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 a b c l. (forall bpr_index_bpifpe_before. (exists bpr_gap_bpifpe_before_bound. bpr_gap_bpifpe_before_bound + S (bpr_index_bpifpe_before) = l) -> exists bpr_value_bpifpe_before. ((((exists bpr_height_bpifpe_before_decoded. bpr_height_bpifpe_before_decoded + S (bpr_value_bpifpe_before) = S ((S (bpr_index_bpifpe_before)) * c)) /\ exists bpr_quotient_bpifpe_before_decoded. b = bpr_quotient_bpifpe_before_decoded * S ((S (bpr_index_bpifpe_before)) * c) + (bpr_value_bpifpe_before))) /\ (((((~(S (a + bpr_index_bpifpe_before) = 1) /\ forall bpr_left_bpifpe_before_choice_prime bpr_right_bpifpe_before_choice_prime. S (a + bpr_index_bpifpe_before) = bpr_left_bpifpe_before_choice_prime * bpr_right_bpifpe_before_choice_prime -> bpr_left_bpifpe_before_choice_prime = 1 \/ bpr_right_bpifpe_before_choice_prime = 1)) /\ bpr_value_bpifpe_before = S (a + bpr_index_bpifpe_before)) \/ (~((~(S (a + bpr_index_bpifpe_before) = 1) /\ forall bpr_left_bpifpe_before_choice_prime bpr_right_bpifpe_before_choice_prime. S (a + bpr_index_bpifpe_before) = bpr_left_bpifpe_before_choice_prime * bpr_right_bpifpe_before_choice_prime -> bpr_left_bpifpe_before_choice_prime = 1 \/ bpr_right_bpifpe_before_choice_prime = 1)) /\ bpr_value_bpifpe_before = 1))))) -> exists d e. (forall bpr_index_bpifpe_after. (exists bpr_gap_bpifpe_after_bound. bpr_gap_bpifpe_after_bound + S (bpr_index_bpifpe_after) = S l) -> exists bpr_value_bpifpe_after. ((((exists bpr_height_bpifpe_after_decoded. bpr_height_bpifpe_after_decoded + S (bpr_value_bpifpe_after) = S ((S (bpr_index_bpifpe_after)) * e)) /\ exists bpr_quotient_bpifpe_after_decoded. d = bpr_quotient_bpifpe_after_decoded * S ((S (bpr_index_bpifpe_after)) * e) + (bpr_value_bpifpe_after))) /\ (((((~(S (a + bpr_index_bpifpe_after) = 1) /\ forall bpr_left_bpifpe_after_choice_prime bpr_right_bpifpe_after_choice_prime. S (a + bpr_index_bpifpe_after) = bpr_left_bpifpe_after_choice_prime * bpr_right_bpifpe_after_choice_prime -> bpr_left_bpifpe_after_choice_prime = 1 \/ bpr_right_bpifpe_after_choice_prime = 1)) /\ bpr_value_bpifpe_after = S (a + bpr_index_bpifpe_after)) \/ (~((~(S (a + bpr_index_bpifpe_after) = 1) /\ forall bpr_left_bpifpe_after_choice_prime bpr_right_bpifpe_after_choice_prime. S (a + bpr_index_bpifpe_after) = bpr_left_bpifpe_after_choice_prime * bpr_right_bpifpe_after_choice_prime -> bpr_left_bpifpe_after_choice_prime = 1 \/ bpr_right_bpifpe_after_choice_prime = 1)) /\ bpr_value_bpifpe_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

BT00US primorial_interval_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

43 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro hprefix
02Establish hchoiceL6–7

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

  1. L6
    have hchoice : ∃ x. Prime(S (a + l)) ∧ x = S (a + l) ∨ ¬Prime(S (a + l)) ∧ x = 1Definitions: Prime(S (a + l))Original native command in the exact edition
  2. L7
    apply primorial_factor_choice_exists
03Separate the logical casesL8–8

