TS000S · theorem body

finite_bounded_into_oversized_collision

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

An oversized beta-coded map into a bounded finite interval has an actual existentially witnessed collision.

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. ∀ l. ∀ m. (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,m)) → Lt(m,l) → ∃ x. ∃ y. ∃ z. Lt(x,l) ∧ (Lt(y,l) ∧ (¬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 l m. (forall fom_index_fpcd_bounded. (exists fom_gap_fpcd_bounded_index_bound. fom_gap_fpcd_bounded_index_bound + S (fom_index_fpcd_bounded) = l) -> exists fom_value_fpcd_bounded. ((((exists fom_beta_height_fpcd_bounded_entry. fom_beta_height_fpcd_bounded_entry + S (fom_value_fpcd_bounded) = S ((S (fom_index_fpcd_bounded)) * c)) /\ exists fom_beta_quotient_fpcd_bounded_entry. b = fom_beta_quotient_fpcd_bounded_entry * S ((S (fom_index_fpcd_bounded)) * c) + (fom_value_fpcd_bounded))) /\ (exists fom_gap_fpcd_bounded_value_bound. fom_gap_fpcd_bounded_value_bound + S (fom_value_fpcd_bounded) = m))) -> (exists ftsp_gap_fpcd_overflow. ftsp_gap_fpcd_overflow + S (m) = l) -> (exists ftsp_first_fpcd_generic ftsp_second_fpcd_generic ftsp_value_fpcd_generic. ((exists ftsp_gap_fpcd_generic_first. ftsp_gap_fpcd_generic_first + S (ftsp_first_fpcd_generic) = l) /\ ((exists ftsp_gap_fpcd_generic_second. ftsp_gap_fpcd_generic_second + S (ftsp_second_fpcd_generic) = l) /\ (~(ftsp_first_fpcd_generic = ftsp_second_fpcd_generic) /\ ((((exists ff_h_ftsp_fpcd_generic_left. ff_h_ftsp_fpcd_generic_left + S (ftsp_value_fpcd_generic) = S ((S (ftsp_first_fpcd_generic)) * c)) /\ exists ff_q_ftsp_fpcd_generic_left. b = ff_q_ftsp_fpcd_generic_left * S ((S (ftsp_first_fpcd_generic)) * c) + (ftsp_value_fpcd_generic))) /\ (((exists ff_h_ftsp_fpcd_generic_right. ff_h_ftsp_fpcd_generic_right + S (ftsp_value_fpcd_generic) = S ((S (ftsp_second_fpcd_generic)) * c)) /\ exists ff_q_ftsp_fpcd_generic_right. b = ff_q_ftsp_fpcd_generic_right * S ((S (ftsp_second_fpcd_generic)) * c) + (ftsp_value_fpcd_generic))))))))

Proof neighborhood

Direct theorem prerequisites

TS000R finite_prefix_collision_or_injective TS000N finite_bounded_into_collision_from_constructive_decision

Direct theorem dependents

TS000T floor_square_oversized_bounded_grid_collision

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

17 script commands · 3 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.

Named ingredients (2)
finite prefix collision or injectivefinite bounded into collision from constructive decision
01Fix variables and assumptionsL1–6

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro m
  5. L5
    intro hbounded
  6. L6
    intro hoverflow
02Use earlier factsL7–16

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

  1. L7
    specialize finite_bounded_into_collision_from_constructive_decision b
  2. L8
    specialize finite_bounded_into_collision_from_constructive_decision c
  3. L9
    specialize finite_bounded_into_collision_from_constructive_decision l
  4. L10
    specialize finite_bounded_into_collision_from_constructive_decision m
  5. L11
    apply finite_bounded_into_collision_from_constructive_decision
  6. L12
    exact hbounded
  7. L13
    exact hoverflow
  8. L14
    specialize finite_prefix_collision_or_injective b
  9. L15
    specialize finite_prefix_collision_or_injective c
  10. L16
    specialize finite_prefix_collision_or_injective l
03Use earlier factsL17–17

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

  1. L17
    exact finite_prefix_collision_or_injective

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro m
  5. 0005intro hbounded
  6. 0006intro hoverflow
  7. 0007specialize finite_bounded_into_collision_from_constructive_decision b
  8. 0008specialize finite_bounded_into_collision_from_constructive_decision c
  9. 0009specialize finite_bounded_into_collision_from_constructive_decision l
  10. 0010specialize finite_bounded_into_collision_from_constructive_decision m
  11. 0011apply finite_bounded_into_collision_from_constructive_decision
  12. 0012exact hbounded
  13. 0013exact hoverflow
  14. 0014specialize finite_prefix_collision_or_injective b
  15. 0015specialize finite_prefix_collision_or_injective c
  16. 0016specialize finite_prefix_collision_or_injective l
  17. 0017exact finite_prefix_collision_or_injective