PF001Y

pythagorean_positive_primitive_leg_swap

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

Exact expanded first-order arithmetic 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)))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 18 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

PF000C pythagorean_primitive_leg_swap

Direct dependents

PF0022 pythagorean_positive_primitive_from_parameters

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

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.

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 exact 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