PF002L

fermat_four_primitive_descent

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites and contains 31 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

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 dependents

PF002M fermat_four_strict_descent_proved

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 : (((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))
  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 : (((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)))
  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 exact command ledger · 31 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hprimitive
  5. 0005have 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 : (((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