FS0021 · theorem body

four_square_descent_nonunit_proper_factor_not_prime

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

A prime has no divisor k that is simultaneously nonunit and strictly smaller than the prime.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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. ∀ k. ∀ q. Prime(p) → ¬k = 1 → Lt(k,p) → ¬p = k · q

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

PD0002 Lt PD0004 Prime

In local proof propositions

none
Exact expanded first-order statement
forall p k q. ((~(p = 1) /\ forall frm_prime_left_fsd_p frm_prime_right_fsd_p. p = frm_prime_left_fsd_p * frm_prime_right_fsd_p -> frm_prime_left_fsd_p = 1 \/ frm_prime_right_fsd_p = 1)) -> ~(k = 1) -> (exists gap. gap + S k = p) -> ~(p = k * q)

Proof neighborhood

Direct theorem prerequisites

mul_one · Stable closed lt_irrefl_expanded · Stable closed

Direct theorem dependents

FS0023 four_square_descent_bounded_centered_quotient_nonzero

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

23 script commands · 10 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro q
  4. L4
    intro hprime
  5. L5
    intro hnonunit
  6. L6
    intro hproper
  7. L7
    intro hfactor
02Separate the logical casesL8–8

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

  1. L8
    cases hprime
03Use earlier factsL9–10

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

  1. L9
    specialize hprime_right k
  2. L10
    specialize hprime_right q
04Establish hcasesL11–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprime right.

  1. L11
    have hcases : k = 1 \/ q = 1
  2. L12
    apply hprime_right
  3. L13
    exact hfactor
05Separate the logical casesL14–14

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

  1. L14
    cases hcases
06Use earlier factsL15–16

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

  1. L15
    apply hnonunit
  2. L16
    exact hcases_left
07Calculate and transport equalitiesL17–17

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

  1. L17
    rewrite hcases_right at hfactor
08Use earlier factsL18–18

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

  1. L18
    specialize mul_one k
09Calculate and transport equalitiesL19–20

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

  1. L19
    rewrite mul_one at hfactor
  2. L20
    rewrite hfactor at hproper
10Use earlier factsL21–23

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

  1. L21
    specialize lt_irrefl_expanded k
  2. L22
    apply lt_irrefl_expanded
  3. L23
    exact hproper

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro q
  4. 0004intro hprime
  5. 0005intro hnonunit
  6. 0006intro hproper
  7. 0007intro hfactor
  8. 0008cases hprime
  9. 0009specialize hprime_right k
  10. 0010specialize hprime_right q
  11. 0011have hcases : k = 1 \/ q = 1
  12. 0012apply hprime_right
  13. 0013exact hfactor
  14. 0014cases hcases
  15. 0015apply hnonunit
  16. 0016exact hcases_left
  17. 0017rewrite hcases_right at hfactor
  18. 0018specialize mul_one k
  19. 0019rewrite mul_one at hfactor
  20. 0020rewrite hfactor at hproper
  21. 0021specialize lt_irrefl_expanded k
  22. 0022apply lt_irrefl_expanded
  23. 0023exact hproper