PA0099 · theorem

scaled_inverse_prefix_entry_sound

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

Every decoded scaled-inverse prefix entry satisfies its stored 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. ∀ a. ∀ n. ∀ b. ∀ c. ∀ l. ∀ i. ∀ y. ScaledInversePrefix(p,a,n,b,c,l) → Lt(i,l) → BetaAt(b,c,i,y) → ScaledInverseIndex(p,a,n,i,y)

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 PD0038 ScaledInverseIndex PD0039 ScaledInversePrefix

4 occurrences

In local proof propositions

PD0013 BetaAt PD0038 ScaledInverseIndex

2 occurrences

Exact expanded native-PA statement
forall p a n b c l i y. (forall esip_index_ext_entry_prefix. (exists esip_gap_ext_entry_prefix_prefix_bound. esip_gap_ext_entry_prefix_prefix_bound + S (esip_index_ext_entry_prefix) = l) -> exists esip_mate_ext_entry_prefix. ((((exists ff_h_esip_ext_entry_prefix_entry. ff_h_esip_ext_entry_prefix_entry + S (esip_mate_ext_entry_prefix) = S ((S (esip_index_ext_entry_prefix)) * c)) /\ exists ff_q_esip_ext_entry_prefix_entry. b = ff_q_esip_ext_entry_prefix_entry * S ((S (esip_index_ext_entry_prefix)) * c) + (esip_mate_ext_entry_prefix))) /\ ((exists esip_gap_ext_entry_prefix_relation_index_bound. esip_gap_ext_entry_prefix_relation_index_bound + S (esip_index_ext_entry_prefix) = n) /\ ((((~((S esip_index_ext_entry_prefix) = 0) /\ (exists esip_gap_ext_entry_prefix_relation_scaled_left_bound. esip_gap_ext_entry_prefix_relation_scaled_left_bound + S (S esip_index_ext_entry_prefix) = p))) /\ (((~(esip_mate_ext_entry_prefix = 0) /\ (exists esip_gap_ext_entry_prefix_relation_scaled_right_bound. esip_gap_ext_entry_prefix_relation_scaled_right_bound + S (esip_mate_ext_entry_prefix) = p))) /\ (exists esi_mod_left_ext_entry_prefix_relation_scaled_mod esi_mod_right_ext_entry_prefix_relation_scaled_mod. ((S esip_index_ext_entry_prefix) * esip_mate_ext_entry_prefix) + p * esi_mod_left_ext_entry_prefix_relation_scaled_mod = (a) + p * esi_mod_right_ext_entry_prefix_relation_scaled_mod))))))) -> (exists esip_gap_ext_entry_bound. esip_gap_ext_entry_bound + S (i) = l) -> (((exists ff_h_esipe_entry_at. ff_h_esipe_entry_at + S (y) = S ((S (i)) * c)) /\ exists ff_q_esipe_entry_at. b = ff_q_esipe_entry_at * S ((S (i)) * c) + (y))) -> ((exists esip_gap_ext_entry_relation_index_bound. esip_gap_ext_entry_relation_index_bound + S (i) = n) /\ ((((~((S i) = 0) /\ (exists esip_gap_ext_entry_relation_scaled_left_bound. esip_gap_ext_entry_relation_scaled_left_bound + S (S i) = p))) /\ (((~(y = 0) /\ (exists esip_gap_ext_entry_relation_scaled_right_bound. esip_gap_ext_entry_relation_scaled_right_bound + S (y) = p))) /\ (exists esi_mod_left_ext_entry_relation_scaled_mod esi_mod_right_ext_entry_relation_scaled_mod. ((S i) * y) + p * esi_mod_left_ext_entry_relation_scaled_mod = (a) + p * esi_mod_right_ext_entry_relation_scaled_mod)))))

Proof neighborhood

Direct theorem prerequisites

PA002F beta_at_unique

Direct theorem dependents

PA009A scaled_inverse_prefix_mate_predecessor PA009E scaled_inverse_prefix_involutive PA009H scaled_inverse_prefix_no_fixed_of_not_qres

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)
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 a
  3. L3
    intro n
  4. L4
    intro b
  5. L5
    intro c
  6. L6
    intro l
  7. L7
    intro i
  8. L8
    intro y
  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 hat
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 : ∃ z. BetaAt(b,c,i,z) ∧ ScaledInverseIndex(p,a,n,i,z)Definitions: BetaAt(b,c,i,z)ScaledInverseIndex(p,a,n,i,z)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 beta at unique.

  1. L18
    have heq : y = x
  2. L19
    specialize beta_at_unique b
  3. L20
    specialize beta_at_unique c
  4. L21
    specialize beta_at_unique i
  5. L22
    specialize beta_at_unique y
  6. L23
    specialize beta_at_unique x
  7. L24
    apply beta_at_unique
  8. L25
    exact hat
  9. L26
    exact hstored_witness_left
  10. L27
    rewrite heq
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_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro n
  4. 0004intro b
  5. 0005intro c
  6. 0006intro l
  7. 0007intro i
  8. 0008intro y
  9. 0009intro hprefix
  10. 0010intro hi
  11. 0011intro hat
  12. 0012have hstored : ∃ z. BetaAt(b,c,i,z) ∧ ScaledInverseIndex(p,a,n,i,z)
    Exact native replay linehave hstored : exists z. ((((exists ff_h_esipe_stored_at. ff_h_esipe_stored_at + S (z) = S ((S (i)) * c)) /\ exists ff_q_esipe_stored_at. b = ff_q_esipe_stored_at * S ((S (i)) * c) + (z))) /\ ((exists esip_gap_ext_stored_relation_index_bound. esip_gap_ext_stored_relation_index_bound + S (i) = n) /\ ((((~((S i) = 0) /\ (exists esip_gap_ext_stored_relation_scaled_left_bound. esip_gap_ext_stored_relation_scaled_left_bound + S (S i) = p))) /\ (((~(z = 0) /\ (exists esip_gap_ext_stored_relation_scaled_right_bound. esip_gap_ext_stored_relation_scaled_right_bound + S (z) = p))) /\ (exists esi_mod_left_ext_stored_relation_scaled_mod esi_mod_right_ext_stored_relation_scaled_mod. ((S i) * z) + p * esi_mod_left_ext_stored_relation_scaled_mod = (a) + p * esi_mod_right_ext_stored_relation_scaled_mod))))))
  13. 0013specialize hprefix i
  14. 0014apply hprefix
  15. 0015exact hi
  16. 0016cases hstored
  17. 0017cases hstored_witness
  18. 0018have heq : y = x
  19. 0019specialize beta_at_unique b
  20. 0020specialize beta_at_unique c
  21. 0021specialize beta_at_unique i
  22. 0022specialize beta_at_unique y
  23. 0023specialize beta_at_unique x
  24. 0024apply beta_at_unique
  25. 0025exact hat
  26. 0026exact hstored_witness_left
  27. 0027rewrite heq
  28. 0028rewrite heq
  29. 0029rewrite heq
  30. 0030exact hstored_witness_right