PA004L · theorem

finite_bounded_entry_lt

Stable checked-use theorem · independently closed

Every explicitly decoded entry of a bounded prefix satisfies its value bound.

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. BoundedPrefix(b,c,l) → Lt(i,l) → BetaAt(b,c,i,x) → Lt(x,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

PD0002 Lt PD0013 BetaAt PD0024 BoundedPrefix

4 occurrences

In local proof propositions

PD0002 Lt PD0013 BetaAt

2 occurrences

Exact expanded native-PA statement
forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l

Proof neighborhood

Direct theorem prerequisites

PA002F beta_at_unique

Direct theorem dependents

PA004M finite_swap_last_bounded

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

25 script commands · 6 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 (1)
beta at unique
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 hbounded
  7. L7
    intro hi
  8. L8
    intro hentry
02Use earlier factsL9–9

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

  1. L9
    specialize hbounded i
03Establish hdecodedL10–12

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

  1. L10
    have hdecoded : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,l)Definitions: BetaAt(b,c,i,a)Lt(a,l)Original native command in the exact edition
  2. L11
    apply hbounded
  3. L12
    exact hi
04Separate the logical casesL13–14

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

  1. L13
    cases hdecoded
  2. L14
    cases hdecoded_witness
05Establish hxaL15–24

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

  1. L15
    have hxa : x = x1
  2. L16
    specialize beta_at_unique b
  3. L17
    specialize beta_at_unique c
  4. L18
    specialize beta_at_unique i
  5. L19
    specialize beta_at_unique x
  6. L20
    specialize beta_at_unique x1
  7. L21
    apply beta_at_unique
  8. L22
    exact hentry
  9. L23
    exact hdecoded_witness_left
  10. L24
    rewrite hxa
06Use earlier factsL25–25

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

  1. L25
    exact hdecoded_witness_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro x
  6. 0006intro hbounded
  7. 0007intro hi
  8. 0008intro hentry
  9. 0009specialize hbounded i
  10. 0010have hdecoded : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,l)
    Exact native replay linehave hdecoded : exists a. (((exists h. h + S a = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + a) /\ exists h. h + S a = l)
  11. 0011apply hbounded
  12. 0012exact hi
  13. 0013cases hdecoded
  14. 0014cases hdecoded_witness
  15. 0015have hxa : x = x1
  16. 0016specialize beta_at_unique b
  17. 0017specialize beta_at_unique c
  18. 0018specialize beta_at_unique i
  19. 0019specialize beta_at_unique x
  20. 0020specialize beta_at_unique x1
  21. 0021apply beta_at_unique
  22. 0022exact hentry
  23. 0023exact hdecoded_witness_left
  24. 0024rewrite hxa
  25. 0025exact hdecoded_witness_right