TS000O · theorem body

finite_prefix_collision_succ

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

A witnessed collision in a finite prefix remains a collision after adjoining one more decoded entry.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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. ∀ n. (∃ x. ∃ y. ∃ z. Lt(x,n) ∧ (Lt(y,n) ∧ (¬x = y ∧ (BetaAt(b,c,x,z) ∧ BetaAt(b,c,y,z))))) → ∃ x. ∃ y. ∃ z. Lt(x,S n) ∧ (Lt(y,S n) ∧ (¬x = y ∧ (BetaAt(b,c,x,z) ∧ BetaAt(b,c,y,z))))

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

PD0002 Lt PD0013 BetaAt

In local proof propositions

none
Exact expanded first-order statement
forall b c n. (exists ftsp_first_fpcd_old ftsp_second_fpcd_old ftsp_value_fpcd_old. ((exists ftsp_gap_fpcd_old_first. ftsp_gap_fpcd_old_first + S (ftsp_first_fpcd_old) = n) /\ ((exists ftsp_gap_fpcd_old_second. ftsp_gap_fpcd_old_second + S (ftsp_second_fpcd_old) = n) /\ (~(ftsp_first_fpcd_old = ftsp_second_fpcd_old) /\ ((((exists ff_h_ftsp_fpcd_old_left. ff_h_ftsp_fpcd_old_left + S (ftsp_value_fpcd_old) = S ((S (ftsp_first_fpcd_old)) * c)) /\ exists ff_q_ftsp_fpcd_old_left. b = ff_q_ftsp_fpcd_old_left * S ((S (ftsp_first_fpcd_old)) * c) + (ftsp_value_fpcd_old))) /\ (((exists ff_h_ftsp_fpcd_old_right. ff_h_ftsp_fpcd_old_right + S (ftsp_value_fpcd_old) = S ((S (ftsp_second_fpcd_old)) * c)) /\ exists ff_q_ftsp_fpcd_old_right. b = ff_q_ftsp_fpcd_old_right * S ((S (ftsp_second_fpcd_old)) * c) + (ftsp_value_fpcd_old)))))))) -> (exists ftsp_first_fpcd_next ftsp_second_fpcd_next ftsp_value_fpcd_next. ((exists ftsp_gap_fpcd_next_first. ftsp_gap_fpcd_next_first + S (ftsp_first_fpcd_next) = S n) /\ ((exists ftsp_gap_fpcd_next_second. ftsp_gap_fpcd_next_second + S (ftsp_second_fpcd_next) = S n) /\ (~(ftsp_first_fpcd_next = ftsp_second_fpcd_next) /\ ((((exists ff_h_ftsp_fpcd_next_left. ff_h_ftsp_fpcd_next_left + S (ftsp_value_fpcd_next) = S ((S (ftsp_first_fpcd_next)) * c)) /\ exists ff_q_ftsp_fpcd_next_left. b = ff_q_ftsp_fpcd_next_left * S ((S (ftsp_first_fpcd_next)) * c) + (ftsp_value_fpcd_next))) /\ (((exists ff_h_ftsp_fpcd_next_right. ff_h_ftsp_fpcd_next_right + S (ftsp_value_fpcd_next) = S ((S (ftsp_second_fpcd_next)) * c)) /\ exists ff_q_ftsp_fpcd_next_right. b = ff_q_ftsp_fpcd_next_right * S ((S (ftsp_second_fpcd_next)) * c) + (ftsp_value_fpcd_next))))))))

Proof neighborhood

Direct theorem prerequisites

le_succ · Stable closed

Direct theorem dependents

TS000R finite_prefix_collision_or_injective

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

31 script commands · 12 reading checkpoints · 0 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 assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro n
  4. L4
    intro hcollision
02Separate the logical casesL5–11

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

  1. L5
    cases hcollision
  2. L6
    cases hcollision_witness
  3. L7
    cases hcollision_witness_witness
  4. L8
    cases hcollision_witness_witness_witness
  5. L9
    cases hcollision_witness_witness_witness_right
  6. L10
    cases hcollision_witness_witness_witness_right_right
  7. L11
    cases hcollision_witness_witness_witness_right_right_right
03Construct an explicit witnessL12–14

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

  1. L12
    exists x
  2. L13
    exists x1
  3. L14
    exists x2
04Separate the logical casesL15–15

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

  1. L15
    split
05Use earlier factsL16–19

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

  1. L16
    specialize le_succ (S x)
  2. L17
    specialize le_succ n
  3. L18
    apply le_succ
  4. L19
    exact hcollision_witness_witness_witness_left
06Separate the logical casesL20–20

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

  1. L20
    split
07Use earlier factsL21–24

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

  1. L21
    specialize le_succ (S x1)
  2. L22
    specialize le_succ n
  3. L23
    apply le_succ
  4. L24
    exact hcollision_witness_witness_witness_right_left
08Separate the logical casesL25–25

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

  1. L25
    split
09Fix variables and assumptionsL26–26

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

  1. L26
    intro hequal
10Use earlier factsL27–28

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

  1. L27
    apply hcollision_witness_witness_witness_right_right_left
  2. L28
    exact hequal
11Separate the logical casesL29–29

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

  1. L29
    split
12Use earlier factsL30–31

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

  1. L30
    exact hcollision_witness_witness_witness_right_right_right_left
  2. L31
    exact hcollision_witness_witness_witness_right_right_right_right

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro n
  4. 0004intro hcollision
  5. 0005cases hcollision
  6. 0006cases hcollision_witness
  7. 0007cases hcollision_witness_witness
  8. 0008cases hcollision_witness_witness_witness
  9. 0009cases hcollision_witness_witness_witness_right
  10. 0010cases hcollision_witness_witness_witness_right_right
  11. 0011cases hcollision_witness_witness_witness_right_right_right
  12. 0012exists x
  13. 0013exists x1
  14. 0014exists x2
  15. 0015split
  16. 0016specialize le_succ (S x)
  17. 0017specialize le_succ n
  18. 0018apply le_succ
  19. 0019exact hcollision_witness_witness_witness_left
  20. 0020split
  21. 0021specialize le_succ (S x1)
  22. 0022specialize le_succ n
  23. 0023apply le_succ
  24. 0024exact hcollision_witness_witness_witness_right_left
  25. 0025split
  26. 0026intro hequal
  27. 0027apply hcollision_witness_witness_witness_right_right_left
  28. 0028exact hequal
  29. 0029split
  30. 0030exact hcollision_witness_witness_witness_right_right_right_left
  31. 0031exact hcollision_witness_witness_witness_right_right_right_right