HL000D

beta_horner_simple_root_at_congruent_point

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

Every lifted root in a simple residue class has an actual evaluated derivative that remains coprime to the base.

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 b c a r l n d p M. ~(p = 0) -> (exists ff_u_hd_hpl_pair ff_v_hd_hpl_pair ff_d_hd_hpl_pair ff_e_hd_hpl_pair. ((((((exists fs_h_ph_hd_hpl_pair_body_value_start. fs_h_ph_hd_hpl_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_start. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_start * S ((S (0)) * ff_v_hd_hpl_pair) + (0))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_terminal. fs_h_ph_hd_hpl_pair_body_value_terminal + S (n) = S ((S (l)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_terminal. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_pair) + (n))) /\ forall ff_i_ph_hd_hpl_pair_body_value_steps. (exists ph_bound_hd_hpl_pair_body_value_steps. ph_bound_hd_hpl_pair_body_value_steps + S ff_i_ph_hd_hpl_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_pair_body_value_steps ff_previous_ph_hd_hpl_pair_body_value_steps ff_current_ph_hd_hpl_pair_body_value_steps. ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_coefficient. fs_h_ph_hd_hpl_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_pair_body_value_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_coefficient. b = fs_q_ph_hd_hpl_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_hpl_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_before. fs_h_ph_hd_hpl_pair_body_value_steps_before + S (ff_previous_ph_hd_hpl_pair_body_value_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_before. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair) + (ff_previous_ph_hd_hpl_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_after. fs_h_ph_hd_hpl_pair_body_value_steps_after + S (ff_current_ph_hd_hpl_pair_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_after. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair) + (ff_current_ph_hd_hpl_pair_body_value_steps))) /\ ff_current_ph_hd_hpl_pair_body_value_steps = ff_previous_ph_hd_hpl_pair_body_value_steps * a + ff_coefficient_ph_hd_hpl_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_pair_body_derivative_start. fs_h_ph_hd_hpl_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_start. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_pair) + (0))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_terminal. fs_h_ph_hd_hpl_pair_body_derivative_terminal + S (d) = S ((S (l)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_terminal. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_pair) + (d))) /\ forall ff_i_ph_hd_hpl_pair_body_derivative_steps. (exists ph_bound_hd_hpl_pair_body_derivative_steps. ph_bound_hd_hpl_pair_body_derivative_steps + S ff_i_ph_hd_hpl_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_pair_body_derivative_steps ff_previous_ph_hd_hpl_pair_body_derivative_steps ff_current_ph_hd_hpl_pair_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_coefficient. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_v_hd_hpl_pair) + (ff_coefficient_ph_hd_hpl_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_before. fs_h_ph_hd_hpl_pair_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_before. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair) + (ff_previous_ph_hd_hpl_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_after. fs_h_ph_hd_hpl_pair_body_derivative_steps_after + S (ff_current_ph_hd_hpl_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_after. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair) + (ff_current_ph_hd_hpl_pair_body_derivative_steps))) /\ ff_current_ph_hd_hpl_pair_body_derivative_steps = ff_previous_ph_hd_hpl_pair_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_pair_body_derivative_steps)))))))) -> (forall hmi_divisor_hpl_coprime. (exists hmi_left_factor_hpl_coprime. d = hmi_divisor_hpl_coprime * hmi_left_factor_hpl_coprime) -> (exists hmi_right_factor_hpl_coprime. p = hmi_divisor_hpl_coprime * hmi_right_factor_hpl_coprime) -> hmi_divisor_hpl_coprime = 1) -> (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. a + p * hgcrt_mod_left_hpl_mod = r + p * hgcrt_mod_right_hpl_mod) -> (exists hpl_value_root. ((exists ff_u_ph_hpl_root ff_v_ph_hpl_root. ((((exists fs_h_ph_hpl_root_body_start. fs_h_ph_hpl_root_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_start. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_start * S ((S (0)) * ff_v_ph_hpl_root) + (0))) /\ ((((exists fs_h_ph_hpl_root_body_terminal. fs_h_ph_hpl_root_body_terminal + S (hpl_value_root) = S ((S (l)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_terminal. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_terminal * S ((S (l)) * ff_v_ph_hpl_root) + (hpl_value_root))) /\ forall ff_i_ph_hpl_root_body_steps. (exists ph_bound_hpl_root_body_steps. ph_bound_hpl_root_body_steps + S ff_i_ph_hpl_root_body_steps = l) -> exists ff_coefficient_ph_hpl_root_body_steps ff_previous_ph_hpl_root_body_steps ff_current_ph_hpl_root_body_steps. ((((exists fs_h_ph_hpl_root_body_steps_coefficient. fs_h_ph_hpl_root_body_steps_coefficient + S (ff_coefficient_ph_hpl_root_body_steps) = S ((S (ff_i_ph_hpl_root_body_steps)) * c)) /\ exists fs_q_ph_hpl_root_body_steps_coefficient. b = fs_q_ph_hpl_root_body_steps_coefficient * S ((S (ff_i_ph_hpl_root_body_steps)) * c) + (ff_coefficient_ph_hpl_root_body_steps))) /\ ((((exists fs_h_ph_hpl_root_body_steps_before. fs_h_ph_hpl_root_body_steps_before + S (ff_previous_ph_hpl_root_body_steps) = S ((S (ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_steps_before. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_steps_before * S ((S (ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root) + (ff_previous_ph_hpl_root_body_steps))) /\ ((((exists fs_h_ph_hpl_root_body_steps_after. fs_h_ph_hpl_root_body_steps_after + S (ff_current_ph_hpl_root_body_steps) = S ((S (S ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_steps_after. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_steps_after * S ((S (S ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root) + (ff_current_ph_hpl_root_body_steps))) /\ ff_current_ph_hpl_root_body_steps = ff_previous_ph_hpl_root_body_steps * r + ff_coefficient_ph_hpl_root_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_root hgcrt_mod_right_hpl_root. hpl_value_root + M * hgcrt_mod_left_hpl_root = 0 + M * hgcrt_mod_right_hpl_root))) -> (exists hpl_value_simple hpl_derivative_simple. ((exists ff_u_hd_hpl_simple ff_v_hd_hpl_simple ff_d_hd_hpl_simple ff_e_hd_hpl_simple. ((((((exists fs_h_ph_hd_hpl_simple_body_value_start. fs_h_ph_hd_hpl_simple_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_value_start. ff_u_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_value_start * S ((S (0)) * ff_v_hd_hpl_simple) + (0))) /\ ((((exists fs_h_ph_hd_hpl_simple_body_value_terminal. fs_h_ph_hd_hpl_simple_body_value_terminal + S (hpl_value_simple) = S ((S (l)) * ff_v_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_value_terminal. ff_u_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_simple) + (hpl_value_simple))) /\ forall ff_i_ph_hd_hpl_simple_body_value_steps. (exists ph_bound_hd_hpl_simple_body_value_steps. ph_bound_hd_hpl_simple_body_value_steps + S ff_i_ph_hd_hpl_simple_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_simple_body_value_steps ff_previous_ph_hd_hpl_simple_body_value_steps ff_current_ph_hd_hpl_simple_body_value_steps. ((((exists fs_h_ph_hd_hpl_simple_body_value_steps_coefficient. fs_h_ph_hd_hpl_simple_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_simple_body_value_steps) = S ((S (ff_i_ph_hd_hpl_simple_body_value_steps)) * c)) /\ exists fs_q_ph_hd_hpl_simple_body_value_steps_coefficient. b = fs_q_ph_hd_hpl_simple_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_simple_body_value_steps)) * c) + (ff_coefficient_ph_hd_hpl_simple_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_simple_body_value_steps_before. fs_h_ph_hd_hpl_simple_body_value_steps_before + S (ff_previous_ph_hd_hpl_simple_body_value_steps) = S ((S (ff_i_ph_hd_hpl_simple_body_value_steps)) * ff_v_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_value_steps_before. ff_u_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_simple_body_value_steps)) * ff_v_hd_hpl_simple) + (ff_previous_ph_hd_hpl_simple_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_simple_body_value_steps_after. fs_h_ph_hd_hpl_simple_body_value_steps_after + S (ff_current_ph_hd_hpl_simple_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_simple_body_value_steps)) * ff_v_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_value_steps_after. ff_u_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_simple_body_value_steps)) * ff_v_hd_hpl_simple) + (ff_current_ph_hd_hpl_simple_body_value_steps))) /\ ff_current_ph_hd_hpl_simple_body_value_steps = ff_previous_ph_hd_hpl_simple_body_value_steps * r + ff_coefficient_ph_hd_hpl_simple_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_simple_body_derivative_start. fs_h_ph_hd_hpl_simple_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_derivative_start. ff_d_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_simple) + (0))) /\ ((((exists fs_h_ph_hd_hpl_simple_body_derivative_terminal. fs_h_ph_hd_hpl_simple_body_derivative_terminal + S (hpl_derivative_simple) = S ((S (l)) * ff_e_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_derivative_terminal. ff_d_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_simple) + (hpl_derivative_simple))) /\ forall ff_i_ph_hd_hpl_simple_body_derivative_steps. (exists ph_bound_hd_hpl_simple_body_derivative_steps. ph_bound_hd_hpl_simple_body_derivative_steps + S ff_i_ph_hd_hpl_simple_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_simple_body_derivative_steps ff_previous_ph_hd_hpl_simple_body_derivative_steps ff_current_ph_hd_hpl_simple_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_simple_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_simple_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_simple_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_simple_body_derivative_steps)) * ff_v_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_derivative_steps_coefficient. ff_u_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_simple_body_derivative_steps)) * ff_v_hd_hpl_simple) + (ff_coefficient_ph_hd_hpl_simple_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_simple_body_derivative_steps_before. fs_h_ph_hd_hpl_simple_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_simple_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_simple_body_derivative_steps)) * ff_e_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_derivative_steps_before. ff_d_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_simple_body_derivative_steps)) * ff_e_hd_hpl_simple) + (ff_previous_ph_hd_hpl_simple_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_simple_body_derivative_steps_after. fs_h_ph_hd_hpl_simple_body_derivative_steps_after + S (ff_current_ph_hd_hpl_simple_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_simple_body_derivative_steps)) * ff_e_hd_hpl_simple)) /\ exists fs_q_ph_hd_hpl_simple_body_derivative_steps_after. ff_d_hd_hpl_simple = fs_q_ph_hd_hpl_simple_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_simple_body_derivative_steps)) * ff_e_hd_hpl_simple) + (ff_current_ph_hd_hpl_simple_body_derivative_steps))) /\ ff_current_ph_hd_hpl_simple_body_derivative_steps = ff_previous_ph_hd_hpl_simple_body_derivative_steps * r + ff_coefficient_ph_hd_hpl_simple_body_derivative_steps)))))))) /\ ((exists hgcrt_mod_left_hpl_simple hgcrt_mod_right_hpl_simple. hpl_value_simple + M * hgcrt_mod_left_hpl_simple = 0 + M * hgcrt_mod_right_hpl_simple) /\ (forall hmi_divisor_hpl_simple. (exists hmi_left_factor_hpl_simple. hpl_derivative_simple = hmi_divisor_hpl_simple * hmi_left_factor_hpl_simple) -> (exists hmi_right_factor_hpl_simple. p = hmi_divisor_hpl_simple * hmi_right_factor_hpl_simple) -> hmi_divisor_hpl_simple = 1))))

