PF001A

square_lt_reflect

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

A witnessed strict inequality between natural squares reflects to their roots.

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

Constructive proof overview

Generated structural guide

A witnessed strict inequality between natural squares reflects to their roots.

The unchanged tactic script uses 3 declared prerequisites and contains 16 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

le_or_lt Stable theorem; checked-use authorized natural_square_monotone_expanded Alpha theorem; checked-use authorized lt_not_le Stable theorem; checked-use authorized

Direct dependents

PF001I pythagorean_leg_strictly_below_hypotenuse PF001T pythagorean_positive_gap_orders_parameters

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

16 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.

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 factsL4–5

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

  1. L4
    specialize le_or_lt b
  2. L5
    specialize le_or_lt a
03Separate the logical casesL6–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    cases le_or_lt
  2. L7
    exfalso
04Use earlier factsL8–16

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

  1. L8
    specialize lt_not_le (a * a)
  2. L9
    specialize lt_not_le (b * b)
  3. L10
    apply lt_not_le
  4. L11
    exact hlt
  5. L12
    specialize natural_square_monotone_expanded b
  6. L13
    specialize natural_square_monotone_expanded a
  7. L14
    apply natural_square_monotone_expanded
  8. L15
    exact le_or_lt_left
  9. L16
    exact le_or_lt_right

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hlt
  4. 0004specialize le_or_lt b
  5. 0005specialize le_or_lt a
  6. 0006cases le_or_lt
  7. 0007exfalso
  8. 0008specialize lt_not_le (a * a)
  9. 0009specialize lt_not_le (b * b)
  10. 0010apply lt_not_le
  11. 0011exact hlt
  12. 0012specialize natural_square_monotone_expanded b
  13. 0013specialize natural_square_monotone_expanded a
  14. 0014apply natural_square_monotone_expanded
  15. 0015exact le_or_lt_left
  16. 0016exact le_or_lt_right