PH0005

beta_horner_eval_exists_unique

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

Every coded natural polynomial has exactly one witnessed evaluation.

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 t l. exists n. ((exists ff_u_ph_root_value ff_v_ph_root_value. ((((exists fs_h_ph_root_value_body_start. fs_h_ph_root_value_body_start + S (0) = S ((S (0)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_start. ff_u_ph_root_value = fs_q_ph_root_value_body_start * S ((S (0)) * ff_v_ph_root_value) + (0))) /\ ((((exists fs_h_ph_root_value_body_terminal. fs_h_ph_root_value_body_terminal + S (n) = S ((S (l)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_terminal. ff_u_ph_root_value = fs_q_ph_root_value_body_terminal * S ((S (l)) * ff_v_ph_root_value) + (n))) /\ forall ff_i_ph_root_value_body_steps. (exists ph_bound_root_value_body_steps. ph_bound_root_value_body_steps + S ff_i_ph_root_value_body_steps = l) -> exists ff_coefficient_ph_root_value_body_steps ff_previous_ph_root_value_body_steps ff_current_ph_root_value_body_steps. ((((exists fs_h_ph_root_value_body_steps_coefficient. fs_h_ph_root_value_body_steps_coefficient + S (ff_coefficient_ph_root_value_body_steps) = S ((S (ff_i_ph_root_value_body_steps)) * c)) /\ exists fs_q_ph_root_value_body_steps_coefficient. b = fs_q_ph_root_value_body_steps_coefficient * S ((S (ff_i_ph_root_value_body_steps)) * c) + (ff_coefficient_ph_root_value_body_steps))) /\ ((((exists fs_h_ph_root_value_body_steps_before. fs_h_ph_root_value_body_steps_before + S (ff_previous_ph_root_value_body_steps) = S ((S (ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_steps_before. ff_u_ph_root_value = fs_q_ph_root_value_body_steps_before * S ((S (ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value) + (ff_previous_ph_root_value_body_steps))) /\ ((((exists fs_h_ph_root_value_body_steps_after. fs_h_ph_root_value_body_steps_after + S (ff_current_ph_root_value_body_steps) = S ((S (S ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_steps_after. ff_u_ph_root_value = fs_q_ph_root_value_body_steps_after * S ((S (S ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value) + (ff_current_ph_root_value_body_steps))) /\ ff_current_ph_root_value_body_steps = ff_previous_ph_root_value_body_steps * t + ff_coefficient_ph_root_value_body_steps)))))) /\ forall m. (exists ff_u_ph_root_other ff_v_ph_root_other. ((((exists fs_h_ph_root_other_body_start. fs_h_ph_root_other_body_start + S (0) = S ((S (0)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_start. ff_u_ph_root_other = fs_q_ph_root_other_body_start * S ((S (0)) * ff_v_ph_root_other) + (0))) /\ ((((exists fs_h_ph_root_other_body_terminal. fs_h_ph_root_other_body_terminal + S (m) = S ((S (l)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_terminal. ff_u_ph_root_other = fs_q_ph_root_other_body_terminal * S ((S (l)) * ff_v_ph_root_other) + (m))) /\ forall ff_i_ph_root_other_body_steps. (exists ph_bound_root_other_body_steps. ph_bound_root_other_body_steps + S ff_i_ph_root_other_body_steps = l) -> exists ff_coefficient_ph_root_other_body_steps ff_previous_ph_root_other_body_steps ff_current_ph_root_other_body_steps. ((((exists fs_h_ph_root_other_body_steps_coefficient. fs_h_ph_root_other_body_steps_coefficient + S (ff_coefficient_ph_root_other_body_steps) = S ((S (ff_i_ph_root_other_body_steps)) * c)) /\ exists fs_q_ph_root_other_body_steps_coefficient. b = fs_q_ph_root_other_body_steps_coefficient * S ((S (ff_i_ph_root_other_body_steps)) * c) + (ff_coefficient_ph_root_other_body_steps))) /\ ((((exists fs_h_ph_root_other_body_steps_before. fs_h_ph_root_other_body_steps_before + S (ff_previous_ph_root_other_body_steps) = S ((S (ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_steps_before. ff_u_ph_root_other = fs_q_ph_root_other_body_steps_before * S ((S (ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other) + (ff_previous_ph_root_other_body_steps))) /\ ((((exists fs_h_ph_root_other_body_steps_after. fs_h_ph_root_other_body_steps_after + S (ff_current_ph_root_other_body_steps) = S ((S (S ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_steps_after. ff_u_ph_root_other = fs_q_ph_root_other_body_steps_after * S ((S (S ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other) + (ff_current_ph_root_other_body_steps))) /\ ff_current_ph_root_other_body_steps = ff_previous_ph_root_other_body_steps * t + ff_coefficient_ph_root_other_body_steps)))))) -> n = m)

Constructive proof overview

Generated structural guide

Every coded natural polynomial has exactly one witnessed evaluation.

The unchanged tactic script uses 2 declared prerequisites and contains 23 exact native proof lines.

Alpha v34 checked-use · first admitted v20 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

PH0002 beta_horner_eval_exists PH0004 beta_horner_eval_functional

Direct dependents

none

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

23 script commands · 8 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (2)
beta horner eval existsbeta horner eval functional
01Fix variables and assumptionsL1–4

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
02Use earlier factsL5–8

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

  1. L5
    specialize beta_horner_eval_exists b
  2. L6
    specialize beta_horner_eval_exists c
  3. L7
    specialize beta_horner_eval_exists t
  4. L8
    specialize beta_horner_eval_exists l
03Separate the logical casesL9–9

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

  1. L9
    cases beta_horner_eval_exists
04Construct an explicit witnessL10–10

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

  1. L10
    exists x
05Separate the logical casesL11–11

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

  1. L11
    split
06Use earlier factsL12–12

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

  1. L12
    exact beta_horner_eval_exists_witness
07Fix variables and assumptionsL13–14

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

  1. L13
    intro m
  2. L14
    intro hm
08Use earlier factsL15–23

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

  1. L15
    specialize beta_horner_eval_functional b
  2. L16
    specialize beta_horner_eval_functional c
  3. L17
    specialize beta_horner_eval_functional t
  4. L18
    specialize beta_horner_eval_functional l
  5. L19
    specialize beta_horner_eval_functional x
  6. L20
    specialize beta_horner_eval_functional m
  7. L21
    apply beta_horner_eval_functional
  8. L22
    exact beta_horner_eval_exists_witness
  9. L23
    exact hm

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005specialize beta_horner_eval_exists b
  6. 0006specialize beta_horner_eval_exists c
  7. 0007specialize beta_horner_eval_exists t
  8. 0008specialize beta_horner_eval_exists l
  9. 0009cases beta_horner_eval_exists
  10. 0010exists x
  11. 0011split
  12. 0012exact beta_horner_eval_exists_witness
  13. 0013intro m
  14. 0014intro hm
  15. 0015specialize beta_horner_eval_functional b
  16. 0016specialize beta_horner_eval_functional c
  17. 0017specialize beta_horner_eval_functional t
  18. 0018specialize beta_horner_eval_functional l
  19. 0019specialize beta_horner_eval_functional x
  20. 0020specialize beta_horner_eval_functional m
  21. 0021apply beta_horner_eval_functional
  22. 0022exact beta_horner_eval_exists_witness
  23. 0023exact hm