PF002M

fermat_four_strict_descent_proved

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every positive fourth-power square counterexample constructs an actual positive counterexample with strictly smaller height; no descent premise remains.

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 pff_first_proved pff_second_proved pff_hypotenuse_proved. ((~((pff_first_proved) = 0) /\ (~((pff_second_proved) = 0) /\ (~((pff_hypotenuse_proved) = 0) /\ ((pff_first_proved) * (pff_first_proved) * (pff_first_proved) * (pff_first_proved) + (pff_second_proved) * (pff_second_proved) * (pff_second_proved) * (pff_second_proved) = (pff_hypotenuse_proved) * (pff_hypotenuse_proved)))))) -> exists pff_smaller_first_proved pff_smaller_second_proved pff_smaller_hypotenuse_proved. (((~((pff_smaller_first_proved) = 0) /\ (~((pff_smaller_second_proved) = 0) /\ (~((pff_smaller_hypotenuse_proved) = 0) /\ ((pff_smaller_first_proved) * (pff_smaller_first_proved) * (pff_smaller_first_proved) * (pff_smaller_first_proved) + (pff_smaller_second_proved) * (pff_smaller_second_proved) * (pff_smaller_second_proved) * (pff_smaller_second_proved) = (pff_smaller_hypotenuse_proved) * (pff_smaller_hypotenuse_proved)))))) /\ (exists pff_gap_proved. pff_gap_proved + S pff_smaller_hypotenuse_proved = pff_hypotenuse_proved))

Constructive proof overview

Generated structural guide

Every positive fourth-power square counterexample constructs an actual positive counterexample with strictly smaller height; no descent premise remains.

The unchanged tactic script uses 3 declared prerequisites and contains 35 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

PF002F fermat_four_primitive_normalization PF002L fermat_four_primitive_descent lt_of_lt_of_le Stable theorem; checked-use authorized

Direct dependents

PF002N fermat_four_no_square PF002O fermat_four_no_fourth

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

35 script commands · 8 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 (2)
fermat four primitive normalizationfermat four primitive 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 hcounter
02Establish hnormalizedL5–10

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

  1. L5
    have hnormalized : exists A B H. (((((~((A) = 0) /\ (~((B) = 0) /\ (~((H) = 0) /\ ((A) * (A) * (A) * (A) + (B) * (B) * (B) * (B) = (H) * (H)))))) /\ (forall pff_divisor_strict_normalized. (exists pff_left_strict_normalized. (A) = pff_divisor_strict_normalized * pff_left_strict_normalized) -> (exists pff_right_strict_normalized. (B) = pff_divisor_strict_normalized * pff_right_strict_normalized) -> pff_divisor_strict_normalized = 1))) /\ (exists k. k + H = h))
  2. L6
    specialize fermat_four_primitive_normalization (a)
  3. L7
    specialize fermat_four_primitive_normalization (b)
  4. L8
    specialize fermat_four_primitive_normalization (h)
  5. L9
    apply fermat_four_primitive_normalization
  6. L10
    exact hcounter
03Separate the logical casesL11–14

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

  1. L11
    cases hnormalized
  2. L12
    cases hnormalized_witness
  3. L13
    cases hnormalized_witness_witness
  4. L14
    cases hnormalized_witness_witness_witness
04Establish hsmallerL15–20

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

  1. L15
    have hsmaller : exists A B H. ((((~((A) = 0) /\ (~((B) = 0) /\ (~((H) = 0) /\ ((A) * (A) * (A) * (A) + (B) * (B) * (B) * (B) = (H) * (H)))))) /\ (exists ffd_gap_strict_smaller. ffd_gap_strict_smaller + S H = x2)))
  2. L16
    specialize fermat_four_primitive_descent (x)
  3. L17
    specialize fermat_four_primitive_descent (x1)
  4. L18
    specialize fermat_four_primitive_descent (x2)
  5. L19
    apply fermat_four_primitive_descent
  6. L20
    exact hnormalized_witness_witness_witness_left
05Separate the logical casesL21–24

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

  1. L21
    cases hsmaller
  2. L22
    cases hsmaller_witness
  3. L23
    cases hsmaller_witness_witness
  4. L24
    cases hsmaller_witness_witness_witness
06Construct an explicit witnessL25–27

Supply the displayed value, then prove that it has the required property.

  1. L25
    exists x3
  2. L26
    exists x4
  3. L27
    exists x5
07Separate the logical casesL28–28

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

  1. L28
    split
08Use earlier factsL29–35

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

  1. L29
    exact hsmaller_witness_witness_witness_left
  2. L30
    specialize lt_of_lt_of_le (x5)
  3. L31
    specialize lt_of_lt_of_le (x2)
  4. L32
    specialize lt_of_lt_of_le (h)
  5. L33
    apply lt_of_lt_of_le
  6. L34
    exact hsmaller_witness_witness_witness_right
  7. L35
    exact hnormalized_witness_witness_witness_right

Library-wide reading audit

Original exact command ledger · 35 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hcounter
  5. 0005have hnormalized : exists A B H. (((((~((A) = 0) /\ (~((B) = 0) /\ (~((H) = 0) /\ ((A) * (A) * (A) * (A) + (B) * (B) * (B) * (B) = (H) * (H)))))) /\ (forall pff_divisor_strict_normalized. (exists pff_left_strict_normalized. (A) = pff_divisor_strict_normalized * pff_left_strict_normalized) -> (exists pff_right_strict_normalized. (B) = pff_divisor_strict_normalized * pff_right_strict_normalized) -> pff_divisor_strict_normalized = 1))) /\ (exists k. k + H = h))
  6. 0006specialize fermat_four_primitive_normalization (a)
  7. 0007specialize fermat_four_primitive_normalization (b)
  8. 0008specialize fermat_four_primitive_normalization (h)
  9. 0009apply fermat_four_primitive_normalization
  10. 0010exact hcounter
  11. 0011cases hnormalized
  12. 0012cases hnormalized_witness
  13. 0013cases hnormalized_witness_witness
  14. 0014cases hnormalized_witness_witness_witness
  15. 0015have hsmaller : exists A B H. ((((~((A) = 0) /\ (~((B) = 0) /\ (~((H) = 0) /\ ((A) * (A) * (A) * (A) + (B) * (B) * (B) * (B) = (H) * (H)))))) /\ (exists ffd_gap_strict_smaller. ffd_gap_strict_smaller + S H = x2)))
  16. 0016specialize fermat_four_primitive_descent (x)
  17. 0017specialize fermat_four_primitive_descent (x1)
  18. 0018specialize fermat_four_primitive_descent (x2)
  19. 0019apply fermat_four_primitive_descent
  20. 0020exact hnormalized_witness_witness_witness_left
  21. 0021cases hsmaller
  22. 0022cases hsmaller_witness
  23. 0023cases hsmaller_witness_witness
  24. 0024cases hsmaller_witness_witness_witness
  25. 0025exists x3
  26. 0026exists x4
  27. 0027exists x5
  28. 0028split
  29. 0029exact hsmaller_witness_witness_witness_left
  30. 0030specialize lt_of_lt_of_le (x5)
  31. 0031specialize lt_of_lt_of_le (x2)
  32. 0032specialize lt_of_lt_of_le (h)
  33. 0033apply lt_of_lt_of_le
  34. 0034exact hsmaller_witness_witness_witness_right
  35. 0035exact hnormalized_witness_witness_witness_right