PA00AM · theorem

inverse_prefix_extensional

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

A full inverse relation at a covered index is decoded by the 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

∀ p. ∀ n. ∀ b. ∀ c. ∀ l. ∀ i. ∀ j. p = S n → InversePrefix(p,n,b,c,l) → Lt(i,l) → InverseIndex(p,n,i,j) → BetaAt(b,c,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. p = S n -> (forall wip_index_extensional_prefix. (exists wip_gap_extensional_prefix_prefix_bound. wip_gap_extensional_prefix_prefix_bound + S wip_index_extensional_prefix = l) -> exists wip_mate_extensional_prefix. ((((exists wip_beta_height_extensional_prefix_decoded. wip_beta_height_extensional_prefix_decoded + S (wip_mate_extensional_prefix) = S ((S (wip_index_extensional_prefix)) * c)) /\ exists wip_beta_quotient_extensional_prefix_decoded. b = wip_beta_quotient_extensional_prefix_decoded * S ((S (wip_index_extensional_prefix)) * c) + (wip_mate_extensional_prefix))) /\ ((exists wip_gap_extensional_prefix_inverse_index_bound. wip_gap_extensional_prefix_inverse_index_bound + S wip_index_extensional_prefix = n) /\ ((exists wip_gap_extensional_prefix_inverse_mate_bound. wip_gap_extensional_prefix_inverse_mate_bound + S wip_mate_extensional_prefix = n) /\ (exists wip_mod_left_extensional_prefix_inverse_mod wip_mod_right_extensional_prefix_inverse_mod. ((S wip_index_extensional_prefix) * S wip_mate_extensional_prefix) + p * wip_mod_left_extensional_prefix_inverse_mod = 1 + p * wip_mod_right_extensional_prefix_inverse_mod))))) -> (exists wip_gap_extensional_index_bound. wip_gap_extensional_index_bound + S i = l) -> ((exists wip_gap_extensional_source_index_bound. wip_gap_extensional_source_index_bound + S i = n) /\ ((exists wip_gap_extensional_source_mate_bound. wip_gap_extensional_source_mate_bound + S j = n) /\ (exists wip_mod_left_extensional_source_mod wip_mod_right_extensional_source_mod. ((S i) * S j) + p * wip_mod_left_extensional_source_mod = 1 + p * wip_mod_right_extensional_source_mod))) -> (((exists wip_beta_height_extensional_result. wip_beta_height_extensional_result + S (j) = S ((S (i)) * c)) /\ exists wip_beta_quotient_extensional_result. b = wip_beta_quotient_extensional_result * S ((S (i)) * c) + (j)))

Proof neighborhood

Direct theorem prerequisites

PA00AL bounded_inverse_index_unique

Direct theorem dependents

PA00AN inverse_prefix_involutive PA00AO inverse_prefix_zero_fixed PA00AP inverse_prefix_last_fixed

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 · 7 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)
bounded inverse index 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 hpn
  9. L9
    intro hprefix
  10. L10
    intro hi
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hidx
03Establish hstoredL12–15

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

  1. L12
    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. L13
    specialize hprefix i
  3. L14
    apply hprefix
  4. L15
    exact hi
04Separate the logical casesL16–17

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

  1. L16
    cases hstored
  2. L17
    cases hstored_witness
05Establish heqL18–27

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

  1. L18
    have heq : j = x
  2. L19
    specialize bounded_inverse_index_unique p
  3. L20
    specialize bounded_inverse_index_unique n
  4. L21
    specialize bounded_inverse_index_unique i
  5. L22
    specialize bounded_inverse_index_unique j
  6. L23
    specialize bounded_inverse_index_unique x
  7. L24
    apply bounded_inverse_index_unique
  8. L25
    exact hpn
  9. L26
    exact hidx
  10. L27
    exact hstored_witness_right
06Calculate and transport equalitiesL28–29

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

  1. L28
    rewrite heq
  2. L29
    rewrite heq
07Use earlier factsL30–30

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

  1. L30
    exact hstored_witness_left

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro i
  7. 0007intro j
  8. 0008intro hpn
  9. 0009intro hprefix
  10. 0010intro hi
  11. 0011intro hidx
  12. 0012have hstored : ∃ k. BetaAt(b,c,i,k) ∧ InverseIndex(p,n,i,k)
    Exact native replay linehave hstored : exists k. ((((exists wip_beta_height_extensional_stored_at. wip_beta_height_extensional_stored_at + S (k) = S ((S (i)) * c)) /\ exists wip_beta_quotient_extensional_stored_at. b = wip_beta_quotient_extensional_stored_at * S ((S (i)) * c) + (k))) /\ ((exists wip_gap_extensional_stored_inverse_index_bound. wip_gap_extensional_stored_inverse_index_bound + S i = n) /\ ((exists wip_gap_extensional_stored_inverse_mate_bound. wip_gap_extensional_stored_inverse_mate_bound + S k = n) /\ (exists wip_mod_left_extensional_stored_inverse_mod wip_mod_right_extensional_stored_inverse_mod. ((S i) * S k) + p * wip_mod_left_extensional_stored_inverse_mod = 1 + p * wip_mod_right_extensional_stored_inverse_mod))))
  13. 0013specialize hprefix i
  14. 0014apply hprefix
  15. 0015exact hi
  16. 0016cases hstored
  17. 0017cases hstored_witness
  18. 0018have heq : j = x
  19. 0019specialize bounded_inverse_index_unique p
  20. 0020specialize bounded_inverse_index_unique n
  21. 0021specialize bounded_inverse_index_unique i
  22. 0022specialize bounded_inverse_index_unique j
  23. 0023specialize bounded_inverse_index_unique x
  24. 0024apply bounded_inverse_index_unique
  25. 0025exact hpn
  26. 0026exact hidx
  27. 0027exact hstored_witness_right
  28. 0028rewrite heq
  29. 0029rewrite heq
  30. 0030exact hstored_witness_left