BD0011

binary_digit_operation_count_functional

The independently witnessed initialization/square/multiply operation count is functional for every fixed beta-coded digit prefix.

Alpha v34 checked-use · first admitted v23 · 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.

The exact G102 milestone is fully proved for every natural exponent and every modulus greater than one, including actual canonical digits, a beta-coded accumulator execution, modular-power correctness, and the formal bound k≤3·BitLen(e)+2. The independent T13 determinant/rank/integer-span substrate is now closed in the separate Alpha-v27 integer-linear-algebra branch. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ s. ∀ t. BinaryExecutionOperationCount(b,c,l,s) → BinaryExecutionOperationCount(b,c,l,t) → s = t

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

Definition DAG

BinaryExecutionOperationCount(b,c,ell,k) · 2

Actual proof prerequisites

bit_count_functional · checked external prerequisite
Original expanded first-order statement 
forall b c l s t. (exists ff_ones_bd_cost_left. ((((exists ff_u_bd_cost_left_count_sum ff_v_bd_cost_left_count_sum. ((((exists ff_h_bd_cost_left_count_sum_start. ff_h_bd_cost_left_count_sum_start + S (0) = S ((S (0)) * ff_v_bd_cost_left_count_sum)) /\ exists ff_q_bd_cost_left_count_sum_start. ff_u_bd_cost_left_count_sum = ff_q_bd_cost_left_count_sum_start * S ((S (0)) * ff_v_bd_cost_left_count_sum) + (0))) /\ ((((exists ff_h_bd_cost_left_count_sum_terminal. ff_h_bd_cost_left_count_sum_terminal + S (ff_ones_bd_cost_left) = S ((S (l)) * ff_v_bd_cost_left_count_sum)) /\ exists ff_q_bd_cost_left_count_sum_terminal. ff_u_bd_cost_left_count_sum = ff_q_bd_cost_left_count_sum_terminal * S ((S (l)) * ff_v_bd_cost_left_count_sum) + (ff_ones_bd_cost_left))) /\ forall ff_i_bd_cost_left_count_sum. (exists ff_lt_bd_cost_left_count_sum_bound. ff_lt_bd_cost_left_count_sum_bound + S ff_i_bd_cost_left_count_sum = l) -> exists ff_a_bd_cost_left_count_sum ff_r_bd_cost_left_count_sum ff_s_bd_cost_left_count_sum. ((((exists ff_h_bd_cost_left_count_sum_summand. ff_h_bd_cost_left_count_sum_summand + S (ff_a_bd_cost_left_count_sum) = S ((S (ff_i_bd_cost_left_count_sum)) * c)) /\ exists ff_q_bd_cost_left_count_sum_summand. b = ff_q_bd_cost_left_count_sum_summand * S ((S (ff_i_bd_cost_left_count_sum)) * c) + (ff_a_bd_cost_left_count_sum))) /\ ((((exists ff_h_bd_cost_left_count_sum_partial. ff_h_bd_cost_left_count_sum_partial + S (ff_r_bd_cost_left_count_sum) = S ((S (ff_i_bd_cost_left_count_sum)) * ff_v_bd_cost_left_count_sum)) /\ exists ff_q_bd_cost_left_count_sum_partial. ff_u_bd_cost_left_count_sum = ff_q_bd_cost_left_count_sum_partial * S ((S (ff_i_bd_cost_left_count_sum)) * ff_v_bd_cost_left_count_sum) + (ff_r_bd_cost_left_count_sum))) /\ ((((exists ff_h_bd_cost_left_count_sum_successor. ff_h_bd_cost_left_count_sum_successor + S (ff_s_bd_cost_left_count_sum) = S ((S (S ff_i_bd_cost_left_count_sum)) * ff_v_bd_cost_left_count_sum)) /\ exists ff_q_bd_cost_left_count_sum_successor. ff_u_bd_cost_left_count_sum = ff_q_bd_cost_left_count_sum_successor * S ((S (S ff_i_bd_cost_left_count_sum)) * ff_v_bd_cost_left_count_sum) + (ff_s_bd_cost_left_count_sum))) /\ ff_s_bd_cost_left_count_sum = ff_r_bd_cost_left_count_sum + ff_a_bd_cost_left_count_sum)))))) /\ (forall ff_i_bd_cost_left_count_bits. (exists ff_lt_bd_cost_left_count_bits_bound. ff_lt_bd_cost_left_count_bits_bound + S ff_i_bd_cost_left_count_bits = l) -> exists