TS001K

shared_beta_collision_remainders_equal

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

Both independently decoded affine remainders equal the same witnessed beta-collision value.

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 b c k k2 t t2 v. (((exists ff_h_ftpr_left_entry. ff_h_ftpr_left_entry + S (t) = S ((S (k)) * c)) /\ exists ff_q_ftpr_left_entry. b = ff_q_ftpr_left_entry * S ((S (k)) * c) + (t))) -> (((exists ff_h_ftpr_left_shared. ff_h_ftpr_left_shared + S (v) = S ((S (k)) * c)) /\ exists ff_q_ftpr_left_shared. b = ff_q_ftpr_left_shared * S ((S (k)) * c) + (v))) -> (((exists ff_h_ftpr_right_entry. ff_h_ftpr_right_entry + S (t2) = S ((S (k2)) * c)) /\ exists ff_q_ftpr_right_entry. b = ff_q_ftpr_right_entry * S ((S (k2)) * c) + (t2))) -> (((exists ff_h_ftpr_right_shared. ff_h_ftpr_right_shared + S (v) = S ((S (k2)) * c)) /\ exists ff_q_ftpr_right_shared. b = ff_q_ftpr_right_shared * S ((S (k2)) * c) + (v))) -> t = t2

Constructive proof overview

Generated structural guide

Both independently decoded affine remainders equal the same witnessed beta-collision value.

The unchanged tactic script uses 1 declared prerequisite and contains 33 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

TS001N prime_floor_affine_grid_collision_represents_prime

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

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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro k
  4. L4
    intro k2
  5. L5
    intro t
  6. L6
    intro t2
  7. L7
    intro v
  8. L8
    intro hfirst
  9. L9
    intro hfirst_shared
  10. L10
    intro hsecond
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hsecond_shared
03Establish hleftL12–20

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

  1. L12
    have hleft : t = v
  2. L13
    specialize beta_at_unique b
  3. L14
    specialize beta_at_unique c
  4. L15
    specialize beta_at_unique k
  5. L16
    specialize beta_at_unique t
  6. L17
    specialize beta_at_unique v
  7. L18
    apply beta_at_unique
  8. L19
    exact hfirst
  9. L20
    exact hfirst_shared
04Establish hrightL21–30

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

  1. L21
    have hright : t2 = v
  2. L22
    specialize beta_at_unique b
  3. L23
    specialize beta_at_unique c
  4. L24
    specialize beta_at_unique k2
  5. L25
    specialize beta_at_unique t2
  6. L26
    specialize beta_at_unique v
  7. L27
    apply beta_at_unique
  8. L28
    exact hsecond
  9. L29
    exact hsecond_shared
  10. L30
    trans v
05Use earlier factsL31–31

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

  1. L31
    exact hleft
06Calculate and transport equalitiesL32–32

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

  1. L32
    symm
07Use earlier factsL33–33

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

  1. L33
    exact hright

Library-wide reading audit

Original exact command ledger · 33 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro k
  4. 0004intro k2
  5. 0005intro t
  6. 0006intro t2
  7. 0007intro v
  8. 0008intro hfirst
  9. 0009intro hfirst_shared
  10. 0010intro hsecond
  11. 0011intro hsecond_shared
  12. 0012have hleft : t = v
  13. 0013specialize beta_at_unique b
  14. 0014specialize beta_at_unique c
  15. 0015specialize beta_at_unique k
  16. 0016specialize beta_at_unique t
  17. 0017specialize beta_at_unique v
  18. 0018apply beta_at_unique
  19. 0019exact hfirst
  20. 0020exact hfirst_shared
  21. 0021have hright : t2 = v
  22. 0022specialize beta_at_unique b
  23. 0023specialize beta_at_unique c
  24. 0024specialize beta_at_unique k2
  25. 0025specialize beta_at_unique t2
  26. 0026specialize beta_at_unique v
  27. 0027apply beta_at_unique
  28. 0028exact hsecond
  29. 0029exact hsecond_shared
  30. 0030trans v
  31. 0031exact hleft
  32. 0032symm
  33. 0033exact hright