HL000A

beta_horner_simple_root_hensel_lift_exists_unique

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

Every unrestricted natural-polynomial simple root has a unique canonical lift, with no supplied correction or representative bound.

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 l n d m p s. ~(p = 0) -> ~(m = 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)))))))) -> m = p * s -> (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. n + m * hgcrt_mod_left_hpl_mod = 0 + m * hgcrt_mod_right_hpl_mod) -> (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 r. ((((exists hpl_gap_lift. hpl_gap_lift + S (r) = (p * m)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. r + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * r + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + (p * m) * hgcrt_mod_left_hpl_lift = 0 + (p * m) * hgcrt_mod_right_hpl_lift)))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (p * m)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. z + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * z + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + (p * m) * hgcrt_mod_left_hpl_lift = 0 + (p * m) * hgcrt_mod_right_hpl_lift)))))) -> z = r)

Constructive proof overview

Generated structural guide

Every unrestricted natural-polynomial simple root has a unique canonical lift, with no supplied correction or representative bound.

The unchanged tactic script uses 5 declared prerequisites and contains 103 exact native proof lines.

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

Proof neighborhood

Direct dependencies

HL0009 beta_horner_simple_canonical_representative mod_eq_zero_to_dvd_nonzero Alpha theorem; checked-use authorized HL0008 hensel_canonical_horner_lift_exists_unique mod_eq_trans Stable theorem; checked-use authorized mod_eq_symm Stable theorem; checked-use authorized

Direct dependents

HL0010 beta_horner_prime_power_hensel_lift_exists_unique HL0012 beta_horner_hensel_iterated_exists_unique

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

103 script commands · 23 reading checkpoints · 3 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 (2)
beta horner simple canonical representativehensel canonical horner lift exists unique

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 l
  5. L5
    intro n
  6. L6
    intro d
  7. L7
    intro m
  8. L8
    intro p
  9. L9
    intro s
  10. L10
    intro hp
02Fix variables and assumptionsL11–15

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

  1. L11
    intro hm
  2. L12
    intro hpair
  3. L13
    intro hfactor
  4. L14
    intro hroot
  5. L15
    intro hcop
03Establish hnormalizedL16–25

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

  1. L16
    have hnormalized : ∃ r. Lt(r,m) ∧ (ModEq(m,r,a) ∧ SimpleHornerRoot(b,c,r,l,m,p))Definitions: SimpleHornerRootLtModEq
  2. L17
    specialize beta_horner_simple_canonical_representative b
  3. L18
    specialize beta_horner_simple_canonical_representative c
  4. L19
    specialize beta_horner_simple_canonical_representative a
  5. L20
    specialize beta_horner_simple_canonical_representative l
  6. L21
    specialize beta_horner_simple_canonical_representative n
  7. L22
    specialize beta_horner_simple_canonical_representative d
  8. L23
    specialize beta_horner_simple_canonical_representative m
  9. L24
    specialize beta_horner_simple_canonical_representative p
  10. L25
    specialize beta_horner_simple_canonical_representative s
04Use earlier factsL26–32

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

  1. L26
    apply beta_horner_simple_canonical_representative
  2. L27
    exact hp
  3. L28
    exact hm
  4. L29
    exact hpair
  5. L30
    exact hfactor
  6. L31
    exact hroot
  7. L32
    exact hcop
05Separate the logical casesL33–39

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

  1. L33
    cases hnormalized
  2. L34
    cases hnormalized_witness
  3. L35
    cases hnormalized_witness_right
  4. L36
    cases hnormalized_witness_right_right
  5. L37
    cases hnormalized_witness_right_right_witness
  6. L38
    cases hnormalized_witness_right_right_witness_witness
  7. L39
    cases hnormalized_witness_right_right_witness_witness_right
06Establish hquotientL40–45

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq zero to dvd nonzero.

  1. L40
    have hquotient : exists q. x1 = m * q
  2. L41
    specialize mod_eq_zero_to_dvd_nonzero m
  3. L42
    specialize mod_eq_zero_to_dvd_nonzero x1
  4. L43
    apply mod_eq_zero_to_dvd_nonzero
  5. L44
    exact hm
  6. L45
    exact hnormalized_witness_right_right_witness_witness_right_left
07Separate the logical casesL46–46

