PA0030 · theorem

beta_range_exists

Stable checked-use theorem · independently closed

Every start and length admit a beta-coded consecutive range.

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. Range(b,c,a,l)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0018 Range

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 + ff_i_r) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a + ff_i_r))))

Proof neighborhood

Direct theorem prerequisites

PA0006 beta_range_empty PA002Y beta_range_succ_extend

Direct theorem dependents

PA0060 factorial_exists PA00BE prime_wilson_terminal_product_package_exists PA00D7 odd_prime_gauss_eisenstein_orientation_data_exists

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

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 range emptybeta range 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_range_empty 0
  2. L6
    specialize beta_range_empty 0
  3. L7
    specialize beta_range_empty a
  4. L8
    specialize beta_range_empty 0
  5. L9
    apply beta_range_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_range_succ_extend x
  2. L14
    specialize beta_range_succ_extend x1
  3. L15
    specialize beta_range_succ_extend a
  4. L16
    specialize beta_range_succ_extend l
  5. L17
    specialize beta_range_succ_extend (S l)
  6. L18
    apply beta_range_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_range_empty 0
  6. 0006specialize beta_range_empty 0
  7. 0007specialize beta_range_empty a
  8. 0008specialize beta_range_empty 0
  9. 0009apply beta_range_empty
  10. 0010refl
  11. 0011cases IH
  12. 0012cases IH_witness
  13. 0013specialize beta_range_succ_extend x
  14. 0014specialize beta_range_succ_extend x1
  15. 0015specialize beta_range_succ_extend a
  16. 0016specialize beta_range_succ_extend l
  17. 0017specialize beta_range_succ_extend (S l)
  18. 0018apply beta_range_succ_extend
  19. 0019refl
  20. 0020exact IH_witness_witness