HD000A

beta_horner_derivative_second_component_functional

The formal derivative component is independent of every possible choice of coupled beta trace.

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. ∀ m. ∀ w. HornerDerivative(b,c,t,l,n,z) → HornerDerivative(b,c,t,l,m,w) → z = w

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

Definition DAG

HornerDerivative(b,c,t,l,n,z) · 2

Actual proof prerequisites

beta_horner_derivative_functional
Original expanded first-order statement 
forall b c t l n z m w. (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_hd_other ff_v_hd_other ff_d_hd_other ff_e_hd_other. ((((((exists fs_h_ph_hd_other_body_value_start. fs_h_ph_hd_other_body_value_start + S (0) = S ((S (0)) * ff_v_hd_other)) /\ exists fs_q_ph_hd_other_body_value_start. ff_u_hd_other = fs_q_ph_hd_other_body_value_start * S ((S (0)) * ff_v_hd_other) + (0))) /\ ((((exists fs_h_ph_hd_other_body_value_terminal. fs_h_ph_hd_other_body_value_terminal + S (m) = S ((S (l)) * ff_v_hd_other)) /\ exists fs_q_ph_hd_other_body_value_terminal. ff_u_hd_other = fs_q_ph_hd_other_body_value_terminal * S ((S (l)) * ff_v_hd_other) + (m))) /\ forall ff_i_ph_hd_other_body_value_steps. (exists ph_bound_hd_other_body_value_steps. ph_bound_hd_other_body_value_steps + S ff_i_ph_hd_other_body_value_steps = l) -> exists ff_coefficient_ph_hd_other_body_value_steps ff_previous_ph_hd_other_body_value_steps ff_current_ph_hd_other_body_value_steps. ((((exists fs_h_ph_hd_other_body_value_steps_coefficient. fs_h_ph_hd_other_body_value_steps_coefficient + S (ff_coefficient_ph_hd_other_body_value_steps) = S ((S (ff_i_ph_hd_other_body_value_steps)) * c)) /\ exists fs_q_ph_hd_other_body_value_steps_coefficient. b = fs_q_ph_hd_other_body_value_steps_coefficient * S ((S (ff_i_ph_hd_other_body_value_steps)) * c) + (ff_coefficient_ph_hd_other_body_value_steps))) /\ ((((exists fs_h_ph_hd_other_body_value_steps_before. fs_h_ph_hd_other_body_value_steps_before + S (ff_previous_ph_hd_other_body_value_steps) = S ((S (ff_i_ph_hd_other_body_value_steps)) * ff_v_hd_other)) /\ exists fs_q_ph_hd_other_body_value_steps_before. ff_u_hd_other = fs_q_ph_hd_other_body_value_steps_before * S ((S (ff_i_ph_hd_other_body_value_steps)) * ff_v_hd_other) + (ff_previous_ph_hd_other_body_value_steps))) /\ ((((exists fs_h_ph_hd_other_body_value_steps_after. fs_h_ph_hd_other_body_value_steps_after + S (ff_current_ph_hd_other_body_value_steps) = S ((S (S ff_i_ph_hd_other_body_value_steps)) * ff_v_hd_other)) /\ exists fs_q_ph_hd_other_body_value_steps_after. ff_u_hd_other = fs_q_ph_hd_other_body_value_steps_after * S ((S (S ff_i_ph_hd_other_body_value_steps)) * ff_v_hd_other) + (ff_current_ph_hd_other_body_value_steps))) /\ ff_current_ph_hd_other_body_value_steps = ff_previous_ph_hd_other_body_value_steps * t + ff_coefficient_ph_hd_other_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_other_body_derivative_start. fs_h_ph_hd_other_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_other)) /\ exists fs_q_ph_hd_other_body_derivative_start. ff_d_hd_other = fs_q_ph_hd_other_body_derivative_start * S ((S (0)) * ff_e_hd_other) + (0))) /\ ((((exists fs_h_ph_hd_other_body_derivative_terminal. fs_h_ph_hd_other_body_derivative_terminal + S (w) = S ((S (l)) * ff_e_hd_other)) /\ exists fs_q_ph_hd_other_body_derivative_terminal. ff_d_hd_other = fs_q_ph_hd_other_body_derivative_terminal * S ((S (l)) * ff_e_hd_other) + (w))) /\ forall ff_i_ph_hd_other_body_derivative_steps. (exists ph_bound_hd_other_body_derivative_steps. ph_bound_hd_other_body_derivative_steps + S ff_i_ph_hd_other_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_other_body_derivative_steps ff_previous_ph_hd_other_body_derivative_steps ff_current_ph_hd_other_body_derivative_steps. ((((exists fs_h_ph_hd_other_body_derivative_steps_coefficient. fs_h_ph_hd_other_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_other_body_derivative_steps) = S ((S (ff_i_ph_hd_other_body_derivative_steps)) * ff_v_hd_other)) /\ exists fs_q_ph_hd_other_body_derivative_steps_coefficient. ff_u_hd_other = fs_q_ph_hd_other_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_other_body_derivative_steps)) * ff_v_hd_other) + (ff_coefficient_ph_hd_other_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_other_body_derivative_steps_before. fs_h_ph_hd_other_body_derivative_steps_before + S (ff_previous_ph_hd_other_body_derivative_steps) = S ((S (ff_i_ph_hd_other_body_derivative_steps)) * ff_e_hd_other)) /\ exists fs_q_ph_hd_other_body_derivative_steps_before. ff_d_hd_other = fs_q_ph_hd_other_body_derivative_steps_before * S ((S (ff_i_ph_hd_other_body_derivative_steps)) * ff_e_hd_other) + (ff_previous_ph_hd_other_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_other_body_derivative_steps_after. fs_h_ph_hd_other_body_derivative_steps_after + S (ff_current_ph_hd_other_body_derivative_steps) = S ((S (S ff_i_ph_hd_other_body_derivative_steps)) * ff_e_hd_other)) /\ exists fs_q_ph_hd_other_body_derivative_steps_after. ff_d_hd_other = fs_q_ph_hd_other_body_derivative_steps_after * S ((S (S ff_i_ph_hd_other_body_derivative_steps)) * ff_e_hd_other) + (ff_current_ph_hd_other_body_derivative_steps))) /\ ff_current_ph_hd_other_body_derivative_steps = ff_previous_ph_hd_other_body_derivative_steps * t + ff_coefficient_ph_hd_other_body_derivative_steps)))))))) -> z = w

