TS003L · theorem body

two_square_common_squared_factor_divides_norm

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

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.

Statement with defined notation

∀ p. ∀ a. ∀ b. Dvd(p,a) → Dvd(p,b) → Dvd(p · p,a · a + b · b)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0003 Dvd

In local proof propositions

none
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

TS003K two_square_common_divisor_extracts_squared_factor

Direct theorem dependents

none

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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