TS0037 · theorem body

three_mod_four_number_not_equal_represented

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

A number congruent to three modulo four cannot equal any explicitly represented two-square number.

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. ∀ q. Mod4Three(p) → (∃ x. ∃ y. q = x · x + y · y) → ¬p = 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

PD0012 Mod4Three

In local proof propositions

none
Exact expanded first-order statement
forall p q. (exists ftsc_four_three_ftsp_bad_distinct. (p) = 4 * ftsc_four_three_ftsp_bad_distinct + 3) -> (exists ftsc_first_ftsp_good_prime ftsc_second_ftsp_good_prime. (q) = ftsc_first_ftsp_good_prime * ftsc_first_ftsp_good_prime + ftsc_second_ftsp_good_prime * ftsc_second_ftsp_good_prime) -> ~(p = q)

Proof neighborhood

Direct theorem prerequisites

TS0004 sum_two_squares_not_four_mod_three

Direct theorem dependents

TS0038 all_bad_prime_even_valuations_strip_represented_prime

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

19 script commands · 8 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)
sum two squares not four mod three
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro hthree
  4. L4
    intro hrepresented
  5. L5
    intro hequal
02Separate the logical casesL6–8

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

  1. L6
    cases hthree
  2. L7
    cases hrepresented
  3. L8
    cases hrepresented_witness
03Use earlier factsL9–11

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

  1. L9
    specialize sum_two_squares_not_four_mod_three x1
  2. L10
    specialize sum_two_squares_not_four_mod_three x2
  3. L11
    apply sum_two_squares_not_four_mod_three
04Construct an explicit witnessL12–12

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

  1. L12
    exists x
05Calculate and transport equalitiesL13–14

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

  1. L13
    trans q
  2. L14
    symm
06Use earlier factsL15–15

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

  1. L15
    exact hrepresented_witness_witness
07Calculate and transport equalitiesL16–17

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

  1. L16
    trans p
  2. L17
    symm
08Use earlier factsL18–19

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

  1. L18
    exact hequal
  2. L19
    exact hthree_witness

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro hthree
  4. 0004intro hrepresented
  5. 0005intro hequal
  6. 0006cases hthree
  7. 0007cases hrepresented
  8. 0008cases hrepresented_witness
  9. 0009specialize sum_two_squares_not_four_mod_three x1
  10. 0010specialize sum_two_squares_not_four_mod_three x2
  11. 0011apply sum_two_squares_not_four_mod_three
  12. 0012exists x
  13. 0013trans q
  14. 0014symm
  15. 0015exact hrepresented_witness_witness
  16. 0016trans p
  17. 0017symm
  18. 0018exact hequal
  19. 0019exact hthree_witness