BT0041 · Bertrand theorem

beta_at_exists

Stable checked-use theorem · independently kernel verified

Every Gödel-beta position has a bounded decoded residue.

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. ∀ i. ∃ x. BetaAt(b,c,i,x)

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

PD0013 BetaAt

1 occurrences

In local proof propositions

PD0007 DivRem

1 occurrences

Exact expanded native-PA statement
forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

Proof neighborhood

Direct theorem prerequisites

BT003Y beta_modulus_nonzero BT0006 mul_comm BT001P division_remainder_exists

Direct theorem dependents

BT005A beta_exclusive_recode_congruence_step BT005E beta_prefix_product_trace_exists BT005F beta_product_exists BT0089 beta_prefix_sum_trace_exists BT008A beta_sum_exists BT00T4 beta_pascal_row_step_extend BT00T8 choose_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

24 script commands · 13 reading checkpoints · 2 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)
beta modulus nonzeromul commdivision remainder exists
01Fix variables and assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro i
02Establish hm0L4–9

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

  1. L4
    have hm0 : ~(S ((S i) * c) = 0)
  2. L5
    specialize beta_modulus_nonzero c
  3. L6
    specialize beta_modulus_nonzero i
  4. L7
    exact beta_modulus_nonzero
  5. L8
    specialize division_remainder_exists (S ((S i) * c))
  6. L9
    specialize division_remainder_exists b
03Establish hdivL10–12

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

  1. L10
    have hdiv : ∃ q. ∃ r. DivRem(b,S (S i · c),q,r)Definitions: DivRem(b,S (S i · c),q,r)Original native command in the exact edition
  2. L11
    apply division_remainder_exists
  3. L12
    exact hm0
04Separate the logical casesL13–15

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

  1. L13
    cases hdiv
  2. L14
    cases hdiv_witness
  3. L15
    cases hdiv_witness_witness
05Construct an explicit witnessL16–16

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

  1. L16
    exists x1
06Separate the logical casesL17–17

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

  1. L17
    split
07Use earlier factsL18–18

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

  1. L18
    exact hdiv_witness_witness_right
08Construct an explicit witnessL19–19

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

  1. L19
    exists x
09Calculate and transport equalitiesL20–20

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

  1. L20
    trans S ((S i) * c) * x + x1
10Use earlier factsL21–21

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

  1. L21
    exact hdiv_witness_witness_left
11Calculate and transport equalitiesL22–22

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

  1. L22
    congr
12Use earlier factsL23–23

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

  1. L23
    apply mul_comm
13Calculate and transport equalitiesL24–24

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

  1. L24
    refl

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro i
  4. 0004have hm0 : ~(S ((S i) * c) = 0)
  5. 0005specialize beta_modulus_nonzero c
  6. 0006specialize beta_modulus_nonzero i
  7. 0007exact beta_modulus_nonzero
  8. 0008specialize division_remainder_exists (S ((S i) * c))
  9. 0009specialize division_remainder_exists b
  10. 0010have hdiv : ∃ q. ∃ r. DivRem(b,S (S i · c),q,r)
    Exact native replay linehave hdiv : exists q r. b = S ((S i) * c) * q + r /\ exists h. h + S r = S ((S i) * c)
  11. 0011apply division_remainder_exists
  12. 0012exact hm0
  13. 0013cases hdiv
  14. 0014cases hdiv_witness
  15. 0015cases hdiv_witness_witness
  16. 0016exists x1
  17. 0017split
  18. 0018exact hdiv_witness_witness_right
  19. 0019exists x
  20. 0020trans S ((S i) * c) * x + x1
  21. 0021exact hdiv_witness_witness_left
  22. 0022congr
  23. 0023apply mul_comm
  24. 0024refl