BT007W · Bertrand theorem

beta_repeat_exists

Stable checked-use theorem · independently kernel verified

Every value and length admit a beta-coded constant 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

∀ a. ∀ l. ∃ b. ∃ c. Repeat(b,c,a,l)

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

PD0019 Repeat

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a l. exists b c. (forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> (((exists ff_h_r_decoded. ff_h_r_decoded + S (a) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a))))

Proof neighborhood

Direct theorem prerequisites

BT007U beta_repeat_empty BT007V beta_repeat_succ_extend

Direct theorem dependents

BT0080 pow_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

20 script commands · 9 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 (2)
beta repeat emptybeta repeat succ extend
01Fix variables and assumptionsL1–1

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

  1. L1
    intro a
02Induction on lL2–2

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

  1. L2
    induction l
03Construct an explicit witnessL3–4

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

  1. L3
    exists 0
  2. L4
    exists 0
04Use earlier factsL5–9

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

  1. L5
    specialize beta_repeat_empty 0
  2. L6
    specialize beta_repeat_empty 0
  3. L7
    specialize beta_repeat_empty a
  4. L8
    specialize beta_repeat_empty 0
  5. L9
    apply beta_repeat_empty
05Calculate and transport equalitiesL10–10

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

  1. L10
    refl
06Separate the logical casesL11–12

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

  1. L11
    cases IH
  2. L12
    cases IH_witness
07Use earlier factsL13–18

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

  1. L13
    specialize beta_repeat_succ_extend x
  2. L14
    specialize beta_repeat_succ_extend x1
  3. L15
    specialize beta_repeat_succ_extend a
  4. L16
    specialize beta_repeat_succ_extend l
  5. L17
    specialize beta_repeat_succ_extend (S l)
  6. L18
    apply beta_repeat_succ_extend
08Calculate and transport equalitiesL19–19

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

  1. L19
    refl
09Use earlier factsL20–20

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

  1. L20
    exact IH_witness_witness

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro a
  2. 0002induction l
  3. 0003exists 0
  4. 0004exists 0
  5. 0005specialize beta_repeat_empty 0
  6. 0006specialize beta_repeat_empty 0
  7. 0007specialize beta_repeat_empty a
  8. 0008specialize beta_repeat_empty 0
  9. 0009apply beta_repeat_empty
  10. 0010refl
  11. 0011cases IH
  12. 0012cases IH_witness
  13. 0013specialize beta_repeat_succ_extend x
  14. 0014specialize beta_repeat_succ_extend x1
  15. 0015specialize beta_repeat_succ_extend a
  16. 0016specialize beta_repeat_succ_extend l
  17. 0017specialize beta_repeat_succ_extend (S l)
  18. 0018apply beta_repeat_succ_extend
  19. 0019refl
  20. 0020exact IH_witness_witness