PF0019

square_le_reflect

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

A witnessed weak 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 + a * a = b * b) -> exists k. k + a = b

Constructive proof overview

Generated structural guide

A witnessed weak 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 PF0018 square_lt_strict lt_not_le Stable theorem; checked-use authorized

Direct dependents

PF001B square_eq_injective

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 · 6 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 (1)
square lt strict
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 hle
02Use earlier factsL4–5

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

  1. L4
    specialize le_or_lt a
  2. L5
    specialize le_or_lt b
03Separate the logical casesL6–6

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

  1. L6
    cases le_or_lt
04Use earlier factsL7–7

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

  1. L7
    exact le_or_lt_left
05Separate the logical casesL8–8

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

  1. L8
    exfalso
06Use earlier factsL9–16

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

  1. L9
    specialize lt_not_le (b * b)
  2. L10
    specialize lt_not_le (a * a)
  3. L11
    apply lt_not_le
  4. L12
    specialize square_lt_strict b
  5. L13
    specialize square_lt_strict a
  6. L14
    apply square_lt_strict
  7. L15
    exact le_or_lt_right
  8. L16
    exact hle

Library-wide reading audit

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