TS003L

two_square_common_squared_factor_divides_norm

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Any common coordinate divisor has its square as an actual divisor of the two-square norm.

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 ftcn_factor_ftsv_square_divides. (a * a + b * b) = (p * p) * ftcn_factor_ftsv_square_divides)

Constructive proof overview

Generated structural guide

Any common coordinate divisor has its square as an actual divisor of the two-square norm.

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

TS003K two_square_common_divisor_extracts_squared_factor

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

18 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 (1)
two square common divisor extracts squared factor
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
02Establish hextractionL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply two square common divisor extracts squared factor.

  1. L6
    have hextraction : 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))))
  2. L7
    specialize two_square_common_divisor_extracts_squared_factor p
  3. L8
    specialize two_square_common_divisor_extracts_squared_factor a
  4. L9
    specialize two_square_common_divisor_extracts_squared_factor b
  5. L10
    apply two_square_common_divisor_extracts_squared_factor
  6. L11
    exact hfirst
  7. L12
    exact hsecond
03Separate the logical casesL13–16

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

  1. L13
    cases hextraction
  2. L14
    cases hextraction_witness
  3. L15
    cases hextraction_witness_witness
  4. L16
    cases hextraction_witness_witness_right
04Construct an explicit witnessL17–17

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

  1. L17
    exists x * x + x1 * x1
05Use earlier factsL18–18

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

  1. L18
    exact hextraction_witness_witness_right_right

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. 0006have hextraction : 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))))
  7. 0007specialize two_square_common_divisor_extracts_squared_factor p
  8. 0008specialize two_square_common_divisor_extracts_squared_factor a
  9. 0009specialize two_square_common_divisor_extracts_squared_factor b
  10. 0010apply two_square_common_divisor_extracts_squared_factor
  11. 0011exact hfirst
  12. 0012exact hsecond
  13. 0013cases hextraction
  14. 0014cases hextraction_witness
  15. 0015cases hextraction_witness_witness
  16. 0016cases hextraction_witness_witness_right
  17. 0017exists x * x + x1 * x1
  18. 0018exact hextraction_witness_witness_right_right