PA00FH · theorem

gauss_count_sum_mod_two_from_quotient_sums

Alpha v34 checked-use theorem · independently closed; not Stable

Two oriented count/quotient congruences plus the exact floor-sum identity give e+f == h*k modulo two.

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

∀ e. ∀ f. ∀ Q. ∀ U. ∀ h. ∀ k. ModEq(2,e,Q) → ModEq(2,f,U) → Q + U = h · k → ModEq(2,e + f,h · k)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0008 ModEq

3 occurrences

In local proof propositions

PD0008 ModEq

1 occurrences

Exact expanded native-PA statement
forall e f Q U h k. (exists gcsp_u_e_q gcsp_v_e_q. e + 2 * gcsp_u_e_q = Q + 2 * gcsp_v_e_q) -> (exists gcsp_u_f_u gcsp_v_f_u. f + 2 * gcsp_u_f_u = U + 2 * gcsp_v_f_u) -> Q + U = h * k -> (exists gcsp_u_sum_product gcsp_v_sum_product. e + f + 2 * gcsp_u_sum_product = h * k + 2 * gcsp_v_sum_product)

Proof neighborhood

Direct theorem prerequisites

PA0022 mod_eq_add

Direct theorem dependents

PA00FP conditional_qres_same_status_from_oriented_gauss_counts PA00FV conditional_qres_opposite_status_from_oriented_gauss_counts

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

20 script commands · 3 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)
mod eq add
01Fix variables and assumptionsL1–9

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

  1. L1
    intro e
  2. L2
    intro f
  3. L3
    intro Q
  4. L4
    intro U
  5. L5
    intro h
  6. L6
    intro k
  7. L7
    intro heq
  8. L8
    intro hfu
  9. L9
    intro hsum
02Establish hjoinedL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq add.

  1. L10
    have hjoined : ModEq(2,e + f,Q + U)Definitions: ModEq(2,e + f,Q + U)Original native command in the exact edition
  2. L11
    specialize mod_eq_add 2
  3. L12
    specialize mod_eq_add e
  4. L13
    specialize mod_eq_add Q
  5. L14
    specialize mod_eq_add f
  6. L15
    specialize mod_eq_add U
  7. L16
    apply mod_eq_add
  8. L17
    exact heq
  9. L18
    exact hfu
  10. L19
    rewrite hsum at hjoined
03Use earlier factsL20–20

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

  1. L20
    exact hjoined

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro e
  2. 0002intro f
  3. 0003intro Q
  4. 0004intro U
  5. 0005intro h
  6. 0006intro k
  7. 0007intro heq
  8. 0008intro hfu
  9. 0009intro hsum
  10. 0010have hjoined : ModEq(2,e + f,Q + U)
    Exact native replay linehave hjoined : exists gcsp_u_sum_qu gcsp_v_sum_qu. e + f + 2 * gcsp_u_sum_qu = Q + U + 2 * gcsp_v_sum_qu
  11. 0011specialize mod_eq_add 2
  12. 0012specialize mod_eq_add e
  13. 0013specialize mod_eq_add Q
  14. 0014specialize mod_eq_add f
  15. 0015specialize mod_eq_add U
  16. 0016apply mod_eq_add
  17. 0017exact heq
  18. 0018exact hfu
  19. 0019rewrite hsum at hjoined
  20. 0020exact hjoined