PF000D · theorem body

fermat_four_counterexample_is_pythagorean

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

A positive fourth-power square counterexample necessarily supplies a Pythagorean triple whose two legs are natural 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. ∀ b. ∀ h. a · a · a · a + b · b · b · b = h · h → Pythagorean(a · a,b · b,h)

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

CF0011 Pythagorean

In local proof propositions

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

Proof neighborhood

Direct theorem prerequisites

fourth_power_regroup · Stable closed

Direct theorem dependents

PF0028 fermat_four_scaled_equation PF002I fermat_four_primitive_square_triangle

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

11 script commands · 3 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
  4. L4
    intro hequation
02Establish hfirstL5–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply fourth power regroup.

  1. L5
    have hfirst : a * a * a * a = (a * a) * (a * a)
  2. L6
    apply fourth_power_regroup
03Establish hsecondL7–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply fourth power regroup.

  1. L7
    have hsecond : b * b * b * b = (b * b) * (b * b)
  2. L8
    apply fourth_power_regroup
  3. L9
    rewrite hfirst at hequation
  4. L10
    rewrite hsecond at hequation
  5. L11
    exact hequation

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hequation
  5. 0005have hfirst : a * a * a * a = (a * a) * (a * a)
  6. 0006apply fourth_power_regroup
  7. 0007have hsecond : b * b * b * b = (b * b) * (b * b)
  8. 0008apply fourth_power_regroup
  9. 0009rewrite hfirst at hequation
  10. 0010rewrite hsecond at hequation
  11. 0011exact hequation