TS000T

floor_square_oversized_bounded_grid_collision

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

Every residue-bounded beta map on the floor-square oversized grid contains explicit distinct colliding indices.

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 l p s. l = S s * S s -> (((exists bcs_sqrt_lower_gap_fpcd_floor. bcs_sqrt_lower_gap_fpcd_floor + (s) * (s) = (p)) /\ exists bcs_sqrt_upper_gap_fpcd_floor. bcs_sqrt_upper_gap_fpcd_floor + S (p) = S (s) * S (s))) -> (forall fom_index_fpcd_grid. (exists fom_gap_fpcd_grid_index_bound. fom_gap_fpcd_grid_index_bound + S (fom_index_fpcd_grid) = l) -> exists fom_value_fpcd_grid. ((((exists fom_beta_height_fpcd_grid_entry. fom_beta_height_fpcd_grid_entry + S (fom_value_fpcd_grid) = S ((S (fom_index_fpcd_grid)) * c)) /\ exists fom_beta_quotient_fpcd_grid_entry. b = fom_beta_quotient_fpcd_grid_entry * S ((S (fom_index_fpcd_grid)) * c) + (fom_value_fpcd_grid))) /\ (exists fom_gap_fpcd_grid_value_bound. fom_gap_fpcd_grid_value_bound + S (fom_value_fpcd_grid) = p))) -> (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))))))))

Constructive proof overview

Generated structural guide

Every residue-bounded beta map on the floor-square oversized grid contains explicit distinct colliding indices.

The unchanged tactic script uses 2 declared prerequisites and contains 19 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

TS000I floor_square_successor_grid_strictly_exceeds_input TS000S finite_bounded_into_oversized_collision

Direct dependents

TS000Z prime_floor_affine_residue_grid_collision

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

19 script commands · 4 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.

Named ingredients (2)
floor square successor grid strictly exceeds inputfinite bounded into oversized collision
01Fix variables and assumptionsL1–8

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 p
  5. L5
    intro s
  6. L6
    intro hlength
  7. L7
    intro hfloor
  8. L8
    intro hbounded
02Use earlier factsL9–14

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

  1. L9
    specialize finite_bounded_into_oversized_collision b
  2. L10
    specialize finite_bounded_into_oversized_collision c
  3. L11
    specialize finite_bounded_into_oversized_collision l
  4. L12
    specialize finite_bounded_into_oversized_collision p
  5. L13
    apply finite_bounded_into_oversized_collision
  6. L14
    exact hbounded
03Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite hlength
04Use earlier factsL16–19

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

  1. L16
    specialize floor_square_successor_grid_strictly_exceeds_input p
  2. L17
    specialize floor_square_successor_grid_strictly_exceeds_input s
  3. L18
    apply floor_square_successor_grid_strictly_exceeds_input
  4. L19
    exact hfloor

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro p
  5. 0005intro s
  6. 0006intro hlength
  7. 0007intro hfloor
  8. 0008intro hbounded
  9. 0009specialize finite_bounded_into_oversized_collision b
  10. 0010specialize finite_bounded_into_oversized_collision c
  11. 0011specialize finite_bounded_into_oversized_collision l
  12. 0012specialize finite_bounded_into_oversized_collision p
  13. 0013apply finite_bounded_into_oversized_collision
  14. 0014exact hbounded
  15. 0015rewrite hlength
  16. 0016specialize floor_square_successor_grid_strictly_exceeds_input p
  17. 0017specialize floor_square_successor_grid_strictly_exceeds_input s
  18. 0018apply floor_square_successor_grid_strictly_exceeds_input
  19. 0019exact hfloor