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

  1. L8
    cases hchoice
04Establish hextL9–10

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

  1. L9
    have hext : ∃ d. ∃ e. BetaAt(d,e,l,x) ∧ (∀ y. ∀ z. Lt(y,l) → BetaAt(b,c,y,z) → BetaAt(d,e,y,z))Definitions: BetaAt(d,e,l,x)Lt(y,l)BetaAt(b,c,y,z)BetaAt(d,e,y,z)Original native command in the exact edition
  2. L10
    apply beta_prefix_extend
05Separate the logical casesL11–13

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

  1. L11
    cases hext
  2. L12
    cases hext_witness
  3. L13
    cases hext_witness_witness
06Construct an explicit witnessL14–15

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

  1. L14
    exists x1
  2. L15
    exists x2
07Fix variables and assumptionsL16–17

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

  1. L16
    intro i
  2. L17
    intro hi
08Establish hsplitL18–20

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. L18
    have hsplit : i = l ∨ Lt(i,l)Definitions: Lt(i,l)Original native command in the exact edition
  2. L19
    apply finite_lt_succ_eq_or_lt
  3. L20
    exact hi
09Separate the logical casesL21–21

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

  1. L21
    cases hsplit
10Calculate and transport equalitiesL22–28

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

  1. L22
    rewrite hsplit_left
  2. L23
    rewrite hsplit_left
  3. L24
    rewrite hsplit_left
  4. L25
    rewrite hsplit_left
  5. L26
    rewrite hsplit_left
  6. L27
    rewrite hsplit_left
  7. L28
    rewrite hsplit_left
11Construct an explicit witnessL29–29

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

  1. L29
    exists x
12Separate the logical casesL30–30

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

  1. L30
    split
13Use earlier factsL31–32

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

  1. L31
    exact hext_witness_witness_left
  2. L32
    exact hchoice_witness
14Establish holdL33–35

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

  1. L33
    have hold : ∃ p. BetaAt(b,c,i,p) ∧ (Prime(S (a + i)) ∧ p = S (a + i) ∨ ¬Prime(S (a + i)) ∧ p = 1)Definitions: BetaAt(b,c,i,p)Prime(S (a + i))Original native command in the exact edition
  2. L34
    apply hprefix
  3. L35
    exact hsplit_right
15Separate the logical casesL36–37

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

  1. L36
    cases hold
  2. L37
    cases hold_witness
16Construct an explicit witnessL38–38

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

  1. L38
    exists x3
17Separate the logical casesL39–39

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

  1. L39
    split
18Use earlier factsL40–43

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

  1. L40
    apply hext_witness_witness_right
  2. L41
    exact hsplit_right
  3. L42
    exact hold_witness_left
  4. L43
    exact hold_witness_right

