TS000M

floor_square_oversized_bounded_grid_not_injective

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Every beta-coded prime-residue-bounded map on the oversized floor-square grid has a collision obstruction.

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_ftsp_prime_floor. bcs_sqrt_lower_gap_ftsp_prime_floor + (s) * (s) = (p)) /\ exists bcs_sqrt_upper_gap_ftsp_prime_floor. bcs_sqrt_upper_gap_ftsp_prime_floor + S (p) = S (s) * S (s))) -> (forall fom_index_ftsp_grid_bounded. (exists fom_gap_ftsp_grid_bounded_index_bound. fom_gap_ftsp_grid_bounded_index_bound + S (fom_index_ftsp_grid_bounded) = l) -> exists fom_value_ftsp_grid_bounded. ((((exists fom_beta_height_ftsp_grid_bounded_entry. fom_beta_height_ftsp_grid_bounded_entry + S (fom_value_ftsp_grid_bounded) = S ((S (fom_index_ftsp_grid_bounded)) * c)) /\ exists fom_beta_quotient_ftsp_grid_bounded_entry. b = fom_beta_quotient_ftsp_grid_bounded_entry * S ((S (fom_index_ftsp_grid_bounded)) * c) + (fom_value_ftsp_grid_bounded))) /\ (exists fom_gap_ftsp_grid_bounded_value_bound. fom_gap_ftsp_grid_bounded_value_bound + S (fom_value_ftsp_grid_bounded) = p))) -> ~(forall fp_i_ftsp_grid_injective fp_j_ftsp_grid_injective fp_value_ftsp_grid_injective. (exists fp_gap_ftsp_grid_injective_i. fp_gap_ftsp_grid_injective_i + S fp_i_ftsp_grid_injective = l) -> (exists fp_gap_ftsp_grid_injective_j. fp_gap_ftsp_grid_injective_j + S fp_j_ftsp_grid_injective = l) -> (((exists ff_h_ftsp_grid_injective_left. ff_h_ftsp_grid_injective_left + S (fp_value_ftsp_grid_injective) = S ((S (fp_i_ftsp_grid_injective)) * c)) /\ exists ff_q_ftsp_grid_injective_left. b = ff_q_ftsp_grid_injective_left * S ((S (fp_i_ftsp_grid_injective)) * c) + (fp_value_ftsp_grid_injective))) -> (((exists ff_h_ftsp_grid_injective_right. ff_h_ftsp_grid_injective_right + S (fp_value_ftsp_grid_injective) = S ((S (fp_j_ftsp_grid_injective)) * c)) /\ exists ff_q_ftsp_grid_injective_right. b = ff_q_ftsp_grid_injective_right * S ((S (fp_j_ftsp_grid_injective)) * c) + (fp_value_ftsp_grid_injective))) -> fp_i_ftsp_grid_injective = fp_j_ftsp_grid_injective)

Constructive proof overview

Generated structural guide

Every beta-coded prime-residue-bounded map on the oversized floor-square grid has a collision obstruction.

The unchanged tactic script uses 2 declared prerequisites and contains 23 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 TS000L finite_bounded_into_oversized_not_injective

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

23 script commands · 3 reading checkpoints · 1 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 not injective
01Fix variables and assumptionsL1–9

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
  9. L9
    intro hinjective
02Establish hoverflowL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply floor square successor grid strictly exceeds input.

  1. L10
    have hoverflow : exists k. k + S p = l
  2. L11
    rewrite hlength
  3. L12
    specialize floor_square_successor_grid_strictly_exceeds_input p
  4. L13
    specialize floor_square_successor_grid_strictly_exceeds_input s
  5. L14
    apply floor_square_successor_grid_strictly_exceeds_input
  6. L15
    exact hfloor
  7. L16
    specialize finite_bounded_into_oversized_not_injective b
  8. L17
    specialize finite_bounded_into_oversized_not_injective c
  9. L18
    specialize finite_bounded_into_oversized_not_injective l
  10. L19
    specialize finite_bounded_into_oversized_not_injective p
03Use earlier factsL20–23

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

  1. L20
    apply finite_bounded_into_oversized_not_injective
  2. L21
    exact hbounded
  3. L22
    exact hoverflow
  4. L23
    exact hinjective

Library-wide reading audit

Original exact command ledger · 23 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. 0009intro hinjective
  10. 0010have hoverflow : exists k. k + S p = l
  11. 0011rewrite hlength
  12. 0012specialize floor_square_successor_grid_strictly_exceeds_input p
  13. 0013specialize floor_square_successor_grid_strictly_exceeds_input s
  14. 0014apply floor_square_successor_grid_strictly_exceeds_input
  15. 0015exact hfloor
  16. 0016specialize finite_bounded_into_oversized_not_injective b
  17. 0017specialize finite_bounded_into_oversized_not_injective c
  18. 0018specialize finite_bounded_into_oversized_not_injective l
  19. 0019specialize finite_bounded_into_oversized_not_injective p
  20. 0020apply finite_bounded_into_oversized_not_injective
  21. 0021exact hbounded
  22. 0022exact hoverflow
  23. 0023exact hinjective