PF002J · theorem body

fermat_four_primitive_counterexample_swap

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

Swapping the positive fourth-power bases preserves the primitive counterexample relation.

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) → PrimitiveFermatFourCounterexample(b,a,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

In local proof propositions

none
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_swap_source. (exists pff_left_swap_source. (a) = pff_divisor_swap_source * pff_left_swap_source) -> (exists pff_right_swap_source. (b) = pff_divisor_swap_source * pff_right_swap_source) -> pff_divisor_swap_source = 1))) -> ((((~((b) = 0) /\ (~((a) = 0) /\ (~((h) = 0) /\ ((b) * (b) * (b) * (b) + (a) * (a) * (a) * (a) = (h) * (h)))))) /\ (forall pff_divisor_swap_result. (exists pff_left_swap_result. (b) = pff_divisor_swap_result * pff_left_swap_result) -> (exists pff_right_swap_result. (a) = pff_divisor_swap_result * pff_right_swap_result) -> pff_divisor_swap_result = 1)))

Proof neighborhood

Direct theorem prerequisites

add_comm · Stable closed coprime_symm · Stable closed

Direct theorem dependents

PF002L fermat_four_primitive_descent

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

20 script commands · 9 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.

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
02Separate the logical casesL5–10

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

  1. L5
    cases hprimitive
  2. L6
    cases hprimitive_left
  3. L7
    cases hprimitive_left_right
  4. L8
    cases hprimitive_left_right_right
  5. L9
    split
  6. L10
    split
03Use earlier factsL11–11

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

  1. L11
    exact hprimitive_left_right_left
04Separate the logical casesL12–12

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

  1. L12
    split
05Use earlier factsL13–13

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

  1. L13
    exact hprimitive_left_left
06Separate the logical casesL14–14

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

  1. L14
    split
07Use earlier factsL15–15

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

  1. L15
    exact hprimitive_left_right_right_left
08Calculate and transport equalitiesL16–16

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L16
    trans a * a * a * a + b * b * b * b
09Use earlier factsL17–20

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

  1. L17
    apply add_comm
  2. L18
    exact hprimitive_left_right_right_right
  3. L19
    apply coprime_symm
  4. L20
    exact hprimitive_right

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hprimitive
  5. 0005cases hprimitive
  6. 0006cases hprimitive_left
  7. 0007cases hprimitive_left_right
  8. 0008cases hprimitive_left_right_right
  9. 0009split
  10. 0010split
  11. 0011exact hprimitive_left_right_left
  12. 0012split
  13. 0013exact hprimitive_left_left
  14. 0014split
  15. 0015exact hprimitive_left_right_right_left
  16. 0016trans a * a * a * a + b * b * b * b
  17. 0017apply add_comm
  18. 0018exact hprimitive_left_right_right_right
  19. 0019apply coprime_symm
  20. 0020exact hprimitive_right