PA00FT

qres_opposite_status_from_odd_half_product_mod_two

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

Modulo-two equality with an odd half product gives opposite residue status.

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.

Exact expanded PA statement

forall p q e f h k. (((((exists qr_x_qrp_pq. exists qr_u_qrp_pq qr_v_qrp_pq. qr_x_qrp_pq * qr_x_qrp_pq + p * qr_u_qrp_pq = q + p * qr_v_qrp_pq) -> (exists qrp_even_e_even. e = 2 * qrp_even_e_even)) /\ ((exists qrp_even_e_even. e = 2 * qrp_even_e_even) -> (exists qr_x_qrp_pq. exists qr_u_qrp_pq qr_v_qrp_pq. qr_x_qrp_pq * qr_x_qrp_pq + p * qr_u_qrp_pq = q + p * qr_v_qrp_pq))) /\ (((~(exists qr_x_qrp_pq. exists qr_u_qrp_pq qr_v_qrp_pq. qr_x_qrp_pq * qr_x_qrp_pq + p * qr_u_qrp_pq = q + p * qr_v_qrp_pq)) -> (exists qrp_odd_e_odd. e = 2 * qrp_odd_e_odd + 1)) /\ ((exists qrp_odd_e_odd. e = 2 * qrp_odd_e_odd + 1) -> ~(exists qr_x_qrp_pq. exists qr_u_qrp_pq qr_v_qrp_pq. qr_x_qrp_pq * qr_x_qrp_pq + p * qr_u_qrp_pq = q + p * qr_v_qrp_pq))))) -> (((((exists qr_x_qrp_qp. exists qr_u_qrp_qp qr_v_qrp_qp. qr_x_qrp_qp * qr_x_qrp_qp + q * qr_u_qrp_qp = p + q * qr_v_qrp_qp) -> (exists qrp_even_f_even. f = 2 * qrp_even_f_even)) /\ ((exists qrp_even_f_even. f = 2 * qrp_even_f_even) -> (exists qr_x_qrp_qp. exists qr_u_qrp_qp qr_v_qrp_qp. qr_x_qrp_qp * qr_x_qrp_qp + q * qr_u_qrp_qp = p + q * qr_v_qrp_qp))) /\ (((~(exists qr_x_qrp_qp. exists qr_u_qrp_qp qr_v_qrp_qp. qr_x_qrp_qp * qr_x_qrp_qp + q * qr_u_qrp_qp = p + q * qr_v_qrp_qp)) -> (exists qrp_odd_f_odd. f = 2 * qrp_odd_f_odd + 1)) /\ ((exists qrp_odd_f_odd. f = 2 * qrp_odd_f_odd + 1) -> ~(exists qr_x_qrp_qp. exists qr_u_qrp_qp qr_v_qrp_qp. qr_x_qrp_qp * qr_x_qrp_qp + q * qr_u_qrp_qp = p + q * qr_v_qrp_qp))))) -> (exists qrp_u_count_product qrp_v_count_product. e + f + 2 * qrp_u_count_product = h * k + 2 * qrp_v_count_product) -> (exists qrp_odd_half_product. h * k = 2 * qrp_odd_half_product + 1) -> ((((exists qr_x_qrp_pq. exists qr_u_qrp_pq qr_v_qrp_pq. qr_x_qrp_pq * qr_x_qrp_pq + p * qr_u_qrp_pq = q + p * qr_v_qrp_pq) /\ ~(exists qr_x_qrp_qp. exists qr_u_qrp_qp qr_v_qrp_qp. qr_x_qrp_qp * qr_x_qrp_qp + q * qr_u_qrp_qp = p + q * qr_v_qrp_qp)) \/ (~(exists qr_x_qrp_pq. exists qr_u_qrp_pq qr_v_qrp_pq. qr_x_qrp_pq * qr_x_qrp_pq + p * qr_u_qrp_pq = q + p * qr_v_qrp_pq) /\ (exists qr_x_qrp_qp. exists qr_u_qrp_qp qr_v_qrp_qp. qr_x_qrp_qp * qr_x_qrp_qp + q * qr_u_qrp_qp = p + q * qr_v_qrp_qp))))

Structural proof guide

Generated structural guide

Modulo-two equality with an odd half product gives opposite residue status.

