PF001O · theorem body

pythagorean_coprime_square_roots

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Coprimality of two natural squares implies coprimality of their explicit roots.

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

∀ m. ∀ n. Coprime(m · m,n · n)Coprime(m,n)

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

In local proof propositions

none
Exact expanded first-order statement
forall m n. (forall pff_divisor_pi_root_source. (exists pff_left_pi_root_source. (m * m) = pff_divisor_pi_root_source * pff_left_pi_root_source) -> (exists pff_right_pi_root_source. (n * n) = pff_divisor_pi_root_source * pff_right_pi_root_source) -> pff_divisor_pi_root_source = 1) -> (forall pff_divisor_pi_root_result. (exists pff_left_pi_root_result. (m) = pff_divisor_pi_root_result * pff_left_pi_root_result) -> (exists pff_right_pi_root_result. (n) = pff_divisor_pi_root_result * pff_right_pi_root_result) -> pff_divisor_pi_root_result = 1)

Proof neighborhood

Direct theorem prerequisites

Direct theorem dependents

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

12 script commands · 2 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)
01Fix variables and assumptionsL1–6

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro hcop
  4. L4
    intro divisor
  5. L5
    intro hm
  6. L6
    intro hn
02Use earlier factsL7–12

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

  1. L7
    specialize hcop divisor
  2. L8
    apply hcop
  3. L9
    apply pythagorean_parameter_divisor_divides_square
  4. L10
    exact hm
  5. L11
    apply pythagorean_parameter_divisor_divides_square
  6. L12
    exact hn

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro hcop
  4. 0004intro divisor
  5. 0005intro hm
  6. 0006intro hn
  7. 0007specialize hcop divisor
  8. 0008apply hcop
  9. 0009apply pythagorean_parameter_divisor_divides_square
  10. 0010exact hm
  11. 0011apply pythagorean_parameter_divisor_divides_square
  12. 0012exact hn