PF0019 · theorem body

square_le_reflect

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ a. ∀ b. Le(a · a,b · b) → Le(a,b)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0001 Le

In local proof propositions

none
Exact expanded first-order statement
forall a b. (exists k. k + a * a = b * b) -> exists k. k + a = b

Proof neighborhood

Direct theorem prerequisites

le_or_lt · Stable closed PF0018 square_lt_strict lt_not_le · Stable closed

Direct theorem dependents

PF001B square_eq_injective

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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