PH0007

beta_horner_eval_successor_decompose

A nonempty polynomial splits into its evaluated prefix and final coefficient.

Alpha v34 checked-use · first admitted v20 · 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.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ t. ∀ l. ∀ n. Horner(b,c,t,S l,n) → ∃ x. ∃ y. Beta(b,c,l,x) ∧ (Horner(b,c,t,l,y) ∧ n = y · t + x)

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

Definition DAG

Beta(b,c,i,x) · 1Horner(b,c,x,ell,z) · 2

Actual proof prerequisites

le_refl · checked external prerequisitele_succ · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c t l n. (exists ff_u_ph_successor ff_v_ph_successor. ((((exists fs_h_ph_successor_body_start. fs_h_ph_successor_body_start + S (0) = S ((S (0)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_start. ff_u_ph_successor = fs_q_ph_successor_body_start * S ((S (0)) * ff_v_ph_successor) + (0))) /\ ((((exists fs_h_ph_successor_body_terminal. fs_h_ph_successor_body_terminal + S (n) = S ((S (S l)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_terminal. ff_u_ph_successor = fs_q_ph_successor_body_terminal * S ((S (S l)) * ff_v_ph_successor) + (n))) /\ forall ff_i_ph_successor_body_steps. (exists ph_bound_successor_body_steps. ph_bound_successor_body_steps + S ff_i_ph_successor_body_steps = S l) -> exists ff_coefficient_ph_successor_body_steps ff_previous_ph_successor_body_steps ff_current_ph_successor_body_steps. ((((exists fs_h_ph_successor_body_steps_coefficient. fs_h_ph_successor_body_steps_coefficient + S (ff_coefficient_ph_successor_body_steps) = S ((S (ff_i_ph_successor_body_steps)) * c)) /\ exists fs_q_ph_successor_body_steps_coefficient. b = fs_q_ph_successor_body_steps_coefficient * S ((S (ff_i_ph_successor_body_steps)) * c) + (ff_coefficient_ph_successor_body_steps))) /\ ((((exists fs_h_ph_successor_body_steps_before. fs_h_ph_successor_body_steps_before + S (ff_previous_ph_successor_body_steps) = S ((S (ff_i_ph_successor_body_steps)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_steps_before. ff_u_ph_successor = fs_q_ph_successor_body_steps_before * S ((S (ff_i_ph_successor_body_steps)) * ff_v_ph_successor) + (ff_previous_ph_successor_body_steps))) /\ ((((exists fs_h_ph_successor_body_steps_after. fs_h_ph_successor_body_steps_after + S (ff_current_ph_successor_body_steps) = S ((S (S ff_i_ph_successor_body_steps)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_steps_after. ff_u_ph_successor = fs_q_ph_successor_body_steps_after * S ((S (S ff_i_ph_successor_body_steps)) * ff_v_ph_successor) + (ff_current_ph_successor_body_steps))) /\ ff_current_ph_successor_body_steps = ff_previous_ph_successor_body_steps * t + ff_coefficient_ph_successor_body_steps)))))) -> exists a r. ((((exists fs_h_ph_final_coefficient. fs_h_ph_final_coefficient + S (a) = S ((S (l)) * c)) /\ exists fs_q_ph_final_coefficient. b = fs_q_ph_final_coefficient * S ((S (l)) * c) + (a))) /\ ((exists ff_u_ph_prefix ff_v_ph_prefix. ((((exists fs_h_ph_prefix_body_start. fs_h_ph_prefix_body_start + S (0) = S ((S (0)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_start. ff_u_ph_prefix = fs_q_ph_prefix_body_start * S ((S (0)) * ff_v_ph_prefix) + (0))) /\ ((((exists fs_h_ph_prefix_body_terminal. fs_h_ph_prefix_body_terminal + S (r) = S ((S (l)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_terminal. ff_u_ph_prefix = fs_q_ph_prefix_body_terminal * S ((S (l)) * ff_v_ph_prefix) + (r))) /\ forall ff_i_ph_prefix_body_steps. (exists ph_bound_prefix_body_steps. ph_bound_prefix_body_steps + S ff_i_ph_prefix_body_steps = l) -> exists ff_coefficient_ph_prefix_body_steps ff_previous_ph_prefix_body_steps ff_current_ph_prefix_body_steps. ((((exists fs_h_ph_prefix_body_steps_coefficient. fs_h_ph_prefix_body_steps_coefficient + S (ff_coefficient_ph_prefix_body_steps) = S ((S (ff_i_ph_prefix_body_steps)) * c)) /\ exists fs_q_ph_prefix_body_steps_coefficient. b = fs_q_ph_prefix_body_steps_coefficient * S ((S (ff_i_ph_prefix_body_steps)) * c) + (ff_coefficient_ph_prefix_body_steps))) /\ ((((exists fs_h_ph_prefix_body_steps_before. fs_h_ph_prefix_body_steps_before + S (ff_previous_ph_prefix_body_steps) = S ((S (ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_steps_before. ff_u_ph_prefix = fs_q_ph_prefix_body_steps_before * S ((S (ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix) + (ff_previous_ph_prefix_body_steps))) /\ ((((exists fs_h_ph_prefix_body_steps_after. fs_h_ph_prefix_body_steps_after + S (ff_current_ph_prefix_body_steps) = S ((S (S ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_steps_after. ff_u_ph_prefix = fs_q_ph_prefix_body_steps_after * S ((S (S ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix) + (ff_current_ph_prefix_body_steps))) /\ ff_current_ph_prefix_body_steps = ff_previous_ph_prefix_body_steps * t + ff_coefficient_ph_prefix_body_steps)))))) /\ n = r * t + a))