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

  1. L46
    cases hquotient
08Establish hcanonicalL47–56

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

  1. L47
    have hcanonical : ∃ r. CanonicalHornerLift(b,c,l,m,x,p · m,r) ∧ (∀ y. CanonicalHornerLift(b,c,l,m,x,p · m,y) → y = r)Definitions: CanonicalHornerLift
  2. L48
    specialize hensel_canonical_horner_lift_exists_unique b
  3. L49
    specialize hensel_canonical_horner_lift_exists_unique c
  4. L50
    specialize hensel_canonical_horner_lift_exists_unique x
  5. L51
    specialize hensel_canonical_horner_lift_exists_unique l
  6. L52
    specialize hensel_canonical_horner_lift_exists_unique x1
  7. L53
    specialize hensel_canonical_horner_lift_exists_unique x2
  8. L54
    specialize hensel_canonical_horner_lift_exists_unique m
  9. L55
    specialize hensel_canonical_horner_lift_exists_unique p
  10. L56
    specialize hensel_canonical_horner_lift_exists_unique s
09Use earlier factsL57–65

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

  1. L57
    specialize hensel_canonical_horner_lift_exists_unique x3
  2. L58
    apply hensel_canonical_horner_lift_exists_unique
  3. L59
    exact hp
  4. L60
    exact hm
  5. L61
    exact hnormalized_witness_right_right_witness_witness_left
  6. L62
    exact hfactor
  7. L63
    exact hquotient_witness
  8. L64
    exact hnormalized_witness_right_right_witness_witness_right_right
  9. L65
    exact hnormalized_witness_left
10Separate the logical casesL66–69

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

  1. L66
    cases hcanonical
  2. L67
    cases hcanonical_witness
  3. L68
    cases hcanonical_witness_left
  4. L69
    cases hcanonical_witness_left_right
11Construct an explicit witnessL70–70

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

  1. L70
    exists x4
12Separate the logical casesL71–72

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

  1. L71
    split
  2. L72
    split
13Use earlier factsL73–73

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

  1. L73
    exact hcanonical_witness_left_left
14Separate the logical casesL74–74

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

  1. L74
    split
15Use earlier factsL75–82

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

  1. L75
    specialize mod_eq_trans m
  2. L76
    specialize mod_eq_trans x4
  3. L77
    specialize mod_eq_trans x
  4. L78
    specialize mod_eq_trans a
  5. L79
    apply mod_eq_trans
  6. L80
    exact hcanonical_witness_left_right_left
  7. L81
    exact hnormalized_witness_right_left
  8. L82
    exact hcanonical_witness_left_right_right
16Fix variables and assumptionsL83–84

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

  1. L83
    intro z
  2. L84
    intro hz
17Separate the logical casesL85–86

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

  1. L85
    cases hz
  2. L86
    cases hz_right
18Use earlier factsL87–88

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

  1. L87
    specialize hcanonical_witness_right z
  2. L88
    apply hcanonical_witness_right
19Separate the logical casesL89–89

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

  1. L89
    split
20Use earlier factsL90–90

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

  1. L90
    exact hz_left
21Separate the logical casesL91–91

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

  1. L91
    split
22Use earlier factsL92–101

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

  1. L92
    specialize mod_eq_trans m
  2. L93
    specialize mod_eq_trans z
  3. L94
    specialize mod_eq_trans a
  4. L95
    specialize mod_eq_trans x
  5. L96
    apply mod_eq_trans
  6. L97
    exact hz_right_left
  7. L98
    specialize mod_eq_symm m
  8. L99
    specialize mod_eq_symm x
  9. L100
    specialize mod_eq_symm a
  10. L101
    apply mod_eq_symm
23Use earlier factsL102–103

