PA009D · theorem

scaled_inverse_prefix_extensional

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

A valid scaled inverse at a covered source 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. ∀ a. ∀ n. ∀ b. ∀ c. ∀ l. ∀ i. ∀ y. Prime(p) → ScaledInversePrefix(p,a,n,b,c,l) → Lt(i,l) → ScaledInverseIndex(p,a,n,i,y) → BetaAt(b,c,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 PD0004 Prime PD0013 BetaAt PD0038 ScaledInverseIndex PD0039 ScaledInversePrefix

5 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. ((~(p = 1) /\ forall esi_prime_left_esipe_ext_prime esi_prime_right_esipe_ext_prime. p = esi_prime_left_esipe_ext_prime * esi_prime_right_esipe_ext_prime -> esi_prime_left_esipe_ext_prime = 1 \/ esi_prime_right_esipe_ext_prime = 1)) -> (forall esip_index_extensional_prefix. (exists esip_gap_extensional_prefix_prefix_bound. esip_gap_extensional_prefix_prefix_bound + S (esip_index_extensional_prefix) = l) -> exists esip_mate_extensional_prefix. ((((exists ff_h_esip_extensional_prefix_entry. ff_h_esip_extensional_prefix_entry + S (esip_mate_extensional_prefix) = S ((S (esip_index_extensional_prefix)) * c)) /\ exists ff_q_esip_extensional_prefix_entry. b = ff_q_esip_extensional_prefix_entry * S ((S (esip_index_extensional_prefix)) * c) + (esip_mate_extensional_prefix))) /\ ((exists esip_gap_extensional_prefix_relation_index_bound. esip_gap_extensional_prefix_relation_index_bound + S (esip_index_extensional_prefix) = n) /\ ((((~((S esip_index_extensional_prefix) = 0) /\ (exists esip_gap_extensional_prefix_relation_scaled_left_bound. esip_gap_extensional_prefix_relation_scaled_left_bound + S (S esip_index_extensional_prefix) = p))) /\ (((~(esip_mate_extensional_prefix = 0) /\ (exists esip_gap_extensional_prefix_relation_scaled_right_bound. esip_gap_extensional_prefix_relation_scaled_right_bound + S (esip_mate_extensional_prefix) = p))) /\ (exists esi_mod_left_extensional_prefix_relation_scaled_mod esi_mod_right_extensional_prefix_relation_scaled_mod. ((S esip_index_extensional_prefix) * esip_mate_extensional_prefix) + p * esi_mod_left_extensional_prefix_relation_scaled_mod = (a) + p * esi_mod_right_extensional_prefix_relation_scaled_mod))))))) -> (exists esip_gap_extensional_bound. esip_gap_extensional_bound + S (i) = l) -> ((exists esip_gap_extensional_relation_index_bound. esip_gap_extensional_relation_index_bound + S (i) = n) /\ ((((~((S i) = 0) /\ (exists esip_gap_extensional_relation_scaled_left_bound. esip_gap_extensional_relation_scaled_left_bound + S (S i) = p))) /\ (((~(y = 0) /\ (exists esip_gap_extensional_relation_scaled_right_bound. esip_gap_extensional_relation_scaled_right_bound + S (y) = p))) /\ (exists esi_mod_left_extensional_relation_scaled_mod esi_mod_right_extensional_relation_scaled_mod. ((S i) * y) + p * esi_mod_left_extensional_relation_scaled_mod = (a) + p * esi_mod_right_extensional_relation_scaled_mod))))) -> (((exists ff_h_esipe_extensional_result. ff_h_esipe_extensional_result + S (y) = S ((S (i)) * c)) /\ exists ff_q_esipe_extensional_result. b = ff_q_esipe_extensional_result * S ((S (i)) * c) + (y)))

Proof neighborhood

Direct theorem prerequisites

PA009C prime_scaled_inverse_unique

Direct theorem dependents

PA009E scaled_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

33 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)
prime scaled inverse 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 hp
  10. L10
    intro hprefix
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hi
  2. L12
    intro hrelation
03Establish hstoredL13–16

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

  1. L13
    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. L14
    specialize hprefix i
  3. L15
    apply hprefix
  4. L16
    exact hi
04Separate the logical casesL17–20

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

  1. L17
    cases hstored
  2. L18
    cases hstored_witness
  3. L19
    cases hrelation
  4. L20
    cases hstored_witness_right
05Establish heqL21–30

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

  1. L21
    have heq : y = x
  2. L22
    specialize prime_scaled_inverse_unique p
  3. L23
    specialize prime_scaled_inverse_unique a
  4. L24
    specialize prime_scaled_inverse_unique (S i)
  5. L25
    specialize prime_scaled_inverse_unique y
  6. L26
    specialize prime_scaled_inverse_unique x
  7. L27
    apply prime_scaled_inverse_unique
  8. L28
    exact hp
  9. L29
    exact hrelation_right
  10. L30
    exact hstored_witness_right_right
06Calculate and transport equalitiesL31–32

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

  1. L31
    rewrite heq
  2. L32
    rewrite heq
07Use earlier factsL33–33

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

  1. L33
    exact hstored_witness_left

Library-wide reading audit

Original defined command ledger · 33 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 hp
  10. 0010intro hprefix
  11. 0011intro hi
  12. 0012intro hrelation
  13. 0013have hstored : ∃ z. BetaAt(b,c,i,z) ∧ ScaledInverseIndex(p,a,n,i,z)
    Exact native replay linehave hstored : exists z. ((((exists ff_h_esipe_extensional_stored_at. ff_h_esipe_extensional_stored_at + S (z) = S ((S (i)) * c)) /\ exists ff_q_esipe_extensional_stored_at. b = ff_q_esipe_extensional_stored_at * S ((S (i)) * c) + (z))) /\ ((exists esip_gap_extensional_stored_relation_index_bound. esip_gap_extensional_stored_relation_index_bound + S (i) = n) /\ ((((~((S i) = 0) /\ (exists esip_gap_extensional_stored_relation_scaled_left_bound. esip_gap_extensional_stored_relation_scaled_left_bound + S (S i) = p))) /\ (((~(z = 0) /\ (exists esip_gap_extensional_stored_relation_scaled_right_bound. esip_gap_extensional_stored_relation_scaled_right_bound + S (z) = p))) /\ (exists esi_mod_left_extensional_stored_relation_scaled_mod esi_mod_right_extensional_stored_relation_scaled_mod. ((S i) * z) + p * esi_mod_left_extensional_stored_relation_scaled_mod = (a) + p * esi_mod_right_extensional_stored_relation_scaled_mod))))))
  14. 0014specialize hprefix i
  15. 0015apply hprefix
  16. 0016exact hi
  17. 0017cases hstored
  18. 0018cases hstored_witness
  19. 0019cases hrelation
  20. 0020cases hstored_witness_right
  21. 0021have heq : y = x
  22. 0022specialize prime_scaled_inverse_unique p
  23. 0023specialize prime_scaled_inverse_unique a
  24. 0024specialize prime_scaled_inverse_unique (S i)
  25. 0025specialize prime_scaled_inverse_unique y
  26. 0026specialize prime_scaled_inverse_unique x
  27. 0027apply prime_scaled_inverse_unique
  28. 0028exact hp
  29. 0029exact hrelation_right
  30. 0030exact hstored_witness_right_right
  31. 0031rewrite heq
  32. 0032rewrite heq
  33. 0033exact hstored_witness_left