PF002L · theorem body

fermat_four_primitive_descent

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

Constructive parity selection orients every primitive counterexample and constructs a strictly smaller counterexample.

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. PrimitiveFermatFourCounterexample(a,b,h) → ∃ x. ∃ y. ∃ z. SmallerFermatFourCounterexample(x,y,z,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

ND0071 PrimitiveFermatFourCounterexample ND0072 SmallerFermatFourCounterexample

In local proof propositions

CF0013 PrimitivePythagorean CF0016 OppositeParity
Exact expanded first-order statement
forall a b h. ((((~((a) = 0) /\ (~((b) = 0) /\ (~((h) = 0) /\ ((a) * (a) * (a) * (a) + (b) * (b) * (b) * (b) = (h) * (h)))))) /\ (forall pff_divisor_primitive_descent_source. (exists pff_left_primitive_descent_source. (a) = pff_divisor_primitive_descent_source * pff_left_primitive_descent_source) -> (exists pff_right_primitive_descent_source. (b) = pff_divisor_primitive_descent_source * pff_right_primitive_descent_source) -> pff_divisor_primitive_descent_source = 1))) -> exists A B H. ((((~((A) = 0) /\ (~((B) = 0) /\ (~((H) = 0) /\ ((A) * (A) * (A) * (A) + (B) * (B) * (B) * (B) = (H) * (H)))))) /\ (exists ffd_gap_primitive_descent_result. ffd_gap_primitive_descent_result + S H = h)))

Proof neighborhood

Direct theorem prerequisites

PF002I fermat_four_primitive_square_triangle PF0014 pythagorean_primitive_legs_opposite_parity PF002J fermat_four_primitive_counterexample_swap PF002K fermat_four_primitive_odd_even_descent

Direct theorem dependents

PF002M fermat_four_strict_descent_proved

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

31 script commands · 7 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.

Named ingredients (4)
fermat four primitive square trianglepythagorean primitive legs opposite parityfermat four primitive counterexample swapfermat four primitive odd even descent
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 hprimitive
02Establish htriangleL5–7

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply fermat four primitive square triangle.

  1. L5
    have htriangle : PrimitivePythagorean(a · a,b · b,h)Definitions: PrimitivePythagorean(a · a,b · b,h)Original native command in the exact edition
  2. L6
    apply fermat_four_primitive_square_triangle
  3. L7
    exact hprimitive
03Establish hparityL8–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean primitive legs opposite parity.

  1. L8
    have hparity : OppositeParity(a · a,b · b)Definitions: OppositeParity(a · a,b · b)Original native command in the exact edition
  2. L9
    specialize pythagorean_primitive_legs_opposite_parity (a * a)
  3. L10
    specialize pythagorean_primitive_legs_opposite_parity (b * b)
  4. L11
    specialize pythagorean_primitive_legs_opposite_parity (h)
  5. L12
    apply pythagorean_primitive_legs_opposite_parity
  6. L13
    exact htriangle
04Separate the logical casesL14–15

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    cases hparity
  2. L15
    cases hparity_left
05Use earlier factsL16–23

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

  1. L16
    specialize fermat_four_primitive_odd_even_descent (b)
  2. L17
    specialize fermat_four_primitive_odd_even_descent (a)
  3. L18
    specialize fermat_four_primitive_odd_even_descent (h)
  4. L19
    apply fermat_four_primitive_odd_even_descent
  5. L20
    apply fermat_four_primitive_counterexample_swap
  6. L21
    exact hprimitive
  7. L22
    exact hparity_left_right
  8. L23
    exact hparity_left_left
06Separate the logical casesL24–24

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L24
    cases hparity_right
07Use earlier factsL25–31

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

  1. L25
    specialize fermat_four_primitive_odd_even_descent (a)
  2. L26
    specialize fermat_four_primitive_odd_even_descent (b)
  3. L27
    specialize fermat_four_primitive_odd_even_descent (h)
  4. L28
    apply fermat_four_primitive_odd_even_descent
  5. L29
    exact hprimitive
  6. L30
    exact hparity_right_left
  7. L31
    exact hparity_right_right

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hprimitive
  5. 0005have htriangle : PrimitivePythagorean(a · a,b · b,h)
    Exact native replay linehave htriangle : (((a * a) * (a * a) + (b * b) * (b * b) = (h) * (h)) /\ (forall pff_divisor_ffd_primitive_descent_triangle. (exists pff_left_ffd_primitive_descent_triangle. (a * a) = pff_divisor_ffd_primitive_descent_triangle * pff_left_ffd_primitive_descent_triangle) -> (exists pff_right_ffd_primitive_descent_triangle. (b * b) = pff_divisor_ffd_primitive_descent_triangle * pff_right_ffd_primitive_descent_triangle) -> pff_divisor_ffd_primitive_descent_triangle = 1))
  6. 0006apply fermat_four_primitive_square_triangle
  7. 0007exact hprimitive
  8. 0008have hparity : OppositeParity(a · a,b · b)
    Exact native replay linehave hparity : (((exists pp_even_ffd_primitive_descent_parity_first_even. (a * a) = 2 * pp_even_ffd_primitive_descent_parity_first_even) /\ (exists pp_odd_ffd_primitive_descent_parity_second_odd. (b * b) = 2 * pp_odd_ffd_primitive_descent_parity_second_odd + 1)) \/ ((exists pp_odd_ffd_primitive_descent_parity_first_odd. (a * a) = 2 * pp_odd_ffd_primitive_descent_parity_first_odd + 1) /\ (exists pp_even_ffd_primitive_descent_parity_second_even. (b * b) = 2 * pp_even_ffd_primitive_descent_parity_second_even)))
  9. 0009specialize pythagorean_primitive_legs_opposite_parity (a * a)
  10. 0010specialize pythagorean_primitive_legs_opposite_parity (b * b)
  11. 0011specialize pythagorean_primitive_legs_opposite_parity (h)
  12. 0012apply pythagorean_primitive_legs_opposite_parity
  13. 0013exact htriangle
  14. 0014cases hparity
  15. 0015cases hparity_left
  16. 0016specialize fermat_four_primitive_odd_even_descent (b)
  17. 0017specialize fermat_four_primitive_odd_even_descent (a)
  18. 0018specialize fermat_four_primitive_odd_even_descent (h)
  19. 0019apply fermat_four_primitive_odd_even_descent
  20. 0020apply fermat_four_primitive_counterexample_swap
  21. 0021exact hprimitive
  22. 0022exact hparity_left_right
  23. 0023exact hparity_left_left
  24. 0024cases hparity_right
  25. 0025specialize fermat_four_primitive_odd_even_descent (a)
  26. 0026specialize fermat_four_primitive_odd_even_descent (b)
  27. 0027specialize fermat_four_primitive_odd_even_descent (h)
  28. 0028apply fermat_four_primitive_odd_even_descent
  29. 0029exact hprimitive
  30. 0030exact hparity_right_left
  31. 0031exact hparity_right_right