PH0004

beta_horner_eval_functional

The beta-coded polynomial-evaluation relation is functional.

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. ∀ m. Horner(b,c,t,l,n) → Horner(b,c,t,l,m) → n = m

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

Definition DAG

Horner(b,c,x,ell,z) · 2

Actual proof prerequisites

beta_horner_trace_functional
Original expanded first-order statement 
forall b c t l n m. (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)))))) -> (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

Complete unchanged native tactic proof

All 25 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

25 script commands · 4 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.

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 trace functional
01Fix variables and assumptionsL1–8

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 m
  7. L7
    intro hn
  8. L8
    intro hm
02Separate the logical casesL9–12

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

  1. L9
    cases hn
  2. L10
    cases hn_witness
  3. L11
    cases hm
  4. L12
    cases hm_witness
03Use earlier factsL13–22

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

  1. L13
    specialize beta_horner_trace_functional b
  2. L14
    specialize beta_horner_trace_functional c
  3. L15
    specialize beta_horner_trace_functional t
  4. L16
    specialize beta_horner_trace_functional l
  5. L17
    specialize beta_horner_trace_functional n
  6. L18
    specialize beta_horner_trace_functional x
  7. L19
    specialize beta_horner_trace_functional x1
  8. L20
    specialize beta_horner_trace_functional m
  9. L21
    specialize beta_horner_trace_functional x2
  10. L22
    specialize beta_horner_trace_functional x3
04Use earlier factsL23–25

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

  1. L23
    apply beta_horner_trace_functional
  2. L24
    exact hn_witness_witness
  3. L25
    exact hm_witness_witness

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro l
  5. 0005intro n
  6. 0006intro m
  7. 0007intro hn
  8. 0008intro hm
  9. 0009cases hn
  10. 0010cases hn_witness
  11. 0011cases hm
  12. 0012cases hm_witness
  13. 0013specialize beta_horner_trace_functional b
  14. 0014specialize beta_horner_trace_functional c
  15. 0015specialize beta_horner_trace_functional t
  16. 0016specialize beta_horner_trace_functional l
  17. 0017specialize beta_horner_trace_functional n
  18. 0018specialize beta_horner_trace_functional x
  19. 0019specialize beta_horner_trace_functional x1
  20. 0020specialize beta_horner_trace_functional m
  21. 0021specialize beta_horner_trace_functional x2
  22. 0022specialize beta_horner_trace_functional x3
  23. 0023apply beta_horner_trace_functional
  24. 0024exact hn_witness_witness
  25. 0025exact hm_witness_witness