PA00AF · theorem

inverse_prefix_entry_sound

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

Every decoded inverse-prefix entry satisfies its stored inverse relation.

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

∀ p. ∀ n. ∀ b. ∀ c. ∀ l. ∀ i. ∀ j. InversePrefix(p,n,b,c,l) → Lt(i,l) → BetaAt(b,c,i,j) → InverseIndex(p,n,i,j)

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 PD0036 InverseIndex PD0037 InversePrefix

4 occurrences

In local proof propositions

PD0013 BetaAt PD0036 InverseIndex

2 occurrences

Exact expanded native-PA statement
forall p n b c l i j. (forall wip_index_entry_prefix. (exists wip_gap_entry_prefix_prefix_bound. wip_gap_entry_prefix_prefix_bound + S wip_index_entry_prefix = l) -> exists wip_mate_entry_prefix. ((((exists wip_beta_height_entry_prefix_decoded. wip_beta_height_entry_prefix_decoded + S (wip_mate_entry_prefix) = S ((S (wip_index_entry_prefix)) * c)) /\ exists wip_beta_quotient_entry_prefix_decoded. b = wip_beta_quotient_entry_prefix_decoded * S ((S (wip_index_entry_prefix)) * c) + (wip_mate_entry_prefix))) /\ ((exists wip_gap_entry_prefix_inverse_index_bound. wip_gap_entry_prefix_inverse_index_bound + S wip_index_entry_prefix = n) /\ ((exists wip_gap_entry_prefix_inverse_mate_bound. wip_gap_entry_prefix_inverse_mate_bound + S wip_mate_entry_prefix = n) /\ (exists wip_mod_left_entry_prefix_inverse_mod wip_mod_right_entry_prefix_inverse_mod. ((S wip_index_entry_prefix) * S wip_mate_entry_prefix) + p * wip_mod_left_entry_prefix_inverse_mod = 1 + p * wip_mod_right_entry_prefix_inverse_mod))))) -> (exists wip_gap_entry_index_bound. wip_gap_entry_index_bound + S i = l) -> (((exists wip_beta_height_entry_source. wip_beta_height_entry_source + S (j) = S ((S (i)) * c)) /\ exists wip_beta_quotient_entry_source. b = wip_beta_quotient_entry_source * S ((S (i)) * c) + (j))) -> ((exists wip_gap_entry_result_index_bound. wip_gap_entry_result_index_bound + S i = n) /\ ((exists wip_gap_entry_result_mate_bound. wip_gap_entry_result_mate_bound + S j = n) /\ (exists wip_mod_left_entry_result_mod wip_mod_right_entry_result_mod. ((S i) * S j) + p * wip_mod_left_entry_result_mod = 1 + p * wip_mod_right_entry_result_mod)))

Proof neighborhood

Direct theorem prerequisites

PA002F beta_at_unique

Direct theorem dependents

PA00AH prime_inverse_prefix_fixed_cases PA00AN inverse_prefix_involutive

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

28 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 assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro j
  8. L8
    intro hprefix
  9. L9
    intro hi
  10. L10
    intro hat
02Establish hstoredL11–14

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

  1. L11
    have hstored : ∃ k. BetaAt(b,c,i,k) ∧ InverseIndex(p,n,i,k)Definitions: BetaAt(b,c,i,k)InverseIndex(p,n,i,k)Original native command in the exact edition
  2. L12
    specialize hprefix i
  3. L13
    apply hprefix
  4. L14
    exact hi
03Separate the logical casesL15–16

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

  1. L15
    cases hstored
  2. L16
    cases hstored_witness
04Establish heqL17–26

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

  1. L17
    have heq : j = x
  2. L18
    specialize beta_at_unique b
  3. L19
    specialize beta_at_unique c
  4. L20
    specialize beta_at_unique i
  5. L21
    specialize beta_at_unique j
  6. L22
    specialize beta_at_unique x
  7. L23
    apply beta_at_unique
  8. L24
    exact hat
  9. L25
    exact hstored_witness_left
  10. L26
    rewrite heq
05Calculate and transport equalitiesL27–27

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

  1. L27
    rewrite heq
06Use earlier factsL28–28

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

  1. L28
    exact hstored_witness_right

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro i
  7. 0007intro j
  8. 0008intro hprefix
  9. 0009intro hi
  10. 0010intro hat
  11. 0011have hstored : ∃ k. BetaAt(b,c,i,k) ∧ InverseIndex(p,n,i,k)
    Exact native replay linehave hstored : exists k. ((((exists wip_beta_height_entry_stored_at. wip_beta_height_entry_stored_at + S (k) = S ((S (i)) * c)) /\ exists wip_beta_quotient_entry_stored_at. b = wip_beta_quotient_entry_stored_at * S ((S (i)) * c) + (k))) /\ ((exists wip_gap_entry_stored_inverse_index_bound. wip_gap_entry_stored_inverse_index_bound + S i = n) /\ ((exists wip_gap_entry_stored_inverse_mate_bound. wip_gap_entry_stored_inverse_mate_bound + S k = n) /\ (exists wip_mod_left_entry_stored_inverse_mod wip_mod_right_entry_stored_inverse_mod. ((S i) * S k) + p * wip_mod_left_entry_stored_inverse_mod = 1 + p * wip_mod_right_entry_stored_inverse_mod))))
  12. 0012specialize hprefix i
  13. 0013apply hprefix
  14. 0014exact hi
  15. 0015cases hstored
  16. 0016cases hstored_witness
  17. 0017have heq : j = x
  18. 0018specialize beta_at_unique b
  19. 0019specialize beta_at_unique c
  20. 0020specialize beta_at_unique i
  21. 0021specialize beta_at_unique j
  22. 0022specialize beta_at_unique x
  23. 0023apply beta_at_unique
  24. 0024exact hat
  25. 0025exact hstored_witness_left
  26. 0026rewrite heq
  27. 0027rewrite heq
  28. 0028exact hstored_witness_right