Complete unchanged native tactic proof

All 52 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

52 script commands · 18 reading checkpoints · 2 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.

01Fix variables and assumptionsL1–6

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 hsum
02Separate the logical casesL7–10

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

  1. L7
    cases hsum
  2. L8
    cases hsum_witness
  3. L9
    cases hsum_witness_witness
  4. L10
    cases hsum_witness_witness_right
03Establish hstepL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hsum witness witness right right.

  1. L11
    have hstep : exists a r s. ((((exists fs_h_decomp_factor. fs_h_decomp_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_decomp_factor. b = fs_q_decomp_factor * S ((S (l)) * c) + (a))) /\ ((((exists fs_h_decomp_partial. fs_h_decomp_partial + S (r) = S ((S (l)) * x1)) /\ exists fs_q_decomp_partial. x = fs_q_decomp_partial * S ((S (l)) * x1) + (r))) /\ ((((exists fs_h_decomp_successor. fs_h_decomp_successor + S (s) = S ((S (S l)) * x1)) /\ exists fs_q_decomp_successor. x = fs_q_decomp_successor * S ((S (S l)) * x1) + (s))) /\ s = r * t + a)))
  2. L12
    specialize hsum_witness_witness_right_right l
  3. L13
    apply hsum_witness_witness_right_right
  4. L14
    specialize le_refl (S l)
  5. L15
    exact le_refl
04Separate the logical casesL16–21

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

  1. L16
    cases hstep
  2. L17
    cases hstep_witness
  3. L18
    cases hstep_witness_witness
  4. L19
    cases hstep_witness_witness_witness
  5. L20
    cases hstep_witness_witness_witness_right
  6. L21
    cases hstep_witness_witness_witness_right_right
05Establish hnL22–30

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

  1. L22
    have hn : n = x4
  2. L23
    specialize beta_at_unique x
  3. L24
    specialize beta_at_unique x1
  4. L25
    specialize beta_at_unique (S l)
  5. L26
    specialize beta_at_unique n
  6. L27
    specialize beta_at_unique x4
  7. L28
    apply beta_at_unique
  8. L29
    exact hsum_witness_witness_right_left
  9. L30
    exact hstep_witness_witness_witness_right_right_left
06Construct an explicit witnessL31–32