Library-wide reading audit

Original defined command ledger · 43 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hprefix
  6. 0006have hchoice : ∃ x. Prime(S (a + l)) ∧ x = S (a + l) ∨ ¬Prime(S (a + l)) ∧ x = 1
    Exact native replay linehave hchoice : exists x. (((((~(S (a + l) = 1) /\ forall bpr_left_bpifpe_last_choice_prime bpr_right_bpifpe_last_choice_prime. S (a + l) = bpr_left_bpifpe_last_choice_prime * bpr_right_bpifpe_last_choice_prime -> bpr_left_bpifpe_last_choice_prime = 1 \/ bpr_right_bpifpe_last_choice_prime = 1)) /\ x = S (a + l)) \/ (~((~(S (a + l) = 1) /\ forall bpr_left_bpifpe_last_choice_prime bpr_right_bpifpe_last_choice_prime. S (a + l) = bpr_left_bpifpe_last_choice_prime * bpr_right_bpifpe_last_choice_prime -> bpr_left_bpifpe_last_choice_prime = 1 \/ bpr_right_bpifpe_last_choice_prime = 1)) /\ x = 1)))
  7. 0007apply primorial_factor_choice_exists
  8. 0008cases hchoice
  9. 0009have hext : ∃ d. ∃ e. BetaAt(d,e,l,x) ∧ (∀ y. ∀ z. Lt(y,l) → BetaAt(b,c,y,z) → BetaAt(d,e,y,z))
    Exact native replay linehave hext : exists d e. ((((exists bpr_height_bpifpe_append. bpr_height_bpifpe_append + S (x) = S ((S (l)) * e)) /\ exists bpr_quotient_bpifpe_append. d = bpr_quotient_bpifpe_append * S ((S (l)) * e) + (x))) /\ forall i p. (exists bpr_gap_bpifpe_old_bound. bpr_gap_bpifpe_old_bound + S (i) = l) -> (((exists bpr_height_bpifpe_old. bpr_height_bpifpe_old + S (p) = S ((S (i)) * c)) /\ exists bpr_quotient_bpifpe_old. b = bpr_quotient_bpifpe_old * S ((S (i)) * c) + (p))) -> (((exists bpr_height_bpifpe_new. bpr_height_bpifpe_new + S (p) = S ((S (i)) * e)) /\ exists bpr_quotient_bpifpe_new. d = bpr_quotient_bpifpe_new * S ((S (i)) * e) + (p))))
  10. 0010apply beta_prefix_extend
  11. 0011cases hext
  12. 0012cases hext_witness
  13. 0013cases hext_witness_witness
  14. 0014exists x1
  15. 0015exists x2
  16. 0016intro i
  17. 0017intro hi
  18. 0018have hsplit : i = l ∨ Lt(i,l)
    Exact native replay linehave hsplit : i = l \/ exists gap. gap + S i = l
  19. 0019apply finite_lt_succ_eq_or_lt
  20. 0020exact hi
  21. 0021cases hsplit
  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. 0028rewrite hsplit_left
  29. 0029exists x
  30. 0030split
  31. 0031exact hext_witness_witness_left
  32. 0032exact hchoice_witness
  33. 0033have hold : ∃ p. BetaAt(b,c,i,p) ∧ (Prime(S (a + i)) ∧ p = S (a + i) ∨ ¬Prime(S (a + i)) ∧ p = 1)
    Exact native replay linehave hold : exists p. ((((exists bpr_height_bpifpe_hold_decoded. bpr_height_bpifpe_hold_decoded + S (p) = S ((S (i)) * c)) /\ exists bpr_quotient_bpifpe_hold_decoded. b = bpr_quotient_bpifpe_hold_decoded * S ((S (i)) * c) + (p))) /\ (((((~(S (a + i) = 1) /\ forall bpr_left_bpifpe_hold_choice_prime bpr_right_bpifpe_hold_choice_prime. S (a + i) = bpr_left_bpifpe_hold_choice_prime * bpr_right_bpifpe_hold_choice_prime -> bpr_left_bpifpe_hold_choice_prime = 1 \/ bpr_right_bpifpe_hold_choice_prime = 1)) /\ p = S (a + i)) \/ (~((~(S (a + i) = 1) /\ forall bpr_left_bpifpe_hold_choice_prime bpr_right_bpifpe_hold_choice_prime. S (a + i) = bpr_left_bpifpe_hold_choice_prime * bpr_right_bpifpe_hold_choice_prime -> bpr_left_bpifpe_hold_choice_prime = 1 \/ bpr_right_bpifpe_hold_choice_prime = 1)) /\ p = 1))))
  34. 0034apply hprefix
  35. 0035exact hsplit_right
  36. 0036cases hold
  37. 0037cases hold_witness
  38. 0038exists x3
  39. 0039split
  40. 0040apply hext_witness_witness_right
  41. 0041exact hsplit_right
  42. 0042exact hold_witness_left
  43. 0043exact hold_witness_right