BT005F

beta_product_exists

Stable checked-use theorem · independently kernel verified

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.

Exact expanded 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)))))

Structural proof guide

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

Direct prerequisites: beta_prefix_product_trace_exists, beta_at_exists. The authored body proceeds by case analysis (4), intermediate claims (2).

Proof neighborhood

Direct dependencies

BT005E beta_prefix_product_trace_exists BT0041 beta_at_exists

Direct dependents

BT0080 pow_exists BT008Y factorial_exists BT00UA primorial_exists BT00UQ beta_product_prefix_suffix_split BT00YS prime_contribution_product_exists

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

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.

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 : 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))))
  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 : exists n. ((exists h. h + S n = S ((S l) * x1)) /\ exists q. x = q * S ((S l) * x1) + n)
  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 exact command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have 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 : 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