Complete unchanged native tactic proof

All 24 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

24 script commands · 5 reading checkpoints · 1 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 functional
01Fix variables and assumptionsL1–10

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 m
  8. L8
    intro w
  9. L9
    intro hleft
  10. L10
    intro hright
02Establish hequalL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta horner derivative functional.

  1. L11
    have hequal : (n = m /\ z = w)
  2. L12
    specialize beta_horner_derivative_functional b
  3. L13
    specialize beta_horner_derivative_functional c
  4. L14
    specialize beta_horner_derivative_functional t
  5. L15
    specialize beta_horner_derivative_functional l
  6. L16
    specialize beta_horner_derivative_functional n
  7. L17
    specialize beta_horner_derivative_functional z
  8. L18
    specialize beta_horner_derivative_functional m
  9. L19
    specialize beta_horner_derivative_functional w
  10. L20
    apply beta_horner_derivative_functional
03Use earlier factsL21–22

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

  1. L21
    exact hleft
  2. L22
    exact hright
04Separate the logical casesL23–23

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

  1. L23
    cases hequal
05Use earlier factsL24–24

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

  1. L24
    exact hequal_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005intro n
  6. 0006intro z
  7. 0007intro m
  8. 0008intro w
  9. 0009intro hleft
  10. 0010intro hright
  11. 0011have hequal : (n = m /\ z = w)
  12. 0012specialize beta_horner_derivative_functional b
  13. 0013specialize beta_horner_derivative_functional c
  14. 0014specialize beta_horner_derivative_functional t
  15. 0015specialize beta_horner_derivative_functional l
  16. 0016specialize beta_horner_derivative_functional n
  17. 0017specialize beta_horner_derivative_functional z
  18. 0018specialize beta_horner_derivative_functional m
  19. 0019specialize beta_horner_derivative_functional w
  20. 0020apply beta_horner_derivative_functional
  21. 0021exact hleft
  22. 0022exact hright
  23. 0023cases hequal
  24. 0024exact hequal_right