PF000F · theorem body

fermat_four_no_square_from_descent

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

The stronger no-fourth-powers-sum-to-a-square theorem follows constructively from precisely one explicit, still-unproved strict descent premise.

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

FermatFourStrictDescent() → ∀ x. ∀ y. ∀ z. ¬FermatFourCounterexample(x,y,z)

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

CF0014 FermatFourCounterexample CF0015 FermatFourStrictDescent

In local proof propositions

none
Exact expanded first-order statement
(forall pff_first_bounded pff_second_bounded pff_hypotenuse_bounded. ((~((pff_first_bounded) = 0) /\ (~((pff_second_bounded) = 0) /\ (~((pff_hypotenuse_bounded) = 0) /\ ((pff_first_bounded) * (pff_first_bounded) * (pff_first_bounded) * (pff_first_bounded) + (pff_second_bounded) * (pff_second_bounded) * (pff_second_bounded) * (pff_second_bounded) = (pff_hypotenuse_bounded) * (pff_hypotenuse_bounded)))))) -> exists pff_smaller_first_bounded pff_smaller_second_bounded pff_smaller_hypotenuse_bounded. (((~((pff_smaller_first_bounded) = 0) /\ (~((pff_smaller_second_bounded) = 0) /\ (~((pff_smaller_hypotenuse_bounded) = 0) /\ ((pff_smaller_first_bounded) * (pff_smaller_first_bounded) * (pff_smaller_first_bounded) * (pff_smaller_first_bounded) + (pff_smaller_second_bounded) * (pff_smaller_second_bounded) * (pff_smaller_second_bounded) * (pff_smaller_second_bounded) = (pff_smaller_hypotenuse_bounded) * (pff_smaller_hypotenuse_bounded)))))) /\ (exists pff_gap_bounded. pff_gap_bounded + S pff_smaller_hypotenuse_bounded = pff_hypotenuse_bounded))) -> forall a b h. ~((~((a) = 0) /\ (~((b) = 0) /\ (~((h) = 0) /\ ((a) * (a) * (a) * (a) + (b) * (b) * (b) * (b) = (h) * (h))))))

Proof neighborhood

Direct theorem prerequisites

PF000E fermat_four_bounded_descent le_refl · Stable closed

Direct theorem dependents

PF000G fermat_four_no_fourth_from_descent PF002N fermat_four_no_square

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.

Named ingredients (1)
fermat four bounded descent
01Fix variables and assumptionsL1–5

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

  1. L1
    intro hstep
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro h
  5. L5
    intro hcounter
02Use earlier factsL6–13

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

  1. L6
    specialize fermat_four_bounded_descent h
  2. L7
    specialize fermat_four_bounded_descent h
  3. L8
    specialize fermat_four_bounded_descent a
  4. L9
    specialize fermat_four_bounded_descent b
  5. L10
    apply fermat_four_bounded_descent
  6. L11
    apply le_refl
  7. L12
    exact hstep
  8. L13
    exact hcounter

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro hstep
  2. 0002intro a
  3. 0003intro b
  4. 0004intro h
  5. 0005intro hcounter
  6. 0006specialize fermat_four_bounded_descent h
  7. 0007specialize fermat_four_bounded_descent h
  8. 0008specialize fermat_four_bounded_descent a
  9. 0009specialize fermat_four_bounded_descent b
  10. 0010apply fermat_four_bounded_descent
  11. 0011apply le_refl
  12. 0012exact hstep
  13. 0013exact hcounter