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_classification_positive. (exists pff_left_pi_classification_positive. (a) = pff_divisor_pi_classification_positive * pff_left_pi_classification_positive) -> (exists pff_right_pi_classification_positive. (b) = pff_divisor_pi_classification_positive * pff_right_pi_classification_positive) -> pff_divisor_pi_classification_positive = 1))))))) -> (exists pi_larger_classification_parameters pi_smaller_classification_parameters. ((~((pi_smaller_classification_parameters) = 0) /\ ((exists pi_gap_classification_parameters. pi_gap_classification_parameters + S (pi_smaller_classification_parameters) = (pi_larger_classification_parameters)) /\ ((forall pff_divisor_pi_classification_parameters_coprime. (exists pff_left_pi_classification_parameters_coprime. (pi_larger_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_left_pi_classification_parameters_coprime) -> (exists pff_right_pi_classification_parameters_coprime. (pi_smaller_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_right_pi_classification_parameters_coprime) -> pff_divisor_pi_classification_parameters_coprime = 1) /\ (((((exists pp_even_pi_classification_parameters_parity_first_even. (pi_larger_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_first_even) /\ (exists pp_odd_pi_classification_parameters_parity_second_odd. (pi_smaller_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_second_odd + 1)) \/ ((exists pp_odd_pi_classification_parameters_parity_first_odd. (pi_larger_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_first_odd + 1) /\ (exists pp_even_pi_classification_parameters_parity_second_even. (pi_smaller_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_second_even)))) /\ ((c) = (pi_larger_classification_parameters) * (pi_larger_classification_parameters) + (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) /\ (((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (a) /\ (b) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters))) \/ ((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (b) /\ (a) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters)))))))))))) /\ ((exists pi_larger_classification_parameters pi_smaller_classification_parameters. ((~((pi_smaller_classification_parameters) = 0) /\ ((exists pi_gap_classification_parameters. pi_gap_classification_parameters + S (pi_smaller_classification_parameters) = (pi_larger_classification_parameters)) /\ ((forall pff_divisor_pi_classification_parameters_coprime. (exists pff_left_pi_classification_parameters_coprime. (pi_larger_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_left_pi_classification_parameters_coprime) -> (exists pff_right_pi_classification_parameters_coprime. (pi_smaller_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_right_pi_classification_parameters_coprime) -> pff_divisor_pi_classification_parameters_coprime = 1) /\ (((((exists pp_even_pi_classification_parameters_parity_first_even. (pi_larger_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_first_even) /\ (exists pp_odd_pi_classification_parameters_parity_second_odd. (pi_smaller_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_second_odd + 1)) \/ ((exists pp_odd_pi_classification_parameters_parity_first_odd. (pi_larger_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_first_odd + 1) /\ (exists pp_even_pi_classification_parameters_parity_second_even. (pi_smaller_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_second_even)))) /\ ((c) = (pi_larger_classification_parameters) * (pi_larger_classification_parameters) + (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) /\ (((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (a) /\ (b) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters))) \/ ((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (b) /\ (a) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters))))))))))) -> ((~((a) = 0) /\ (~((b) = 0) /\ (~((c) = 0) /\ ((((a) * (a) + (b) * (b) = (c) * (c)) /\ (forall pff_divisor_pi_classification_positive. (exists pff_left_pi_classification_positive. (a) = pff_divisor_pi_classification_positive * pff_left_pi_classification_positive) -> (exists pff_right_pi_classification_positive. (b) = pff_divisor_pi_classification_positive * pff_right_pi_classification_positive) -> pff_divisor_pi_classification_positive = 1)))))))))Constructive proof overview
Generated structural guide
Complete constructive classification: a positive ordered triple is primitive Pythagorean if and only if it has positive strictly ordered coprime opposite-parity Euclid parameters in either leg orientation.
The unchanged tactic script uses 2 declared prerequisites and contains 12 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
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
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)
01Fix variables and assumptionsL1–3
02Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
split
03Use earlier factsL5–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L5
specialize pythagorean_positive_primitive_inverse a - L6
specialize pythagorean_positive_primitive_inverse b - L7
specialize pythagorean_positive_primitive_inverse c - L8
exact pythagorean_positive_primitive_inverse - L9
specialize pythagorean_positive_primitive_from_parameters a - L10
specialize pythagorean_positive_primitive_from_parameters b - L11
specialize pythagorean_positive_primitive_from_parameters c - L12
exact pythagorean_positive_primitive_from_parameters
Original exact command ledger · 12 lines
- 0001
intro a - 0002
intro b - 0003
intro c - 0004
split - 0005
specialize pythagorean_positive_primitive_inverse a - 0006
specialize pythagorean_positive_primitive_inverse b - 0007
specialize pythagorean_positive_primitive_inverse c - 0008
exact pythagorean_positive_primitive_inverse - 0009
specialize pythagorean_positive_primitive_from_parameters a - 0010
specialize pythagorean_positive_primitive_from_parameters b - 0011
specialize pythagorean_positive_primitive_from_parameters c - 0012
exact pythagorean_positive_primitive_from_parameters