HD0003

beta_horner_derivative_value_projection

The first component of simultaneous formal differentiation is exactly the preexisting polynomial Horner value.

Alpha v34 checked-use · first admitted v24 · independently kernel and Lean verified; not Stable

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.

Historical partial components only: this chapter proves arbitrary natural polynomial values and unique formal derivatives. G095 is now closed in the separate Alpha-v27 hensel-lifting branch for integer polynomials, unrestricted input roots, unique canonical lifts, and every positive prime power. Full G095 proof · Alpha v27

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ t. ∀ l. ∀ n. ∀ z. HornerDerivative(b,c,t,l,n,z) → Horner(b,c,t,l,n)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Horner(b,c,x,ell,z) · 1HornerDerivative(b,c,t,l,n,z) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c t l n z. (exists ff_u_hd_pair ff_v_hd_pair ff_d_hd_pair ff_e_hd_pair. ((((((exists fs_h_ph_hd_pair_body_value_start. fs_h_ph_hd_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_pair)) /\ exists fs_q_ph_hd_pair_body_value_start. ff_u_hd_pair = fs_q_ph_hd_pair_body_value_start * S ((S (0)) * ff_v_hd_pair) + (0))) /\ ((((exists fs_h_ph_hd_pair_body_value_terminal. fs_h_ph_hd_pair_body_value_terminal + S (n) = S ((S (l)) * ff_v_hd_pair)) /\ exists fs_q_ph_hd_pair_body_value_terminal. ff_u_hd_pair = fs_q_ph_hd_pair_body_value_terminal * S ((S (l)) * ff_v_hd_pair) + (n))) /\ forall ff_i_ph_hd_pair_body_value_steps. (exists ph_bound_hd_pair_body_value_steps. ph_bound_hd_pair_body_value_steps + S ff_i_ph_hd_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_pair_body_value_steps ff_previous_ph_hd_pair_body_value_steps ff_current_ph_hd_pair_body_value_steps. ((((exists fs_h_ph_hd_pair_body_value_steps_coefficient. fs_h_ph_hd_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_pair_body_value_steps) = S ((S (ff_i_ph_hd_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_pair_body_value_steps_coefficient. b = fs_q_ph_hd_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_pair_body_value_steps_before. fs_h_ph_hd_pair_body_value_steps_before + S (ff_previous_ph_hd_pair_body_value_steps) = S ((S (ff_i_ph_hd_pair_body_value_steps)) * ff_v_hd_pair)) /\ exists fs_q_ph_hd_pair_body_value_steps_before. ff_u_hd_pair = fs_q_ph_hd_pair_body_value_steps_before * S ((S (ff_i_ph_hd_pair_body_value_steps)) * ff_v_hd_pair) + (ff_previous_ph_hd_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_pair_body_value_steps_after. fs_h_ph_hd_pair_body_value_steps_after + S (ff_current_ph_hd_pair_body_value_steps) = S ((S (S ff_i_ph_hd_pair_body_value_steps)) * ff_v_hd_pair)) /\ exists fs_q_ph_hd_pair_body_value_steps_after. ff_u_hd_pair = fs_q_ph_hd_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_pair_body_value_steps)) * ff_v_hd_pair) + (ff_current_ph_hd_pair_body_value_steps))) /\ ff_current_ph_hd_pair_body_value_steps = ff_previous_ph_hd_pair_body_value_steps * t + ff_coefficient_ph_hd_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_pair_body_derivative_start. fs_h_ph_hd_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_pair)) /\ exists fs_q_ph_hd_pair_body_derivative_start. ff_d_hd_pair = fs_q_ph_hd_pair_body_derivative_start * S ((S (0)) * ff_e_hd_pair) + (0))) /\ ((((exists fs_h_ph_hd_pair_body_derivative_terminal. fs_h_ph_hd_pair_body_derivative_terminal + S (z) = S ((S (l)) * ff_e_hd_pair)) /\ exists fs_q_ph_hd_pair_body_derivative_terminal. ff_d_hd_pair = fs_q_ph_hd_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_pair) + (z))) /\ forall ff_i_ph_hd_pair_body_derivative_steps. (exists ph_bound_hd_pair_body_derivative_steps. ph_bound_hd_pair_body_derivative_steps + S ff_i_ph_hd_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_pair_body_derivative_steps ff_previous_ph_hd_pair_body_derivative_steps ff_current_ph_hd_pair_body_derivative_steps. ((((exists fs_h_ph_hd_pair_body_derivative_steps_coefficient. fs_h_ph_hd_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_pair_body_derivative_steps)) * ff_v_hd_pair)) /\ exists fs_q_ph_hd_pair_body_derivative_steps_coefficient. ff_u_hd_pair = fs_q_ph_hd_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_pair_body_derivative_steps)) * ff_v_hd_pair) + (ff_coefficient_ph_hd_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_pair_body_derivative_steps_before. fs_h_ph_hd_pair_body_derivative_steps_before + S (ff_previous_ph_hd_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_pair_body_derivative_steps)) * ff_e_hd_pair)) /\ exists fs_q_ph_hd_pair_body_derivative_steps_before. ff_d_hd_pair = fs_q_ph_hd_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_pair_body_derivative_steps)) * ff_e_hd_pair) + (ff_previous_ph_hd_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_pair_body_derivative_steps_after. fs_h_ph_hd_pair_body_derivative_steps_after + S (ff_current_ph_hd_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_pair_body_derivative_steps)) * ff_e_hd_pair)) /\ exists fs_q_ph_hd_pair_body_derivative_steps_after. ff_d_hd_pair = fs_q_ph_hd_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_pair_body_derivative_steps)) * ff_e_hd_pair) + (ff_current_ph_hd_pair_body_derivative_steps))) /\ ff_current_ph_hd_pair_body_derivative_steps = ff_previous_ph_hd_pair_body_derivative_steps * t + ff_coefficient_ph_hd_pair_body_derivative_steps)))))))) -> (exists ff_u_ph_hd_value ff_v_ph_hd_value. ((((exists fs_h_ph_hd_value_body_start. fs_h_ph_hd_value_body_start + S (0) = S ((S (0)) * ff_v_ph_hd_value)) /\ exists fs_q_ph_hd_value_body_start. ff_u_ph_hd_value = fs_q_ph_hd_value_body_start * S ((S (0)) * ff_v_ph_hd_value) + (0))) /\ ((((exists fs_h_ph_hd_value_body_terminal. fs_h_ph_hd_value_body_terminal + S (n) = S ((S (l)) * ff_v_ph_hd_value)) /\ exists fs_q_ph_hd_value_body_terminal. ff_u_ph_hd_value = fs_q_ph_hd_value_body_terminal * S ((S (l)) * ff_v_ph_hd_value) + (n))) /\ forall ff_i_ph_hd_value_body_steps. (exists ph_bound_hd_value_body_steps. ph_bound_hd_value_body_steps + S ff_i_ph_hd_value_body_steps = l) -> exists ff_coefficient_ph_hd_value_body_steps ff_previous_ph_hd_value_body_steps ff_current_ph_hd_value_body_steps. ((((exists fs_h_ph_hd_value_body_steps_coefficient. fs_h_ph_hd_value_body_steps_coefficient + S (ff_coefficient_ph_hd_value_body_steps) = S ((S (ff_i_ph_hd_value_body_steps)) * c)) /\ exists fs_q_ph_hd_value_body_steps_coefficient. b = fs_q_ph_hd_value_body_steps_coefficient * S ((S (ff_i_ph_hd_value_body_steps)) * c) + (ff_coefficient_ph_hd_value_body_steps))) /\ ((((exists fs_h_ph_hd_value_body_steps_before. fs_h_ph_hd_value_body_steps_before + S (ff_previous_ph_hd_value_body_steps) = S ((S (ff_i_ph_hd_value_body_steps)) * ff_v_ph_hd_value)) /\ exists fs_q_ph_hd_value_body_steps_before. ff_u_ph_hd_value = fs_q_ph_hd_value_body_steps_before * S ((S (ff_i_ph_hd_value_body_steps)) * ff_v_ph_hd_value) + (ff_previous_ph_hd_value_body_steps))) /\ ((((exists fs_h_ph_hd_value_body_steps_after. fs_h_ph_hd_value_body_steps_after + S (ff_current_ph_hd_value_body_steps) = S ((S (S ff_i_ph_hd_value_body_steps)) * ff_v_ph_hd_value)) /\ exists fs_q_ph_hd_value_body_steps_after. ff_u_ph_hd_value = fs_q_ph_hd_value_body_steps_after * S ((S (S ff_i_ph_hd_value_body_steps)) * ff_v_ph_hd_value) + (ff_current_ph_hd_value_body_steps))) /\ ff_current_ph_hd_value_body_steps = ff_previous_ph_hd_value_body_steps * t + ff_coefficient_ph_hd_value_body_steps))))))

Complete unchanged native tactic proof

All 15 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

15 script commands · 4 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro t
  4. L4
    intro l
  5. L5
    intro n
  6. L6
    intro z
  7. L7
    intro hpair
02Separate the logical casesL8–12

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

  1. L8
    cases hpair
  2. L9
    cases hpair_witness
  3. L10
    cases hpair_witness_witness
  4. L11
    cases hpair_witness_witness_witness
  5. L12
    cases hpair_witness_witness_witness_witness
03Construct an explicit witnessL13–14

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

  1. L13
    exists x
  2. L14
    exists x1
04Use earlier factsL15–15

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

  1. L15
    exact hpair_witness_witness_witness_witness_left

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005intro n
  6. 0006intro z
  7. 0007intro hpair
  8. 0008cases hpair
  9. 0009cases hpair_witness
  10. 0010cases hpair_witness_witness
  11. 0011cases hpair_witness_witness_witness
  12. 0012cases hpair_witness_witness_witness_witness
  13. 0013exists x
  14. 0014exists x1
  15. 0015exact hpair_witness_witness_witness_witness_left