ff_bit_bd_cost_left_count_bits. ((((exists ff_h_bd_cost_left_count_bits_decoded. ff_h_bd_cost_left_count_bits_decoded + S (ff_bit_bd_cost_left_count_bits) = S ((S (ff_i_bd_cost_left_count_bits)) * c)) /\ exists ff_q_bd_cost_left_count_bits_decoded. b = ff_q_bd_cost_left_count_bits_decoded * S ((S (ff_i_bd_cost_left_count_bits)) * c) + (ff_bit_bd_cost_left_count_bits))) /\ (ff_bit_bd_cost_left_count_bits = 0 \/ ff_bit_bd_cost_left_count_bits = 1))))) /\ s = (2 + (l + l)) + ff_ones_bd_cost_left)) -> (exists ff_ones_bd_cost_right. ((((exists ff_u_bd_cost_right_count_sum ff_v_bd_cost_right_count_sum. ((((exists ff_h_bd_cost_right_count_sum_start. ff_h_bd_cost_right_count_sum_start + S (0) = S ((S (0)) * ff_v_bd_cost_right_count_sum)) /\ exists ff_q_bd_cost_right_count_sum_start. ff_u_bd_cost_right_count_sum = ff_q_bd_cost_right_count_sum_start * S ((S (0)) * ff_v_bd_cost_right_count_sum) + (0))) /\ ((((exists ff_h_bd_cost_right_count_sum_terminal. ff_h_bd_cost_right_count_sum_terminal + S (ff_ones_bd_cost_right) = S ((S (l)) * ff_v_bd_cost_right_count_sum)) /\ exists ff_q_bd_cost_right_count_sum_terminal. ff_u_bd_cost_right_count_sum = ff_q_bd_cost_right_count_sum_terminal * S ((S (l)) * ff_v_bd_cost_right_count_sum) + (ff_ones_bd_cost_right))) /\ forall ff_i_bd_cost_right_count_sum. (exists ff_lt_bd_cost_right_count_sum_bound. ff_lt_bd_cost_right_count_sum_bound + S ff_i_bd_cost_right_count_sum = l) -> exists ff_a_bd_cost_right_count_sum ff_r_bd_cost_right_count_sum ff_s_bd_cost_right_count_sum. ((((exists ff_h_bd_cost_right_count_sum_summand. ff_h_bd_cost_right_count_sum_summand + S (ff_a_bd_cost_right_count_sum) = S ((S (ff_i_bd_cost_right_count_sum)) * c)) /\ exists ff_q_bd_cost_right_count_sum_summand. b = ff_q_bd_cost_right_count_sum_summand * S ((S (ff_i_bd_cost_right_count_sum)) * c) + (ff_a_bd_cost_right_count_sum))) /\ ((((exists ff_h_bd_cost_right_count_sum_partial. ff_h_bd_cost_right_count_sum_partial + S (ff_r_bd_cost_right_count_sum) = S ((S (ff_i_bd_cost_right_count_sum)) * ff_v_bd_cost_right_count_sum)) /\ exists ff_q_bd_cost_right_count_sum_partial. ff_u_bd_cost_right_count_sum = ff_q_bd_cost_right_count_sum_partial * S ((S (ff_i_bd_cost_right_count_sum)) * ff_v_bd_cost_right_count_sum) + (ff_r_bd_cost_right_count_sum))) /\ ((((exists ff_h_bd_cost_right_count_sum_successor. ff_h_bd_cost_right_count_sum_successor + S (ff_s_bd_cost_right_count_sum) = S ((S (S ff_i_bd_cost_right_count_sum)) * ff_v_bd_cost_right_count_sum)) /\ exists ff_q_bd_cost_right_count_sum_successor. ff_u_bd_cost_right_count_sum = ff_q_bd_cost_right_count_sum_successor * S ((S (S ff_i_bd_cost_right_count_sum)) * ff_v_bd_cost_right_count_sum) + (ff_s_bd_cost_right_count_sum))) /\ ff_s_bd_cost_right_count_sum = ff_r_bd_cost_right_count_sum + ff_a_bd_cost_right_count_sum)))))) /\ (forall ff_i_bd_cost_right_count_bits. (exists ff_lt_bd_cost_right_count_bits_bound. ff_lt_bd_cost_right_count_bits_bound + S ff_i_bd_cost_right_count_bits = l) -> exists ff_bit_bd_cost_right_count_bits. ((((exists ff_h_bd_cost_right_count_bits_decoded. ff_h_bd_cost_right_count_bits_decoded + S (ff_bit_bd_cost_right_count_bits) = S ((S (ff_i_bd_cost_right_count_bits)) * c)) /\ exists ff_q_bd_cost_right_count_bits_decoded. b = ff_q_bd_cost_right_count_bits_decoded * S ((S (ff_i_bd_cost_right_count_bits)) * c) + (ff_bit_bd_cost_right_count_bits))) /\ (ff_bit_bd_cost_right_count_bits = 0 \/ ff_bit_bd_cost_right_count_bits = 1))))) /\ t = (2 + (l + l)) + ff_ones_bd_cost_right)) -> s = t

