PA008F · theorem

beta_range_one_entry_eq_succ

Alpha v34 checked-use theorem · independently closed; not Stable

A decoded entry of the range 1,...,l is the successor of its index.

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. ∀ l. ∀ i. ∀ x. Range(b,c,1,l) → Lt(i,l) → BetaAt(b,c,i,x) → x = S i

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

PD0002 Lt PD0013 BetaAt PD0018 Range

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c l i x. (forall ff_i_frp_range_entry. (exists ff_lt_frp_range_entry_bound. ff_lt_frp_range_entry_bound + S ff_i_frp_range_entry = l) -> (((exists ff_h_frp_range_entry_decoded. ff_h_frp_range_entry_decoded + S (1 + ff_i_frp_range_entry) = S ((S (ff_i_frp_range_entry)) * c)) /\ exists ff_q_frp_range_entry_decoded. b = ff_q_frp_range_entry_decoded * S ((S (ff_i_frp_range_entry)) * c) + (1 + ff_i_frp_range_entry)))) -> (exists frp_gap_entry_bound. frp_gap_entry_bound + S i = l) -> (((exists ff_h_frp_entry. ff_h_frp_entry + S (x) = S ((S (i)) * c)) /\ exists ff_q_frp_entry. b = ff_q_frp_entry * S ((S (i)) * c) + (x))) -> x = S i

Proof neighborhood

Direct theorem prerequisites

PA0032 beta_range_entry_eq PA000E add_succ_left PA0001 zero_add

Direct theorem dependents

PA008G beta_successor_range_reindex_aligned PA008H beta_successor_range_scale_mod PA008K prime_range_product_coprime

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

30 script commands · 5 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 range entry eqadd succ leftzero add
01Fix variables and assumptionsL1–8

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro i
  5. L5
    intro x
  6. L6
    intro hrange
  7. L7
    intro hi
  8. L8
    intro hx
02Establish hrawL9–18

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

  1. L9
    have hraw : x = 1 + i
  2. L10
    specialize beta_range_entry_eq b
  3. L11
    specialize beta_range_entry_eq c
  4. L12
    specialize beta_range_entry_eq 1
  5. L13
    specialize beta_range_entry_eq l
  6. L14
    specialize beta_range_entry_eq i
  7. L15
    specialize beta_range_entry_eq x
  8. L16
    apply beta_range_entry_eq
  9. L17
    exact hrange
  10. L18
    exact hi
03Use earlier factsL19–19

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

  1. L19
    exact hx
04Establish honeL20–29

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

  1. L20
    have hone : 1 + i = S i
  2. L21
    trans S (0 + i)
  3. L22
    specialize add_succ_left 0
  4. L23
    specialize add_succ_left i
  5. L24
    exact add_succ_left
  6. L25
    congr
  7. L26
    specialize zero_add i
  8. L27
    exact zero_add
  9. L28
    trans 1 + i
  10. L29
    exact hraw
05Use earlier factsL30–30

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

  1. L30
    exact hone

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro x
  6. 0006intro hrange
  7. 0007intro hi
  8. 0008intro hx
  9. 0009have hraw : x = 1 + i
  10. 0010specialize beta_range_entry_eq b
  11. 0011specialize beta_range_entry_eq c
  12. 0012specialize beta_range_entry_eq 1
  13. 0013specialize beta_range_entry_eq l
  14. 0014specialize beta_range_entry_eq i
  15. 0015specialize beta_range_entry_eq x
  16. 0016apply beta_range_entry_eq
  17. 0017exact hrange
  18. 0018exact hi
  19. 0019exact hx
  20. 0020have hone : 1 + i = S i
  21. 0021trans S (0 + i)
  22. 0022specialize add_succ_left 0
  23. 0023specialize add_succ_left i
  24. 0024exact add_succ_left
  25. 0025congr
  26. 0026specialize zero_add i
  27. 0027exact zero_add
  28. 0028trans 1 + i
  29. 0029exact hraw
  30. 0030exact hone