PF001T · theorem body

pythagorean_positive_gap_orders_parameters

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

The positive odd leg forces the two natural square roots into the required strict Euclidean order.

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. ∀ m. ∀ n. ¬a = 0 → m · m = n · n + a → Lt(n,m)

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

PD0002 Lt
Exact expanded first-order statement
forall a m n. ~(a = 0) -> m * m = n * n + a -> exists gap. gap + S n = m

Proof neighborhood

Direct theorem prerequisites

PF001H pythagorean_positive_add_strict PF001A square_lt_reflect

Direct theorem dependents

PF001W pythagorean_half_factors_extract_parameters

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 · 1 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 (2)
pythagorean positive add strictsquare lt reflect
01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro n
  4. L4
    intro ha
  5. L5
    intro hgap
02Establish hstrictL6–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean positive add strict.

  1. L6
    have hstrict : Lt(n · n,m · m)Definitions: Lt(n · n,m · m)Original native command in the exact edition
  2. L7
    rewrite hgap
  3. L8
    specialize pythagorean_positive_add_strict (n * n)
  4. L9
    specialize pythagorean_positive_add_strict a
  5. L10
    apply pythagorean_positive_add_strict
  6. L11
    exact ha
  7. L12
    apply square_lt_reflect
  8. L13
    exact hstrict

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro n
  4. 0004intro ha
  5. 0005intro hgap
  6. 0006have hstrict : Lt(n · n,m · m)
    Exact native replay linehave hstrict : exists gap. gap + S (n * n) = m * m
  7. 0007rewrite hgap
  8. 0008specialize pythagorean_positive_add_strict (n * n)
  9. 0009specialize pythagorean_positive_add_strict a
  10. 0010apply pythagorean_positive_add_strict
  11. 0011exact ha
  12. 0012apply square_lt_reflect
  13. 0013exact hstrict