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 PA statement
forall b c l. (forall ff_i_a. (exists ff_lt_a_bound. ff_lt_a_bound + S ff_i_a = l) -> exists ff_bit_a. ((((exists ff_h_a_decoded. ff_h_a_decoded + S (ff_bit_a) = S ((S (ff_i_a)) * c)) /\ exists ff_q_a_decoded. b = ff_q_a_decoded * S ((S (ff_i_a)) * c) + (ff_bit_a))) /\ (ff_bit_a = 0 \/ ff_bit_a = 1))) -> exists n. (((exists ff_u_b_sum ff_v_b_sum. ((((exists ff_h_b_sum_start. ff_h_b_sum_start + S (0) = S ((S (0)) * ff_v_b_sum)) /\ exists ff_q_b_sum_start. ff_u_b_sum = ff_q_b_sum_start * S ((S (0)) * ff_v_b_sum) + (0))) /\ ((((exists ff_h_b_sum_terminal. ff_h_b_sum_terminal + S (n) = S ((S (l)) * ff_v_b_sum)) /\ exists ff_q_b_sum_terminal. ff_u_b_sum = ff_q_b_sum_terminal * S ((S (l)) * ff_v_b_sum) + (n))) /\ forall ff_i_b_sum. (exists ff_lt_b_sum_bound. ff_lt_b_sum_bound + S ff_i_b_sum = l) -> exists ff_a_b_sum ff_r_b_sum ff_s_b_sum. ((((exists ff_h_b_sum_summand. ff_h_b_sum_summand + S (ff_a_b_sum) = S ((S (ff_i_b_sum)) * c)) /\ exists ff_q_b_sum_summand. b = ff_q_b_sum_summand * S ((S (ff_i_b_sum)) * c) + (ff_a_b_sum))) /\ ((((exists ff_h_b_sum_partial. ff_h_b_sum_partial + S (ff_r_b_sum) = S ((S (ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_partial. ff_u_b_sum = ff_q_b_sum_partial * S ((S (ff_i_b_sum)) * ff_v_b_sum) + (ff_r_b_sum))) /\ ((((exists ff_h_b_sum_successor. ff_h_b_sum_successor + S (ff_s_b_sum) = S ((S (S ff_i_b_sum)) * ff_v_b_sum)) /\ exists ff_q_b_sum_successor. ff_u_b_sum = ff_q_b_sum_successor * S ((S (S ff_i_b_sum)) * ff_v_b_sum) + (ff_s_b_sum))) /\ ff_s_b_sum = ff_r_b_sum + ff_a_b_sum)))))) /\ (forall ff_i_b_bits. (exists ff_lt_b_bits_bound. ff_lt_b_bits_bound + S ff_i_b_bits = l) -> exists ff_bit_b_bits. ((((exists ff_h_b_bits_decoded. ff_h_b_bits_decoded + S (ff_bit_b_bits) = S ((S (ff_i_b_bits)) * c)) /\ exists ff_q_b_bits_decoded. b = ff_q_b_bits_decoded * S ((S (ff_i_b_bits)) * c) + (ff_bit_b_bits))) /\ (ff_bit_b_bits = 0 \/ ff_bit_b_bits = 1)))))Structural proof guide
Every all-bits prefix has a relational count of its ones.
Direct prerequisites: beta_sum_exists. The authored body proceeds by case analysis (1).
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.
Read the argument
Proof checkpoints
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 (1)
01Fix variables and assumptionsL1–4
02Use earlier factsL5–7
03Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
cases beta_sum_exists
04Construct an explicit witnessL9–9
Supply the displayed value, then prove that it has the required property.
- L9
exists x
05Separate the logical casesL10–10
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L10
split
Original exact command ledger · 12 lines
- 0001
intro b - 0002
intro c - 0003
intro l - 0004
intro hbits - 0005
specialize beta_sum_exists b - 0006
specialize beta_sum_exists c - 0007
specialize beta_sum_exists l - 0008
cases beta_sum_exists - 0009
exists x - 0010
split - 0011
exact beta_sum_exists_witness - 0012
exact hbits