TS001E

flat_square_index_row_not_at_least_width

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

A flat index strictly inside a square grid cannot have row index at least the grid width.

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 w i j k. k = w * i + j -> (exists ftcn_strict_flat_source. ftcn_strict_flat_source + S (k) = (w * w)) -> ~(exists ftcn_bound_flat_forbidden. ftcn_bound_flat_forbidden + (w) = (i))

Constructive proof overview

Generated structural guide

A flat index strictly inside a square grid cannot have row index at least the grid width.

The unchanged tactic script uses 4 declared prerequisites and contains 30 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

mul_le_mul_left Stable theorem; checked-use authorized le_add_right Stable theorem; checked-use authorized le_trans Stable theorem; checked-use authorized lt_not_le Stable theorem; checked-use authorized

Direct dependents

TS001F flat_square_index_row_below_width

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

30 script commands · 5 reading checkpoints · 3 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro w
  2. L2
    intro i
  3. L3
    intro j
  4. L4
    intro k
  5. L5
    intro hindex
  6. L6
    intro hbound
  7. L7
    intro hrow
02Establish hproductL8–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L8
    have hproduct : exists t. t + w * w = w * i
  2. L9
    specialize mul_le_mul_left w
  3. L10
    specialize mul_le_mul_left i
  4. L11
    specialize mul_le_mul_left w
  5. L12
    apply mul_le_mul_left
  6. L13
    exact hrow
03Establish hsumL14–18

Establish this local claim before using it. It is not an additional assumption.

  1. L14
    have hsum : exists t. t + w * i = k
  2. L15
    rewrite hindex
  3. L16
    specialize le_add_right (w * i)
  4. L17
    specialize le_add_right j
  5. L18
    exact le_add_right
04Establish hlargeL19–28

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le trans.

  1. L19
    have hlarge : exists t. t + w * w = k
  2. L20
    specialize le_trans (w * w)
  3. L21
    specialize le_trans (w * i)
  4. L22
    specialize le_trans k
  5. L23
    apply le_trans
  6. L24
    exact hproduct
  7. L25
    exact hsum
  8. L26
    specialize lt_not_le k
  9. L27
    specialize lt_not_le (w * w)
  10. L28
    apply lt_not_le
05Use earlier factsL29–30

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

  1. L29
    exact hbound
  2. L30
    exact hlarge

Library-wide reading audit

Original exact command ledger · 30 lines
  1. 0001intro w
  2. 0002intro i
  3. 0003intro j
  4. 0004intro k
  5. 0005intro hindex
  6. 0006intro hbound
  7. 0007intro hrow
  8. 0008have hproduct : exists t. t + w * w = w * i
  9. 0009specialize mul_le_mul_left w
  10. 0010specialize mul_le_mul_left i
  11. 0011specialize mul_le_mul_left w
  12. 0012apply mul_le_mul_left
  13. 0013exact hrow
  14. 0014have hsum : exists t. t + w * i = k
  15. 0015rewrite hindex
  16. 0016specialize le_add_right (w * i)
  17. 0017specialize le_add_right j
  18. 0018exact le_add_right
  19. 0019have hlarge : exists t. t + w * w = k
  20. 0020specialize le_trans (w * w)
  21. 0021specialize le_trans (w * i)
  22. 0022specialize le_trans k
  23. 0023apply le_trans
  24. 0024exact hproduct
  25. 0025exact hsum
  26. 0026specialize lt_not_le k
  27. 0027specialize lt_not_le (w * w)
  28. 0028apply lt_not_le
  29. 0029exact hbound
  30. 0030exact hlarge