PF001Y · theorem body

pythagorean_positive_primitive_leg_swap

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

Swapping positive primitive legs preserves the full exact positive triple definition.

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. ∀ c. PrimitiveTriple(a,b,c) → PrimitiveTriple(b,a,c)

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

ND0069 PrimitiveTriple

In local proof propositions

none
Exact expanded first-order statement
forall a b c. ((~((a) = 0) /\ (~((b) = 0) /\ (~((c) = 0) /\ ((((a) * (a) + (b) * (b) = (c) * (c)) /\ (forall pff_divisor_pi_swap_source. (exists pff_left_pi_swap_source. (a) = pff_divisor_pi_swap_source * pff_left_pi_swap_source) -> (exists pff_right_pi_swap_source. (b) = pff_divisor_pi_swap_source * pff_right_pi_swap_source) -> pff_divisor_pi_swap_source = 1))))))) -> ((~((b) = 0) /\ (~((a) = 0) /\ (~((c) = 0) /\ ((((b) * (b) + (a) * (a) = (c) * (c)) /\ (forall pff_divisor_pi_swap_result. (exists pff_left_pi_swap_result. (b) = pff_divisor_pi_swap_result * pff_left_pi_swap_result) -> (exists pff_right_pi_swap_result. (a) = pff_divisor_pi_swap_result * pff_right_pi_swap_result) -> pff_divisor_pi_swap_result = 1)))))))

Proof neighborhood

Direct theorem prerequisites

PF000C pythagorean_primitive_leg_swap

Direct theorem dependents

PF0022 pythagorean_positive_primitive_from_parameters

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

18 script commands · 7 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)
pythagorean primitive leg swap
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 c
  4. L4
    intro hp
02Separate the logical casesL5–8

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

  1. L5
    cases hp
  2. L6
    cases hp_right
  3. L7
    cases hp_right_right
  4. L8
    split
03Use earlier factsL9–9

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

  1. L9
    exact hp_right_left
04Separate the logical casesL10–10

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

  1. L10
    split
05Use earlier factsL11–11

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

  1. L11
    exact hp_left
06Separate the logical casesL12–12

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

  1. L12
    split
07Use earlier factsL13–18

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

  1. L13
    exact hp_right_right_left
  2. L14
    specialize pythagorean_primitive_leg_swap a
  3. L15
    specialize pythagorean_primitive_leg_swap b
  4. L16
    specialize pythagorean_primitive_leg_swap c
  5. L17
    apply pythagorean_primitive_leg_swap
  6. L18
    exact hp_right_right_right

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro hp
  5. 0005cases hp
  6. 0006cases hp_right
  7. 0007cases hp_right_right
  8. 0008split
  9. 0009exact hp_right_left
  10. 0010split
  11. 0011exact hp_left
  12. 0012split
  13. 0013exact hp_right_right_left
  14. 0014specialize pythagorean_primitive_leg_swap a
  15. 0015specialize pythagorean_primitive_leg_swap b
  16. 0016specialize pythagorean_primitive_leg_swap c
  17. 0017apply pythagorean_primitive_leg_swap
  18. 0018exact hp_right_right_right