Constructive proof overview

Generated structural guide

Every lifted root in a simple residue class has an actual evaluated derivative that remains coprime to the base.

The unchanged tactic script uses 5 declared prerequisites and contains 73 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_horner_derivative_value_exists Alpha theorem; checked-use authorized beta_horner_derivative_mod_congruence Alpha theorem; checked-use authorized beta_horner_derivative_value_projection Alpha theorem; checked-use authorized beta_horner_eval_functional Alpha theorem; checked-use authorized HL0002 hensel_coprime_mod_transport

Direct dependents

HL0011 beta_horner_simple_lift_preserves_simplicity

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

73 script commands · 16 reading checkpoints · 4 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.

Named ingredients (1)
hensel coprime mod transport

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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 a
  4. L4
    intro r
  5. L5
    intro l
  6. L6
    intro n
  7. L7
    intro d
  8. L8
    intro p
  9. L9
    intro M
  10. L10
    intro hp
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hpair
  2. L12
    intro hcop
  3. L13
    intro hmod
  4. L14
    intro hroot
03Establish hnewpairL15–20

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

  1. L15
    have hnewpair : ∃ v. ∃ e. HornerDerivative(b,c,r,l,v,e)Definitions: HornerDerivative
  2. L16
    specialize beta_horner_derivative_value_exists b
  3. L17
    specialize beta_horner_derivative_value_exists c
  4. L18
    specialize beta_horner_derivative_value_exists r
  5. L19
    specialize beta_horner_derivative_value_exists l
  6. L20
    apply beta_horner_derivative_value_exists
04Separate the logical casesL21–22

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

  1. L21
    cases hnewpair
  2. L22
    cases hnewpair_witness
05Establish hbothL23–32

Establish this local claim before using it. It is not an additional assumption.

  1. L23
    have hboth : (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. n + p * hgcrt_mod_left_hpl_mod = x + p * hgcrt_mod_right_hpl_mod) /\ (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. d + p * hgcrt_mod_left_hpl_mod = x1 + p * hgcrt_mod_right_hpl_mod)
  2. L24
    specialize beta_horner_derivative_mod_congruence b
  3. L25
    specialize beta_horner_derivative_mod_congruence c
  4. L26
    specialize beta_horner_derivative_mod_congruence p
  5. L27
    specialize beta_horner_derivative_mod_congruence a
  6. L28
    specialize beta_horner_derivative_mod_congruence r
  7. L29
    specialize beta_horner_derivative_mod_congruence l
  8. L30
    specialize beta_horner_derivative_mod_congruence n
  9. L31
    specialize beta_horner_derivative_mod_congruence d
  10. L32
    specialize beta_horner_derivative_mod_congruence x
06Use earlier factsL33–37

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

  1. L33
    specialize beta_horner_derivative_mod_congruence x1
  2. L34
    apply beta_horner_derivative_mod_congruence
  3. L35
    exact hmod
  4. L36
    exact hpair
  5. L37
    exact hnewpair_witness_witness
07Separate the logical casesL38–38

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

  1. L38
    cases hboth
08Establish hvalueL39–47

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

  1. L39
    have hvalue : Horner(b,c,r,l,x)Definitions: Horner
  2. L40
    specialize beta_horner_derivative_value_projection b
  3. L41
    specialize beta_horner_derivative_value_projection c
  4. L42
    specialize beta_horner_derivative_value_projection r
  5. L43
    specialize beta_horner_derivative_value_projection l
  6. L44
    specialize beta_horner_derivative_value_projection x
  7. L45
    specialize beta_horner_derivative_value_projection x1
  8. L46
    apply beta_horner_derivative_value_projection
  9. L47
    exact hnewpair_witness_witness
09Separate the logical casesL48–49