Use the direct prerequisites mod_two_preserves_parity, qres_opposite_status_from_odd_count_sum as previously established PA formulas.

The proof proceeds by case analysis (2), intermediate claims (2).

Referenced ingredients

PA00FL mod_two_preserves_parity PA00FS qres_opposite_status_from_odd_count_sum

Proof neighborhood

Direct dependencies

PA00FL mod_two_preserves_parity PA00FS qres_opposite_status_from_odd_count_sum

Direct dependents

PA00FU qres_opposite_status_from_mod_four_three

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

28 script commands · 5 reading checkpoints · 2 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.

Named ingredients (2)
mod two preserves parityqres opposite status from odd count sum

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro e
  4. L4
    intro f
  5. L5
    intro h
  6. L6
    intro k
  7. L7
    intro heclass
  8. L8
    intro hfclass
  9. L9
    intro hmod
  10. L10
    intro hproduct
02Establish htransportL11–15

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

  1. L11
    have htransport : (Even(e + f) → Even(h · k)) ∧ (Even(h · k) → Even(e + f)) ∧ ((Odd(e + f) → Odd(h · k)) ∧ (Odd(h · k) → Odd(e + f)))Definitions: EvenOdd
  2. L12
    specialize mod_two_preserves_parity (e + f)
  3. L13
    specialize mod_two_preserves_parity (h * k)
  4. L14
    apply mod_two_preserves_parity
  5. L15
    exact hmod
03Separate the logical casesL16–17

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

  1. L16
    cases htransport
  2. L17
    cases htransport_right
04Establish hcountL18–27

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

  1. L18
    have hcount : exists qrp_odd_transport_count. e + f = 2 * qrp_odd_transport_count + 1
  2. L19
    apply htransport_right_right
  3. L20
    exact hproduct
  4. L21
    specialize qres_opposite_status_from_odd_count_sum p
  5. L22
    specialize qres_opposite_status_from_odd_count_sum q
  6. L23
    specialize qres_opposite_status_from_odd_count_sum e
  7. L24
    specialize qres_opposite_status_from_odd_count_sum f
  8. L25
    apply qres_opposite_status_from_odd_count_sum
  9. L26
    exact heclass
  10. L27
    exact hfclass
05Use earlier factsL28–28

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

  1. L28
    exact hcount

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro e
  4. 0004intro f
  5. 0005intro h
  6. 0006intro k
  7. 0007intro heclass
  8. 0008intro hfclass
  9. 0009intro hmod
  10. 0010intro hproduct
  11. 0011have htransport : (((((exists qrp_even_transport_count. e + f = 2 * qrp_even_transport_count) -> (exists qrp_even_transport_product. h * k = 2 * qrp_even_transport_product)) /\ ((exists qrp_even_transport_product. h * k = 2 * qrp_even_transport_product) -> (exists qrp_even_transport_count. e + f = 2 * qrp_even_transport_count)))) /\ ((((exists qrp_odd_transport_count. e + f = 2 * qrp_odd_transport_count + 1) -> (exists qrp_odd_transport_product. h * k = 2 * qrp_odd_transport_product + 1)) /\ ((exists qrp_odd_transport_product. h * k = 2 * qrp_odd_transport_product + 1) -> (exists qrp_odd_transport_count. e + f = 2 * qrp_odd_transport_count + 1)))))
  12. 0012specialize mod_two_preserves_parity (e + f)
  13. 0013specialize mod_two_preserves_parity (h * k)
  14. 0014apply mod_two_preserves_parity
  15. 0015exact hmod
  16. 0016cases htransport
  17. 0017cases htransport_right
  18. 0018have hcount : exists qrp_odd_transport_count. e + f = 2 * qrp_odd_transport_count + 1
  19. 0019apply htransport_right_right
  20. 0020exact hproduct
  21. 0021specialize qres_opposite_status_from_odd_count_sum p
  22. 0022specialize qres_opposite_status_from_odd_count_sum q
  23. 0023specialize qres_opposite_status_from_odd_count_sum e
  24. 0024specialize qres_opposite_status_from_odd_count_sum f
  25. 0025apply qres_opposite_status_from_odd_count_sum
  26. 0026exact heclass
  27. 0027exact hfclass
  28. 0028exact hcount