TS001K · theorem body

shared_beta_collision_remainders_equal

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ b. ∀ c. ∀ k. ∀ k2. ∀ t. ∀ t2. ∀ v. BetaAt(b,c,k,t) → BetaAt(b,c,k,v) → BetaAt(b,c,k2,t2) → BetaAt(b,c,k2,v) → t = t2

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0013 BetaAt

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

beta_at_unique · Stable closed

Direct theorem dependents

TS001N prime_floor_affine_grid_collision_represents_prime

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay 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.

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 defined 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