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

  1. L48
    cases hroot
  2. L49
    cases hroot_witness
10Establish heqL50–59

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

  1. L50
    have heq : x = x2
  2. L51
    specialize beta_horner_eval_functional b
  3. L52
    specialize beta_horner_eval_functional c
  4. L53
    specialize beta_horner_eval_functional r
  5. L54
    specialize beta_horner_eval_functional l
  6. L55
    specialize beta_horner_eval_functional x
  7. L56
    specialize beta_horner_eval_functional x2
  8. L57
    apply beta_horner_eval_functional
  9. L58
    exact hvalue
  10. L59
    exact hroot_witness_left
11Construct an explicit witnessL60–61

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

  1. L60
    exists x
  2. L61
    exists x1
12Separate the logical casesL62–62

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

  1. L62
    split
13Use earlier factsL63–63

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

  1. L63
    exact hnewpair_witness_witness
14Separate the logical casesL64–64

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

  1. L64
    split
15Calculate and transport equalitiesL65–65

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L65
    rewrite heq
16Use earlier factsL66–73

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

  1. L66
    exact hroot_witness_right
  2. L67
    specialize hensel_coprime_mod_transport p
  3. L68
    specialize hensel_coprime_mod_transport d
  4. L69
    specialize hensel_coprime_mod_transport x1
  5. L70
    apply hensel_coprime_mod_transport
  6. L71
    exact hp
  7. L72
    exact hcop
  8. L73
    exact hboth_right

