PA003X · theorem

beta_product_exists

Stable checked-use theorem · independently closed

Every finite decoded beta prefix has an exact relational product and a coded trace.

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. ∃ n. Product(b,c,l,n)

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

PD0014 Product

1 occurrences

In local proof propositions

PD0002 Lt PD0013 BetaAt

6 occurrences

Exact expanded native-PA statement
forall b c l. exists n u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ (((exists h. h + S n = S ((S l) * v)) /\ exists q. u = q * S ((S l) * v) + n) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p)))))

Proof neighborhood

Direct theorem prerequisites

PA003W beta_prefix_product_trace_exists PA0029 beta_at_exists

Direct theorem dependents

PA0046 pow_exists PA0060 factorial_exists PA007J beta_sign_factor_product_power_exists PA007O beta_pointwise_mul_product_exists PA007X beta_product_permutation_invariant PA0085 gauss_lemma_power_congruence_exists PA008J prime_mul_residue_product_balance PA00BA paired_pair_order_product_one_exists PA00BE prime_wilson_terminal_product_package_exists PA00BK scaled_pair_order_terminal_power_mod_predecessor

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 · 10 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 (2)
beta prefix product trace existsbeta at exists
01Fix variables and assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
02Establish htraceL4–8

Establish this local claim before using it. It is not an additional assumption.

  1. L4
    have htrace : ∃ u. ∃ v. BetaAt(u,v,0,1) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. BetaAt(b,c,x,y) ∧ (BetaAt(u,v,x,z) ∧ (BetaAt(u,v,S x,n) ∧ n = z · y)))Definitions: BetaAt(u,v,0,1)Lt(x,l)BetaAt(b,c,x,y)BetaAt(u,v,x,z)BetaAt(u,v,S x,n)Original native command in the exact edition
  2. L5
    specialize beta_prefix_product_trace_exists b
  3. L6
    specialize beta_prefix_product_trace_exists c
  4. L7
    specialize beta_prefix_product_trace_exists l
  5. L8
    exact beta_prefix_product_trace_exists
03Separate the logical casesL9–11

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

  1. L9
    cases htrace
  2. L10
    cases htrace_witness
  3. L11
    cases htrace_witness_witness
04Establish hterminalL12–16

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have hterminal : ∃ n. BetaAt(x,x1,l,n)Definitions: BetaAt(x,x1,l,n)Original native command in the exact edition
  2. L13
    specialize beta_at_exists x
  3. L14
    specialize beta_at_exists x1
  4. L15
    specialize beta_at_exists l
  5. L16
    exact beta_at_exists
05Separate the logical casesL17–17

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

  1. L17
    cases hterminal
06Construct an explicit witnessL18–20

Supply the displayed value, then prove that it has the required property.

  1. L18
    exists x2
  2. L19
    exists x
  3. L20
    exists x1
07Separate the logical casesL21–21

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

  1. L21
    split
08Use earlier factsL22–22

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

  1. L22
    exact htrace_witness_witness_left
09Separate the logical casesL23–23

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

  1. L23
    split
10Use earlier factsL24–25

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

  1. L24
    exact hterminal_witness
  2. L25
    exact htrace_witness_witness_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have htrace : ∃ u. ∃ v. BetaAt(u,v,0,1) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ n. BetaAt(b,c,x,y) ∧ (BetaAt(u,v,x,z) ∧ (BetaAt(u,v,S x,n) ∧ n = z · y)))
    Exact native replay linehave htrace : exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\ exists q. u = q * S ((S 0) * v) + 1) /\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + p) /\ (((exists h. h + S r = S ((S i) * v)) /\ exists q. u = q * S ((S i) * v) + r) /\ (((exists h. h + S s = S ((S (S i)) * v)) /\ exists q. u = q * S ((S (S i)) * v) + s) /\ s = r * p))))
  5. 0005specialize beta_prefix_product_trace_exists b
  6. 0006specialize beta_prefix_product_trace_exists c
  7. 0007specialize beta_prefix_product_trace_exists l
  8. 0008exact beta_prefix_product_trace_exists
  9. 0009cases htrace
  10. 0010cases htrace_witness
  11. 0011cases htrace_witness_witness
  12. 0012have hterminal : ∃ n. BetaAt(x,x1,l,n)
    Exact native replay linehave hterminal : exists n. ((exists h. h + S n = S ((S l) * x1)) /\ exists q. x = q * S ((S l) * x1) + n)
  13. 0013specialize beta_at_exists x
  14. 0014specialize beta_at_exists x1
  15. 0015specialize beta_at_exists l
  16. 0016exact beta_at_exists
  17. 0017cases hterminal
  18. 0018exists x2
  19. 0019exists x
  20. 0020exists x1
  21. 0021split
  22. 0022exact htrace_witness_witness_left
  23. 0023split
  24. 0024exact hterminal_witness
  25. 0025exact htrace_witness_witness_right