TS003K · theorem body

two_square_common_divisor_extracts_squared_factor

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ p. ∀ a. ∀ b. Dvd(p,a) → Dvd(p,b) → ∃ x. ∃ y. a = p · x ∧ (b = p · y ∧ a · a + b · b = p · p · (x · x + y · y))

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 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)))))

Proof neighborhood

Direct theorem prerequisites

TS003J two_square_common_factor_norm_identity

Direct theorem dependents

TS003L two_square_common_squared_factor_divides_norm TS003N three_mod_four_prime_two_square_norm_extracts_squared_factor

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 · 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.

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 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 defined 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