Library-wide reading audit

Original exact command ledger · 73 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro a
  4. 0004intro r
  5. 0005intro l
  6. 0006intro n
  7. 0007intro d
  8. 0008intro p
  9. 0009intro M
  10. 0010intro hp
  11. 0011intro hpair
  12. 0012intro hcop
  13. 0013intro hmod
  14. 0014intro hroot
  15. 0015have hnewpair : exists v e. (exists ff_u_hd_hpl_pair ff_v_hd_hpl_pair ff_d_hd_hpl_pair ff_e_hd_hpl_pair. ((((((exists fs_h_ph_hd_hpl_pair_body_value_start. fs_h_ph_hd_hpl_pair_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_start. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_start * S ((S (0)) * ff_v_hd_hpl_pair) + (0))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_terminal. fs_h_ph_hd_hpl_pair_body_value_terminal + S (v) = S ((S (l)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_terminal. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_pair) + (v))) /\ forall ff_i_ph_hd_hpl_pair_body_value_steps. (exists ph_bound_hd_hpl_pair_body_value_steps. ph_bound_hd_hpl_pair_body_value_steps + S ff_i_ph_hd_hpl_pair_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_pair_body_value_steps ff_previous_ph_hd_hpl_pair_body_value_steps ff_current_ph_hd_hpl_pair_body_value_steps. ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_coefficient. fs_h_ph_hd_hpl_pair_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_pair_body_value_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * c)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_coefficient. b = fs_q_ph_hd_hpl_pair_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * c) + (ff_coefficient_ph_hd_hpl_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_before. fs_h_ph_hd_hpl_pair_body_value_steps_before + S (ff_previous_ph_hd_hpl_pair_body_value_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_before. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair) + (ff_previous_ph_hd_hpl_pair_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_value_steps_after. fs_h_ph_hd_hpl_pair_body_value_steps_after + S (ff_current_ph_hd_hpl_pair_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_value_steps_after. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_pair_body_value_steps)) * ff_v_hd_hpl_pair) + (ff_current_ph_hd_hpl_pair_body_value_steps))) /\ ff_current_ph_hd_hpl_pair_body_value_steps = ff_previous_ph_hd_hpl_pair_body_value_steps * r + ff_coefficient_ph_hd_hpl_pair_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_pair_body_derivative_start. fs_h_ph_hd_hpl_pair_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_start. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_pair) + (0))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_terminal. fs_h_ph_hd_hpl_pair_body_derivative_terminal + S (e) = S ((S (l)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_terminal. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_pair) + (e))) /\ forall ff_i_ph_hd_hpl_pair_body_derivative_steps. (exists ph_bound_hd_hpl_pair_body_derivative_steps. ph_bound_hd_hpl_pair_body_derivative_steps + S ff_i_ph_hd_hpl_pair_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_pair_body_derivative_steps ff_previous_ph_hd_hpl_pair_body_derivative_steps ff_current_ph_hd_hpl_pair_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_pair_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_v_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_coefficient. ff_u_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_v_hd_hpl_pair) + (ff_coefficient_ph_hd_hpl_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_before. fs_h_ph_hd_hpl_pair_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_pair_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_before. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair) + (ff_previous_ph_hd_hpl_pair_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_pair_body_derivative_steps_after. fs_h_ph_hd_hpl_pair_body_derivative_steps_after + S (ff_current_ph_hd_hpl_pair_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair)) /\ exists fs_q_ph_hd_hpl_pair_body_derivative_steps_after. ff_d_hd_hpl_pair = fs_q_ph_hd_hpl_pair_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_pair_body_derivative_steps)) * ff_e_hd_hpl_pair) + (ff_current_ph_hd_hpl_pair_body_derivative_steps))) /\ ff_current_ph_hd_hpl_pair_body_derivative_steps = ff_previous_ph_hd_hpl_pair_body_derivative_steps * r + ff_coefficient_ph_hd_hpl_pair_body_derivative_steps))))))))
  16. 0016specialize beta_horner_derivative_value_exists b
  17. 0017specialize beta_horner_derivative_value_exists c
  18. 0018specialize beta_horner_derivative_value_exists r
