PF0018 · theorem body

square_lt_strict

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

Statement with defined notation

∀ a. ∀ b. Lt(a,b) → Lt(a · a,b · 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

PD0002 Lt

In local proof propositions

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

Proof neighborhood

Direct theorem prerequisites

square_lt_successor_square · Alpha closed natural_square_monotone_expanded · Alpha closed lt_of_lt_of_le · Stable closed

Direct theorem dependents

PF0019 square_le_reflect

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

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.

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

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 defined 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