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

  1. L31
    exists x2
  2. L32
    exists x3
07Separate the logical casesL33–33

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

  1. L33
    split
08Use earlier factsL34–34

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

  1. L34
    exact hstep_witness_witness_witness_left
09Separate the logical casesL35–35

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

  1. L35
    split
10Construct an explicit witnessL36–37

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

  1. L36
    exists x
  2. L37
    exists x1
11Separate the logical casesL38–38

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

  1. L38
    split
12Use earlier factsL39–39

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

  1. L39
    exact hsum_witness_witness_left
13Separate the logical casesL40–40

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

  1. L40
    split
14Use earlier factsL41–41

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

  1. L41
    exact hstep_witness_witness_witness_right_left
15Fix variables and assumptionsL42–43

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

  1. L42
    intro i
  2. L43
    intro hi
16Use earlier factsL44–49

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

  1. L44
    specialize hsum_witness_witness_right_right i
  2. L45
    apply hsum_witness_witness_right_right
  3. L46
    specialize le_succ (S i)
  4. L47
    specialize le_succ l
  5. L48
    apply le_succ
  6. L49
    exact hi
17Calculate and transport equalitiesL50–50

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

  1. L50
    trans x4
18Use earlier factsL51–52

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

  1. L51
    exact hn
  2. L52
    exact hstep_witness_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 52 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005intro n
  6. 0006intro hsum
  7. 0007cases hsum
  8. 0008cases hsum_witness
  9. 0009cases hsum_witness_witness
  10. 0010cases hsum_witness_witness_right
  11. 0011have hstep : exists a r s. ((((exists fs_h_decomp_factor. fs_h_decomp_factor + S (a) = S ((S (l)) * c)) /\ exists fs_q_decomp_factor. b = fs_q_decomp_factor * S ((S (l)) * c) + (a))) /\ ((((exists fs_h_decomp_partial. fs_h_decomp_partial + S (r) = S ((S (l)) * x1)) /\ exists fs_q_decomp_partial. x = fs_q_decomp_partial * S ((S (l)) * x1) + (r))) /\ ((((exists fs_h_decomp_successor. fs_h_decomp_successor + S (s) = S ((S (S l)) * x1)) /\ exists fs_q_decomp_successor. x = fs_q_decomp_successor * S ((S (S l)) * x1) + (s))) /\ s = r * t + a)))
  12. 0012specialize hsum_witness_witness_right_right l
  13. 0013apply hsum_witness_witness_right_right
  14. 0014specialize le_refl (S l)
  15. 0015exact le_refl
  16. 0016cases hstep
  17. 0017cases hstep_witness
  18. 0018cases hstep_witness_witness
  19. 0019cases hstep_witness_witness_witness
  20. 0020cases hstep_witness_witness_witness_right
  21. 0021cases hstep_witness_witness_witness_right_right
  22. 0022have hn : n = x4
  23. 0023specialize beta_at_unique x
  24. 0024specialize beta_at_unique x1
  25. 0025specialize beta_at_unique (S l)
  26. 0026specialize beta_at_unique n
  27. 0027specialize beta_at_unique x4
  28. 0028apply beta_at_unique
  29. 0029exact hsum_witness_witness_right_left
  30. 0030exact hstep_witness_witness_witness_right_right_left
  31. 0031exists x2
  32. 0032exists x3
  33. 0033split
  34. 0034exact hstep_witness_witness_witness_left
  35. 0035split
  36. 0036exists x
  37. 0037exists x1
  38. 0038split
  39. 0039exact hsum_witness_witness_left
  40. 0040split
  41. 0041exact hstep_witness_witness_witness_right_left
  42. 0042intro i
  43. 0043intro hi
  44. 0044specialize hsum_witness_witness_right_right i
  45. 0045apply hsum_witness_witness_right_right
  46. 0046specialize le_succ (S i)
  47. 0047specialize le_succ l
  48. 0048apply le_succ
  49. 0049exact hi
  50. 0050trans x4
  51. 0051exact hn
  52. 0052exact hstep_witness_witness_witness_right_right_right