  19. 0019specialize beta_horner_derivative_value_exists l
  20. 0020apply beta_horner_derivative_value_exists
  21. 0021cases hnewpair
  22. 0022cases hnewpair_witness
  23. 0023have hboth : (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. n + p * hgcrt_mod_left_hpl_mod = x + p * hgcrt_mod_right_hpl_mod) /\ (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. d + p * hgcrt_mod_left_hpl_mod = x1 + p * hgcrt_mod_right_hpl_mod)
  24. 0024specialize beta_horner_derivative_mod_congruence b
  25. 0025specialize beta_horner_derivative_mod_congruence c
  26. 0026specialize beta_horner_derivative_mod_congruence p
  27. 0027specialize beta_horner_derivative_mod_congruence a
  28. 0028specialize beta_horner_derivative_mod_congruence r
  29. 0029specialize beta_horner_derivative_mod_congruence l
  30. 0030specialize beta_horner_derivative_mod_congruence n
  31. 0031specialize beta_horner_derivative_mod_congruence d
  32. 0032specialize beta_horner_derivative_mod_congruence x
  33. 0033specialize beta_horner_derivative_mod_congruence x1
  34. 0034apply beta_horner_derivative_mod_congruence
  35. 0035exact hmod
  36. 0036exact hpair
  37. 0037exact hnewpair_witness_witness
  38. 0038cases hboth
  39. 0039have hvalue : exists ff_u_ph_hpl_eval ff_v_ph_hpl_eval. ((((exists fs_h_ph_hpl_eval_body_start. fs_h_ph_hpl_eval_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_eval)) /\ exists fs_q_ph_hpl_eval_body_start. ff_u_ph_hpl_eval = fs_q_ph_hpl_eval_body_start * S ((S (0)) * ff_v_ph_hpl_eval) + (0))) /\ ((((exists fs_h_ph_hpl_eval_body_terminal. fs_h_ph_hpl_eval_body_terminal + S (x) = S ((S (l)) * ff_v_ph_hpl_eval)) /\ exists fs_q_ph_hpl_eval_body_terminal. ff_u_ph_hpl_eval = fs_q_ph_hpl_eval_body_terminal * S ((S (l)) * ff_v_ph_hpl_eval) + (x))) /\ forall ff_i_ph_hpl_eval_body_steps. (exists ph_bound_hpl_eval_body_steps. ph_bound_hpl_eval_body_steps + S ff_i_ph_hpl_eval_body_steps = l) -> exists ff_coefficient_ph_hpl_eval_body_steps ff_previous_ph_hpl_eval_body_steps ff_current_ph_hpl_eval_body_steps. ((((exists fs_h_ph_hpl_eval_body_steps_coefficient. fs_h_ph_hpl_eval_body_steps_coefficient + S (ff_coefficient_ph_hpl_eval_body_steps) = S ((S (ff_i_ph_hpl_eval_body_steps)) * c)) /\ exists fs_q_ph_hpl_eval_body_steps_coefficient. b = fs_q_ph_hpl_eval_body_steps_coefficient * S ((S (ff_i_ph_hpl_eval_body_steps)) * c) + (ff_coefficient_ph_hpl_eval_body_steps))) /\ ((((exists fs_h_ph_hpl_eval_body_steps_before. fs_h_ph_hpl_eval_body_steps_before + S (ff_previous_ph_hpl_eval_body_steps) = S ((S (ff_i_ph_hpl_eval_body_steps)) * ff_v_ph_hpl_eval)) /\ exists fs_q_ph_hpl_eval_body_steps_before. ff_u_ph_hpl_eval = fs_q_ph_hpl_eval_body_steps_before * S ((S (ff_i_ph_hpl_eval_body_steps)) * ff_v_ph_hpl_eval) + (ff_previous_ph_hpl_eval_body_steps))) /\ ((((exists fs_h_ph_hpl_eval_body_steps_after. fs_h_ph_hpl_eval_body_steps_after + S (ff_current_ph_hpl_eval_body_steps) = S ((S (S ff_i_ph_hpl_eval_body_steps)) * ff_v_ph_hpl_eval)) /\ exists fs_q_ph_hpl_eval_body_steps_after. ff_u_ph_hpl_eval = fs_q_ph_hpl_eval_body_steps_after * S ((S (S ff_i_ph_hpl_eval_body_steps)) * ff_v_ph_hpl_eval) + (ff_current_ph_hpl_eval_body_steps))) /\ ff_current_ph_hpl_eval_body_steps = ff_previous_ph_hpl_eval_body_steps * r + ff_coefficient_ph_hpl_eval_body_steps)))))
  40. 0040specialize beta_horner_derivative_value_projection b
  41. 0041specialize beta_horner_derivative_value_projection c
  42. 0042specialize beta_horner_derivative_value_projection r
  43. 0043specialize beta_horner_derivative_value_projection l
  44. 0044specialize beta_horner_derivative_value_projection x
  45. 0045specialize beta_horner_derivative_value_projection x1
  46. 0046apply beta_horner_derivative_value_projection
  47. 0047exact hnewpair_witness_witness
  48. 0048cases hroot
  49. 0049cases hroot_witness
  50. 0050have heq : x = x2
  51. 0051specialize beta_horner_eval_functional b
  52. 0052specialize beta_horner_eval_functional c
  53. 0053specialize beta_horner_eval_functional r
  54. 0054specialize beta_horner_eval_functional l
  55. 0055specialize beta_horner_eval_functional x
  56. 0056specialize beta_horner_eval_functional x2
  57. 0057apply beta_horner_eval_functional
  58. 0058exact hvalue
  59. 0059exact hroot_witness_left
  60. 0060exists x
  61. 0061exists x1
  62. 0062split
  63. 0063exact hnewpair_witness_witness
  64. 0064split
  65. 0065rewrite heq
  66. 0066exact hroot_witness_right
  67. 0067specialize hensel_coprime_mod_transport p
  68. 0068specialize hensel_coprime_mod_transport d
  69. 0069specialize hensel_coprime_mod_transport x1
  70. 0070apply hensel_coprime_mod_transport
  71. 0071exact hp
  72. 0072exact hcop
  73. 0073exact hboth_right