MD0004

beta_dot_product_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Two arbitrary finite coded vectors have an exactly witnessed dot product.

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 first-order arithmetic statement

forall mb mc sb sc l. exists n. (exists ff_code_dot_value ff_scale_dot_value. ((forall fpmp_index_dot_value_pointwise fpmp_left_dot_value_pointwise fpmp_right_dot_value_pointwise fpmp_target_dot_value_pointwise. (exists fpmp_gap_dot_value_pointwise. fpmp_gap_dot_value_pointwise + S fpmp_index_dot_value_pointwise = l) -> (((exists ff_h_fpmp_dot_value_pointwise_left. ff_h_fpmp_dot_value_pointwise_left + S (fpmp_left_dot_value_pointwise) = S ((S (fpmp_index_dot_value_pointwise)) * mc)) /\ exists ff_q_fpmp_dot_value_pointwise_left. mb = ff_q_fpmp_dot_value_pointwise_left * S ((S (fpmp_index_dot_value_pointwise)) * mc) + (fpmp_left_dot_value_pointwise))) -> (((exists ff_h_fpmp_dot_value_pointwise_right. ff_h_fpmp_dot_value_pointwise_right + S (fpmp_right_dot_value_pointwise) = S ((S (fpmp_index_dot_value_pointwise)) * sc)) /\ exists ff_q_fpmp_dot_value_pointwise_right. sb = ff_q_fpmp_dot_value_pointwise_right * S ((S (fpmp_index_dot_value_pointwise)) * sc) + (fpmp_right_dot_value_pointwise))) -> (((exists ff_h_fpmp_dot_value_pointwise_target. ff_h_fpmp_dot_value_pointwise_target + S (fpmp_target_dot_value_pointwise) = S ((S (fpmp_index_dot_value_pointwise)) * ff_scale_dot_value)) /\ exists ff_q_fpmp_dot_value_pointwise_target. ff_code_dot_value = ff_q_fpmp_dot_value_pointwise_target * S ((S (fpmp_index_dot_value_pointwise)) * ff_scale_dot_value) + (fpmp_target_dot_value_pointwise))) -> fpmp_target_dot_value_pointwise = fpmp_left_dot_value_pointwise * fpmp_right_dot_value_pointwise) /\ (exists ff_u_dot_value_sum ff_v_dot_value_sum. ((((exists ff_h_dot_value_sum_start. ff_h_dot_value_sum_start + S (0) = S ((S (0)) * ff_v_dot_value_sum)) /\ exists ff_q_dot_value_sum_start. ff_u_dot_value_sum = ff_q_dot_value_sum_start * S ((S (0)) * ff_v_dot_value_sum) + (0))) /\ ((((exists ff_h_dot_value_sum_terminal. ff_h_dot_value_sum_terminal + S (n) = S ((S (l)) * ff_v_dot_value_sum)) /\ exists ff_q_dot_value_sum_terminal. ff_u_dot_value_sum = ff_q_dot_value_sum_terminal * S ((S (l)) * ff_v_dot_value_sum) + (n))) /\ forall ff_i_dot_value_sum. (exists ff_lt_dot_value_sum_bound. ff_lt_dot_value_sum_bound + S ff_i_dot_value_sum = l) -> exists ff_a_dot_value_sum ff_r_dot_value_sum ff_s_dot_value_sum. ((((exists ff_h_dot_value_sum_summand. ff_h_dot_value_sum_summand + S (ff_a_dot_value_sum) = S ((S (ff_i_dot_value_sum)) * ff_scale_dot_value)) /\ exists ff_q_dot_value_sum_summand. ff_code_dot_value = ff_q_dot_value_sum_summand * S ((S (ff_i_dot_value_sum)) * ff_scale_dot_value) + (ff_a_dot_value_sum))) /\ ((((exists ff_h_dot_value_sum_partial. ff_h_dot_value_sum_partial + S (ff_r_dot_value_sum) = S ((S (ff_i_dot_value_sum)) * ff_v_dot_value_sum)) /\ exists ff_q_dot_value_sum_partial. ff_u_dot_value_sum = ff_q_dot_value_sum_partial * S ((S (ff_i_dot_value_sum)) * ff_v_dot_value_sum) + (ff_r_dot_value_sum))) /\ ((((exists ff_h_dot_value_sum_successor. ff_h_dot_value_sum_successor + S (ff_s_dot_value_sum) = S ((S (S ff_i_dot_value_sum)) * ff_v_dot_value_sum)) /\ exists ff_q_dot_value_sum_successor. ff_u_dot_value_sum = ff_q_dot_value_sum_successor * S ((S (S ff_i_dot_value_sum)) * ff_v_dot_value_sum) + (ff_s_dot_value_sum))) /\ ff_s_dot_value_sum = ff_r_dot_value_sum + ff_a_dot_value_sum))))))))

