GT000C

euclidean_beta_state_functional

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Two actual beta-history states at the same index have identical dividend, divisor, and encoded quotient list.

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 first-order arithmetic statement

forall h e i a b s A B t. (((exists ff_h_cf_egt_left_state. ff_h_cf_egt_left_state + S (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))) = S ((S (i)) * e)) /\ exists ff_q_cf_egt_left_state. h = ff_q_cf_egt_left_state * S ((S (i)) * e) + (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))))) -> (((exists ff_h_cf_egt_right_state. ff_h_cf_egt_right_state + S (((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) * S ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) + ((((B) + (t)) * S ((B) + (t)) + ((t) + (t))) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t))))) = S ((S (i)) * e)) /\ exists ff_q_cf_egt_right_state. h = ff_q_cf_egt_right_state * S ((S (i)) * e) + (((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) * S ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) + ((((B) + (t)) * S ((B) + (t)) + ((t) + (t))) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t))))))) -> a = A /\ (b = B /\ s = t)

Constructive proof overview

Generated structural guide

Two actual beta-history states at the same index have identical dividend, divisor, and encoded quotient list.

The unchanged tactic script uses 2 declared prerequisites and contains 40 exact native proof lines.

Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized pair_code_injective Alpha theorem; checked-use authorized

Direct dependents

GT000D euclidean_trace_prefix_gcd_invariant GT000E euclidean_trace_initial_state_is_gcd

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

40 script commands · 8 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro h
  2. L2
    intro e
  3. L3
    intro i
  4. L4
    intro a
  5. L5
    intro b
  6. L6
    intro s
  7. L7
    intro A
  8. L8
    intro B
  9. L9
    intro t
  10. L10
    intro hleft
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hright
03Establish hpackedL12–20

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

  1. L12
    have hpacked : ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) = ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) * S ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) + ((((B) + (t)) * S ((B) + (t)) + ((t) + (t))) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t))))
  2. L13
    specialize beta_at_unique h
  3. L14
    specialize beta_at_unique e
  4. L15
    specialize beta_at_unique i
  5. L16
    specialize beta_at_unique (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  6. L17
    specialize beta_at_unique (((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) * S ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) + ((((B) + (t)) * S ((B) + (t)) + ((t) + (t))) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))))
  7. L18
    apply beta_at_unique
  8. L19
    exact hleft
  9. L20
    exact hright
04Establish houterL21–29

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

  1. L21
    have houter : a = A /\ ((b) + (s)) * S ((b) + (s)) + ((s) + (s)) = ((B) + (t)) * S ((B) + (t)) + ((t) + (t))
  2. L22
    specialize pair_code_injective (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  3. L23
    specialize pair_code_injective a
  4. L24
    specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  5. L25
    specialize pair_code_injective A
  6. L26
    specialize pair_code_injective (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))
  7. L27
    apply pair_code_injective
  8. L28
    refl
  9. L29
    exact hpacked
05Separate the logical casesL30–31

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

  1. L30
    cases houter
  2. L31
    split
06Use earlier factsL32–38

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

  1. L32
    exact houter_left
  2. L33
    specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  3. L34
    specialize pair_code_injective b
  4. L35
    specialize pair_code_injective s
  5. L36
    specialize pair_code_injective B
  6. L37
    specialize pair_code_injective t
  7. L38
    apply pair_code_injective
07Calculate and transport equalitiesL39–39

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

  1. L39
    refl
08Use earlier factsL40–40

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

  1. L40
    exact houter_right

Library-wide reading audit

Original exact command ledger · 40 lines
  1. 0001intro h
  2. 0002intro e
  3. 0003intro i
  4. 0004intro a
  5. 0005intro b
  6. 0006intro s
  7. 0007intro A
  8. 0008intro B
  9. 0009intro t
  10. 0010intro hleft
  11. 0011intro hright
  12. 0012have hpacked : ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) = ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) * S ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) + ((((B) + (t)) * S ((B) + (t)) + ((t) + (t))) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t))))
  13. 0013specialize beta_at_unique h
  14. 0014specialize beta_at_unique e
  15. 0015specialize beta_at_unique i
  16. 0016specialize beta_at_unique (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  17. 0017specialize beta_at_unique (((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) * S ((A) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))) + ((((B) + (t)) * S ((B) + (t)) + ((t) + (t))) + (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))))
  18. 0018apply beta_at_unique
  19. 0019exact hleft
  20. 0020exact hright
  21. 0021have houter : a = A /\ ((b) + (s)) * S ((b) + (s)) + ((s) + (s)) = ((B) + (t)) * S ((B) + (t)) + ((t) + (t))
  22. 0022specialize pair_code_injective (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  23. 0023specialize pair_code_injective a
  24. 0024specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  25. 0025specialize pair_code_injective A
  26. 0026specialize pair_code_injective (((B) + (t)) * S ((B) + (t)) + ((t) + (t)))
  27. 0027apply pair_code_injective
  28. 0028refl
  29. 0029exact hpacked
  30. 0030cases houter
  31. 0031split
  32. 0032exact houter_left
  33. 0033specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  34. 0034specialize pair_code_injective b
  35. 0035specialize pair_code_injective s
  36. 0036specialize pair_code_injective B
  37. 0037specialize pair_code_injective t
  38. 0038apply pair_code_injective
  39. 0039refl
  40. 0040exact houter_right