HD000C

beta_horner_derivative_only_functional

The exact formal derivative, considered without its accompanying polynomial value, is functional.

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. ∀ w. HornerDerivativeOnly(b,c,t,l,z) → HornerDerivativeOnly(b,c,t,l,w) → z = w

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

Definition DAG

HornerDerivativeOnly(b,c,t,l,z) · 2

Actual proof prerequisites

beta_horner_derivative_second_component_functional
Original expanded first-order statement 
forall b c t l z w. (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))))))))) -> (exists ff_value_hd_only_other. (exists ff_u_hd_only_other_pair ff_v_hd_only_other_pair ff_d_hd_only_other_pair ff_e_hd_only_other_pair. ((((((exists fs_h_ph_hd_only_other_pair_body_value_start. fs_h_ph_hd_only_other_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_start. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_start * S ((S (0)) * ff_v_hd_only_other_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_value_terminal. fs_h_ph_hd_only_other_pair_body_value_terminal + S (ff_value_hd_only_other) = S ((S (l)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_terminal. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_terminal * S ((S (l)) * ff_v_hd_only_other_pair) + (ff_value_hd_only_other))) /\ forall ff_i_ph_hd_only_other_pair_body_value_steps. (exists ph_bound_hd_only_other_pair_body_value_steps. ph_bound_hd_only_other_pair_body_value_steps + S ff_i_ph_hd_only_other_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_only_other_pair_body_value_steps ff_previous_ph_hd_only_other_pair_body_value_steps ff_current_ph_hd_only_other_pair_body_value_steps. ((((exists fs_h_ph_hd_only_other_pair_body_value_steps_coefficient. fs_h_ph_hd_only_other_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_only_other_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_only_other_pair_body_value_steps_coefficient. b = fs_q_ph_hd_only_other_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_only_other_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_value_steps_before. fs_h_ph_hd_only_other_pair_body_value_steps_before + S (ff_previous_ph_hd_only_other_pair_body_value_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_steps_before. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_steps_before * S ((S (ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair) + (ff_previous_ph_hd_only_other_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_value_steps_after. fs_h_ph_hd_only_other_pair_body_value_steps_after + S (ff_current_ph_hd_only_other_pair_body_value_steps) = S ((S (S ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_value_steps_after. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_only_other_pair_body_value_steps)) * ff_v_hd_only_other_pair) + (ff_current_ph_hd_only_other_pair_body_value_steps))) /\ ff_current_ph_hd_only_other_pair_body_value_steps = ff_previous_ph_hd_only_other_pair_body_value_steps * t + ff_coefficient_ph_hd_only_other_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_only_other_pair_body_derivative_start. fs_h_ph_hd_only_other_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_start. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_start * S ((S (0)) * ff_e_hd_only_other_pair) + (0))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_derivative_terminal. fs_h_ph_hd_only_other_pair_body_derivative_terminal + S (w) = S ((S (l)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_terminal. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_only_other_pair) + (w))) /\ forall ff_i_ph_hd_only_other_pair_body_derivative_steps. (exists ph_bound_hd_only_other_pair_body_derivative_steps. ph_bound_hd_only_other_pair_body_derivative_steps + S ff_i_ph_hd_only_other_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_only_other_pair_body_derivative_steps ff_previous_ph_hd_only_other_pair_body_derivative_steps ff_current_ph_hd_only_other_pair_body_derivative_steps. ((((exists fs_h_ph_hd_only_other_pair_body_derivative_steps_coefficient. fs_h_ph_hd_only_other_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_only_other_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_v_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_steps_coefficient. ff_u_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_v_hd_only_other_pair) + (ff_coefficient_ph_hd_only_other_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_derivative_steps_before. fs_h_ph_hd_only_other_pair_body_derivative_steps_before + S (ff_previous_ph_hd_only_other_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_steps_before. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair) + (ff_previous_ph_hd_only_other_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_only_other_pair_body_derivative_steps_after. fs_h_ph_hd_only_other_pair_body_derivative_steps_after + S (ff_current_ph_hd_only_other_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair)) /\ exists fs_q_ph_hd_only_other_pair_body_derivative_steps_after. ff_d_hd_only_other_pair = fs_q_ph_hd_only_other_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_only_other_pair_body_derivative_steps)) * ff_e_hd_only_other_pair) + (ff_current_ph_hd_only_other_pair_body_derivative_steps))) /\ ff_current_ph_hd_only_other_pair_body_derivative_steps = ff_previous_ph_hd_only_other_pair_body_derivative_steps * t + ff_coefficient_ph_hd_only_other_pair_body_derivative_steps))))))))) -> z = w

Complete unchanged native tactic proof

All 21 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

21 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.

Named ingredients (1)
beta horner derivative second component functional
01Fix variables and assumptionsL1–8

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 z
  6. L6
    intro w
  7. L7
    intro hleft
  8. L8
    intro hright
02Separate the logical casesL9–10

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

  1. L9
    cases hleft
  2. L10
    cases hright
03Use earlier factsL11–20

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

  1. L11
    specialize beta_horner_derivative_second_component_functional b
  2. L12
    specialize beta_horner_derivative_second_component_functional c
  3. L13
    specialize beta_horner_derivative_second_component_functional t
  4. L14
    specialize beta_horner_derivative_second_component_functional l
  5. L15
    specialize beta_horner_derivative_second_component_functional x
  6. L16
    specialize beta_horner_derivative_second_component_functional z
  7. L17
    specialize beta_horner_derivative_second_component_functional x1
  8. L18
    specialize beta_horner_derivative_second_component_functional w
  9. L19
    apply beta_horner_derivative_second_component_functional
  10. L20
    exact hleft_witness
04Use earlier factsL21–21

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

  1. L21
    exact hright_witness

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005intro z
  6. 0006intro w
  7. 0007intro hleft
  8. 0008intro hright
  9. 0009cases hleft
  10. 0010cases hright
  11. 0011specialize beta_horner_derivative_second_component_functional b
  12. 0012specialize beta_horner_derivative_second_component_functional c
  13. 0013specialize beta_horner_derivative_second_component_functional t
  14. 0014specialize beta_horner_derivative_second_component_functional l
  15. 0015specialize beta_horner_derivative_second_component_functional x
  16. 0016specialize beta_horner_derivative_second_component_functional z
  17. 0017specialize beta_horner_derivative_second_component_functional x1
  18. 0018specialize beta_horner_derivative_second_component_functional w
  19. 0019apply beta_horner_derivative_second_component_functional
  20. 0020exact hleft_witness
  21. 0021exact hright_witness