TS000D · theorem body

natural_square_monotone_expanded

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Witnessed weak order on natural coordinates transports to their squares.

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. ∀ s. Le(a,s) → Le(a · a,s · s)

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

PD0001 Le
Exact expanded first-order statement
forall a s. (exists ftsp_weak_square_source. ftsp_weak_square_source + (a) = s) -> (exists ftsp_weak_square_result. ftsp_weak_square_result + (a * a) = s * s)

Proof neighborhood

Direct theorem prerequisites

mul_le_mul_right · Stable closed mul_le_mul_left · Stable closed le_trans · Stable closed

Direct theorem dependents

TS000F prime_floor_bounded_coordinate_square_strict

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

21 script commands · 4 reading checkpoints · 2 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 s
  3. L3
    intro hle
02Establish hfirstL4–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul right.

  1. L4
    have hfirst : Le(a · a,s · a)Definitions: Le(a · a,s · a)Original native command in the exact edition
  2. L5
    specialize mul_le_mul_right a
  3. L6
    specialize mul_le_mul_right s
  4. L7
    specialize mul_le_mul_right a
  5. L8
    apply mul_le_mul_right
  6. L9
    exact hle
03Establish hsecondL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L10
    have hsecond : Le(s · a,s · s)Definitions: Le(s · a,s · s)Original native command in the exact edition
  2. L11
    specialize mul_le_mul_left a
  3. L12
    specialize mul_le_mul_left s
  4. L13
    specialize mul_le_mul_left s
  5. L14
    apply mul_le_mul_left
  6. L15
    exact hle
  7. L16
    specialize le_trans (a * a)
  8. L17
    specialize le_trans (s * a)
  9. L18
    specialize le_trans (s * s)
  10. L19
    apply le_trans
04Use earlier factsL20–21

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

  1. L20
    exact hfirst
  2. L21
    exact hsecond

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro s
  3. 0003intro hle
  4. 0004have hfirst : Le(a · a,s · a)
    Exact native replay linehave hfirst : exists k. k + a * a = s * a
  5. 0005specialize mul_le_mul_right a
  6. 0006specialize mul_le_mul_right s
  7. 0007specialize mul_le_mul_right a
  8. 0008apply mul_le_mul_right
  9. 0009exact hle
  10. 0010have hsecond : Le(s · a,s · s)
    Exact native replay linehave hsecond : exists k. k + s * a = s * s
  11. 0011specialize mul_le_mul_left a
  12. 0012specialize mul_le_mul_left s
  13. 0013specialize mul_le_mul_left s
  14. 0014apply mul_le_mul_left
  15. 0015exact hle
  16. 0016specialize le_trans (a * a)
  17. 0017specialize le_trans (s * a)
  18. 0018specialize le_trans (s * s)
  19. 0019apply le_trans
  20. 0020exact hfirst
  21. 0021exact hsecond