HD0005

beta_horner_derivative_only_exists

The exact formal derivative of every arbitrary beta-coded natural polynomial exists.

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. ∃ z. HornerDerivativeOnly(b,c,t,l,z)

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

Definition DAG

HornerDerivativeOnly(b,c,t,l,z) · 1

Actual proof prerequisites

beta_horner_derivative_value_exists
Original expanded first-order statement 
forall b c t l. exists z. (exists ff_value_hd_only. (exists ff_u_hd_only_pair ff_v_hd_only_pair ff_d_hd_only_pair ff_e_hd_only_pair. ((((((exists fs_h_ph_hd_only_pair_body_value_start. fs_h_ph_hd_only_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_start. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_start * S ((S (0)) * ff_v_hd_only_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_pair_body_value_terminal. fs_h_ph_hd_only_pair_body_value_terminal + S (ff_value_hd_only) = S ((S (l)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_terminal. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_terminal * S ((S (l)) * ff_v_hd_only_pair) + (ff_value_hd_only))) /\ forall ff_i_ph_hd_only_pair_body_value_steps. (exists ph_bound_hd_only_pair_body_value_steps. ph_bound_hd_only_pair_body_value_steps + S ff_i_ph_hd_only_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_only_pair_body_value_steps ff_previous_ph_hd_only_pair_body_value_steps ff_current_ph_hd_only_pair_body_value_steps. ((((exists fs_h_ph_hd_only_pair_body_value_steps_coefficient. fs_h_ph_hd_only_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_only_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_only_pair_body_value_steps_coefficient. b = fs_q_ph_hd_only_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_only_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_value_steps_before. fs_h_ph_hd_only_pair_body_value_steps_before + S (ff_previous_ph_hd_only_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_steps_before. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_steps_before * S ((S (ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair) + (ff_previous_ph_hd_only_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_value_steps_after. fs_h_ph_hd_only_pair_body_value_steps_after + S (ff_current_ph_hd_only_pair_body_value_steps) = S ((S (S ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_value_steps_after. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_only_pair_body_value_steps)) * ff_v_hd_only_pair) + (ff_current_ph_hd_only_pair_body_value_steps))) /\ ff_current_ph_hd_only_pair_body_value_steps = ff_previous_ph_hd_only_pair_body_value_steps * t + ff_coefficient_ph_hd_only_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_only_pair_body_derivative_start. fs_h_ph_hd_only_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_start. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_start * S ((S (0)) * ff_e_hd_only_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_pair_body_derivative_terminal. fs_h_ph_hd_only_pair_body_derivative_terminal + S (z) = S ((S (l)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_terminal. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_only_pair) + (z))) /\ forall ff_i_ph_hd_only_pair_body_derivative_steps. (exists ph_bound_hd_only_pair_body_derivative_steps. ph_bound_hd_only_pair_body_derivative_steps + S ff_i_ph_hd_only_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_only_pair_body_derivative_steps ff_previous_ph_hd_only_pair_body_derivative_steps ff_current_ph_hd_only_pair_body_derivative_steps. ((((exists fs_h_ph_hd_only_pair_body_derivative_steps_coefficient. fs_h_ph_hd_only_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_only_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_v_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_steps_coefficient. ff_u_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_v_hd_only_pair) + (ff_coefficient_ph_hd_only_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_derivative_steps_before. fs_h_ph_hd_only_pair_body_derivative_steps_before + S (ff_previous_ph_hd_only_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_steps_before. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair) + (ff_previous_ph_hd_only_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_pair_body_derivative_steps_after. fs_h_ph_hd_only_pair_body_derivative_steps_after + S (ff_current_ph_hd_only_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair)) /\ exists fs_q_ph_hd_only_pair_body_derivative_steps_after. ff_d_hd_only_pair = fs_q_ph_hd_only_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_only_pair_body_derivative_steps)) * ff_e_hd_only_pair) + (ff_current_ph_hd_only_pair_body_derivative_steps))) /\ ff_current_ph_hd_only_pair_body_derivative_steps = ff_previous_ph_hd_only_pair_body_derivative_steps * t + ff_coefficient_ph_hd_only_pair_body_derivative_steps)))))))))

Complete unchanged native tactic proof

All 13 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

13 script commands · 5 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)
beta horner derivative value exists
01Fix variables and assumptionsL1–4

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
02Use earlier factsL5–8

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

  1. L5
    specialize beta_horner_derivative_value_exists b
  2. L6
    specialize beta_horner_derivative_value_exists c
  3. L7
    specialize beta_horner_derivative_value_exists t
  4. L8
    specialize beta_horner_derivative_value_exists l
03Separate the logical casesL9–10

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

  1. L9
    cases beta_horner_derivative_value_exists
  2. L10
    cases beta_horner_derivative_value_exists_witness
04Construct an explicit witnessL11–12

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

  1. L11
    exists x1
  2. L12
    exists x
05Use earlier factsL13–13

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

  1. L13
    exact beta_horner_derivative_value_exists_witness_witness

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005specialize beta_horner_derivative_value_exists b
  6. 0006specialize beta_horner_derivative_value_exists c
  7. 0007specialize beta_horner_derivative_value_exists t
  8. 0008specialize beta_horner_derivative_value_exists l
  9. 0009cases beta_horner_derivative_value_exists
  10. 0010cases beta_horner_derivative_value_exists_witness
  11. 0011exists x1
  12. 0012exists x
  13. 0013exact beta_horner_derivative_value_exists_witness_witness