PF000U — pythagorean_primitive_euclidean_from_order

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 m n. (exists gap. gap + n = m) -> (forall pff_divisor_primitive_parameters. (exists pff_left_primitive_parameters. (m) = pff_divisor_primitive_parameters * pff_left_primitive_parameters) -> (exists pff_right_primitive_parameters. (n) = pff_divisor_primitive_parameters * pff_right_primitive_parameters) -> pff_divisor_primitive_parameters = 1) -> ((((exists pp_even_primitive_parameters_first_even. (m) = 2 * pp_even_primitive_parameters_first_even) /\ (exists pp_odd_primitive_parameters_second_odd. (n) = 2 * pp_odd_primitive_parameters_second_odd + 1)) \/ ((exists pp_odd_primitive_parameters_first_odd. (m) = 2 * pp_odd_primitive_parameters_first_odd + 1) /\ (exists pp_even_primitive_parameters_second_even. (n) = 2 * pp_even_primitive_parameters_second_even)))) -> exists d. ((((d) * (d) + (2 * (m * n)) * (2 * (m * n)) = (m * m + n * n) * (m * m + n * n)) /\ (forall pff_divisor_euclidean_result. (exists pff_left_euclidean_result. (d) = pff_divisor_euclidean_result * pff_left_euclidean_result) -> (exists pff_right_euclidean_result. (2 * (m * n)) = pff_divisor_euclidean_result * pff_right_euclidean_result) -> pff_divisor_euclidean_result = 1)))

Constructive proof overview

Generated structural guide

Every witnessed ordered pair of coprime opposite-parity natural parameters constructs an actual primitive Euclidean Pythagorean triple.

The unchanged tactic script uses 2 declared prerequisites and contains 19 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

PF0008 pythagorean_square_gap_from_order PF000T pythagorean_primitive_euclidean_constructor

Direct dependents

none

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

19 script commands · 5 reading checkpoints · 1 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)

pythagorean square gap from order pythagorean primitive euclidean constructor

01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.