PF0018

square_lt_strict

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

Squaring strictly preserves witnessed order on all natural numbers.

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 a b. (exists k. k + S a = b) -> exists k. k + S (a * a) = b * b

Constructive proof overview

Generated structural guide

Squaring strictly preserves witnessed order on all natural numbers.

The unchanged tactic script uses 3 declared prerequisites and contains 13 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

square_lt_successor_square Alpha theorem; checked-use authorized natural_square_monotone_expanded Alpha theorem; checked-use authorized lt_of_lt_of_le Stable theorem; checked-use authorized

Direct dependents

PF0019 square_le_reflect

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

13 script commands · 2 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro hlt
02Use earlier factsL4–13

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

  1. L4
    specialize lt_of_lt_of_le (a * a)
  2. L5
    specialize lt_of_lt_of_le (S a * S a)
  3. L6
    specialize lt_of_lt_of_le (b * b)
  4. L7
    apply lt_of_lt_of_le
  5. L8
    specialize square_lt_successor_square a
  6. L9
    apply square_lt_successor_square
  7. L10
    specialize natural_square_monotone_expanded (S a)
  8. L11
    specialize natural_square_monotone_expanded b
  9. L12
    apply natural_square_monotone_expanded
  10. L13
    exact hlt

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hlt
  4. 0004specialize lt_of_lt_of_le (a * a)
  5. 0005specialize lt_of_lt_of_le (S a * S a)
  6. 0006specialize lt_of_lt_of_le (b * b)
  7. 0007apply lt_of_lt_of_le
  8. 0008specialize square_lt_successor_square a
  9. 0009apply square_lt_successor_square
  10. 0010specialize natural_square_monotone_expanded (S a)
  11. 0011specialize natural_square_monotone_expanded b
  12. 0012apply natural_square_monotone_expanded
  13. 0013exact hlt