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

  1. L102
    exact hnormalized_witness_right_left
  2. L103
    exact hz_right_right

Library-wide reading audit

Original exact command ledger · 103 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro a
  4. 0004intro l
  5. 0005intro n
  6. 0006intro d
  7. 0007intro m
  8. 0008intro p
  9. 0009intro s
  10. 0010intro hp
  11. 0011intro hm
  12. 0012intro hpair
  13. 0013intro hfactor
  14. 0014intro hroot
  15. 0015intro hcop
  16. 0016have hnormalized : exists r. ((exists hpl_gap_bound. hpl_gap_bound + S (r) = (m)) /\ ((exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. r + m * hgcrt_mod_left_hpl_mod = a + m * hgcrt_mod_right_hpl_mod) /\ (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))))))
  17. 0017specialize beta_horner_simple_canonical_representative b
  18. 0018specialize beta_horner_simple_canonical_representative c
  19. 0019specialize beta_horner_simple_canonical_representative a
  20. 0020specialize beta_horner_simple_canonical_representative l
  21. 0021specialize beta_horner_simple_canonical_representative n
  22. 0022specialize beta_horner_simple_canonical_representative d
  23. 0023specialize beta_horner_simple_canonical_representative m
  24. 0024specialize beta_horner_simple_canonical_representative p
  25. 0025specialize beta_horner_simple_canonical_representative s
  26. 0026apply beta_horner_simple_canonical_representative
  27. 0027exact hp
  28. 0028exact hm
  29. 0029exact hpair
  30. 0030exact hfactor
  31. 0031exact hroot
  32. 0032exact hcop
  33. 0033cases hnormalized
  34. 0034cases hnormalized_witness
  35. 0035cases hnormalized_witness_right
  36. 0036cases hnormalized_witness_right_right
  37. 0037cases hnormalized_witness_right_right_witness
  38. 0038cases hnormalized_witness_right_right_witness_witness
  39. 0039cases hnormalized_witness_right_right_witness_witness_right
  40. 0040have hquotient : exists q. x1 = m * q
  41. 0041specialize mod_eq_zero_to_dvd_nonzero m
  42. 0042specialize mod_eq_zero_to_dvd_nonzero x1
  43. 0043apply mod_eq_zero_to_dvd_nonzero
  44. 0044exact hm
  45. 0045exact hnormalized_witness_right_right_witness_witness_right_left
  46. 0046cases hquotient
  47. 0047have hcanonical : exists r. ((((exists hpl_gap_lift. hpl_gap_lift + S (r) = (p * m)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. r + m * hgcrt_mod_left_hpl_lift = x + m * hgcrt_mod_right_hpl_lift) /\ (exists hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * r + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + (p * m) * hgcrt_mod_left_hpl_lift = 0 + (p * m) * hgcrt_mod_right_hpl_lift)))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (p * m)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. z + m * hgcrt_mod_left_hpl_lift = x + m * hgcrt_mod_right_hpl_lift) /\ (exists hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * z + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + (p * m) * hgcrt_mod_left_hpl_lift = 0 + (p * m) * hgcrt_mod_right_hpl_lift)))))) -> z = r)
  48. 0048specialize hensel_canonical_horner_lift_exists_unique b
  49. 0049specialize hensel_canonical_horner_lift_exists_unique c
  50. 0050specialize hensel_canonical_horner_lift_exists_unique x
  51. 0051specialize hensel_canonical_horner_lift_exists_unique l
  52. 0052specialize hensel_canonical_horner_lift_exists_unique x1
  53. 0053specialize hensel_canonical_horner_lift_exists_unique x2
  54. 0054specialize hensel_canonical_horner_lift_exists_unique m
  55. 0055specialize hensel_canonical_horner_lift_exists_unique p
  56. 0056specialize hensel_canonical_horner_lift_exists_unique s
  57. 0057specialize hensel_canonical_horner_lift_exists_unique x3
  58. 0058apply hensel_canonical_horner_lift_exists_unique
  59. 0059exact hp
  60. 0060exact hm
  61. 0061exact hnormalized_witness_right_right_witness_witness_left
  62. 0062exact hfactor
  63. 0063exact hquotient_witness
  64. 0064exact hnormalized_witness_right_right_witness_witness_right_right
  65. 0065exact hnormalized_witness_left
  66. 0066cases hcanonical
  67. 0067cases hcanonical_witness
  68. 0068cases hcanonical_witness_left
  69. 0069cases hcanonical_witness_left_right
  70. 0070exists x4
  71. 0071split
  72. 0072split
  73. 0073exact hcanonical_witness_left_left
  74. 0074split
  75. 0075specialize mod_eq_trans m
  76. 0076specialize mod_eq_trans x4
  77. 0077specialize mod_eq_trans x
  78. 0078specialize mod_eq_trans a
  79. 0079apply mod_eq_trans
  80. 0080exact hcanonical_witness_left_right_left
  81. 0081exact hnormalized_witness_right_left
  82. 0082exact hcanonical_witness_left_right_right
  83. 0083intro z
  84. 0084intro hz
  85. 0085cases hz
  86. 0086cases hz_right
  87. 0087specialize hcanonical_witness_right z
  88. 0088apply hcanonical_witness_right
  89. 0089split
  90. 0090exact hz_left
  91. 0091split
  92. 0092specialize mod_eq_trans m
  93. 0093specialize mod_eq_trans z
  94. 0094specialize mod_eq_trans a
  95. 0095specialize mod_eq_trans x
  96. 0096apply mod_eq_trans
  97. 0097exact hz_right_left
  98. 0098specialize mod_eq_symm m
  99. 0099specialize mod_eq_symm x
  100. 0100specialize mod_eq_symm a
  101. 0101apply mod_eq_symm
  102. 0102exact hnormalized_witness_right_left
  103. 0103exact hz_right_right