Constructive proof overview

Generated structural guide

Two arbitrary finite coded vectors have an exactly witnessed dot product.

The unchanged tactic script uses 2 declared prerequisites and contains 22 exact native proof lines.

Alpha v34 checked-use · first admitted v20 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_pointwise_mul_prefix_exists Alpha theorem; checked-use authorized beta_sum_exists Stable theorem; checked-use authorized

Direct dependents

MD0006 beta_dot_product_exists_unique

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

22 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro mb
  2. L2
    intro mc
  3. L3
    intro sb
  4. L4
    intro sc
  5. L5
    intro l
02Use earlier factsL6–10

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

  1. L6
    specialize beta_pointwise_mul_prefix_exists mb
  2. L7
    specialize beta_pointwise_mul_prefix_exists mc
  3. L8
    specialize beta_pointwise_mul_prefix_exists sb
  4. L9
    specialize beta_pointwise_mul_prefix_exists sc
  5. L10
    specialize beta_pointwise_mul_prefix_exists l
03Separate the logical casesL11–12

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

  1. L11
    cases beta_pointwise_mul_prefix_exists
  2. L12
    cases beta_pointwise_mul_prefix_exists_witness
04Use earlier factsL13–15

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

  1. L13
    specialize beta_sum_exists x
  2. L14
    specialize beta_sum_exists x1
  3. L15
    specialize beta_sum_exists l
05Separate the logical casesL16–16

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

  1. L16
    cases beta_sum_exists
06Construct an explicit witnessL17–19

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

  1. L17
    exists x2
  2. L18
    exists x
  3. L19
    exists x1
07Separate the logical casesL20–20

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

  1. L20
    split
08Use earlier factsL21–22

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

  1. L21
    exact beta_pointwise_mul_prefix_exists_witness_witness
  2. L22
    exact beta_sum_exists_witness

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro mb
  2. 0002intro mc
  3. 0003intro sb
  4. 0004intro sc
  5. 0005intro l
  6. 0006specialize beta_pointwise_mul_prefix_exists mb
  7. 0007specialize beta_pointwise_mul_prefix_exists mc
  8. 0008specialize beta_pointwise_mul_prefix_exists sb
  9. 0009specialize beta_pointwise_mul_prefix_exists sc
  10. 0010specialize beta_pointwise_mul_prefix_exists l
  11. 0011cases beta_pointwise_mul_prefix_exists
  12. 0012cases beta_pointwise_mul_prefix_exists_witness
  13. 0013specialize beta_sum_exists x
  14. 0014specialize beta_sum_exists x1
  15. 0015specialize beta_sum_exists l
  16. 0016cases beta_sum_exists
  17. 0017exists x2
  18. 0018exists x
  19. 0019exists x1
  20. 0020split
  21. 0021exact beta_pointwise_mul_prefix_exists_witness_witness
  22. 0022exact beta_sum_exists_witness

Separate complete second-wave branches: Full T13 proof · Alpha v27.