TS003K

two_square_common_divisor_extracts_squared_factor

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Two witnessed coordinate divisors provide both quotient coordinates and the exact squared-factor norm identity.

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 p a b. (exists ftcn_factor_ftsv_first_divides. (a) = (p) * ftcn_factor_ftsv_first_divides) -> (exists ftcn_factor_ftsv_second_divides. (b) = (p) * ftcn_factor_ftsv_second_divides) -> (exists ftsv_first_extracted ftsv_second_extracted. ((a = p * ftsv_first_extracted) /\ ((b = p * ftsv_second_extracted) /\ (a * a + b * b = (p * p) * (ftsv_first_extracted * ftsv_first_extracted + ftsv_second_extracted * ftsv_second_extracted)))))

Constructive proof overview

Generated structural guide

Two witnessed coordinate divisors provide both quotient coordinates and the exact squared-factor norm identity.

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

TS003J two_square_common_factor_norm_identity

Direct dependents

TS003L two_square_common_squared_factor_divides_norm TS003N three_mod_four_prime_two_square_norm_extracts_squared_factor

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 · 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.

Named ingredients (1)
two square common factor norm identity
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hfirst
  5. L5
    intro hsecond
02Separate the logical casesL6–7

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

  1. L6
    cases hfirst
  2. L7
    cases hsecond
03Construct an explicit witnessL8–9

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists x
  2. L9
    exists x1
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 hfirst_witness
06Separate the logical casesL12–12

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

  1. L12
    split
07Use earlier factsL13–13

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

  1. L13
    exact hsecond_witness
08Calculate and transport equalitiesL14–17

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

  1. L14
    rewrite hfirst_witness
  2. L15
    rewrite hfirst_witness
  3. L16
    rewrite hsecond_witness
  4. L17
    rewrite hsecond_witness
09Use earlier factsL18–18

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

  1. L18
    apply two_square_common_factor_norm_identity

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hfirst
  5. 0005intro hsecond
  6. 0006cases hfirst
  7. 0007cases hsecond
  8. 0008exists x
  9. 0009exists x1
  10. 0010split
  11. 0011exact hfirst_witness
  12. 0012split
  13. 0013exact hsecond_witness
  14. 0014rewrite hfirst_witness
  15. 0015rewrite hfirst_witness
  16. 0016rewrite hsecond_witness
  17. 0017rewrite hsecond_witness
  18. 0018apply two_square_common_factor_norm_identity