Complete unchanged native tactic proof

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

Read the argument

Proof checkpoints

25 script commands · 7 reading checkpoints · 1 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 assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro s
  5. L5
    intro t
  6. L6
    intro hleft
  7. L7
    intro hright
02Separate the logical casesL8–11

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

  1. L8
    cases hleft
  2. L9
    cases hleft_witness
  3. L10
    cases hright
  4. L11
    cases hright_witness
03Establish honesL12–21

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

  1. L12
    have hones : x = x1
  2. L13
    specialize bit_count_functional b
  3. L14
    specialize bit_count_functional c
  4. L15
    specialize bit_count_functional l
  5. L16
    specialize bit_count_functional x
  6. L17
    specialize bit_count_functional x1
  7. L18
    apply bit_count_functional
  8. L19
    exact hleft_witness_left
  9. L20
    exact hright_witness_left
  10. L21
    rewrite hones at hleft_witness_right
04Calculate and transport equalitiesL22–22

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

  1. L22
    trans (2 + (l + l)) + x1
05Use earlier factsL23–23

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

  1. L23
    exact hleft_witness_right
06Calculate and transport equalitiesL24–24

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

  1. L24
    symm
07Use earlier factsL25–25

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

  1. L25
    exact hright_witness_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro s
  5. 0005intro t
  6. 0006intro hleft
  7. 0007intro hright
  8. 0008cases hleft
  9. 0009cases hleft_witness
  10. 0010cases hright
  11. 0011cases hright_witness
  12. 0012have hones : x = x1
  13. 0013specialize bit_count_functional b
  14. 0014specialize bit_count_functional c
  15. 0015specialize bit_count_functional l
  16. 0016specialize bit_count_functional x
  17. 0017specialize bit_count_functional x1
  18. 0018apply bit_count_functional
  19. 0019exact hleft_witness_left
  20. 0020exact hright_witness_left
  21. 0021rewrite hones at hleft_witness_right
  22. 0022trans (2 + (l + l)) + x1
  23. 0023exact hleft_witness_right
  24. 0024symm
  25. 0025exact hright_witness_right