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 p. ((exists ff_u_fms_count ff_v_fms_count. ((((exists ff_h_fms_count_start. ff_h_fms_count_start + S (0) = S ((S (0)) * ff_v_fms_count)) /\ exists ff_q_fms_count_start. ff_u_fms_count = ff_q_fms_count_start * S ((S (0)) * ff_v_fms_count) + (0))) /\ ((((exists ff_h_fms_count_terminal. ff_h_fms_count_terminal + S ((0)) = S ((S ((p))) * ff_v_fms_count)) /\ exists ff_q_fms_count_terminal. ff_u_fms_count = ff_q_fms_count_terminal * S ((S ((p))) * ff_v_fms_count) + ((0)))) /\ forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (p)) -> exists ff_a_fms_count ff_r_fms_count ff_s_fms_count. ((((exists ff_h_fms_count_summand. ff_h_fms_count_summand + S (ff_a_fms_count) = S ((S (ff_i_fms_count)) * (0))) /\ exists ff_q_fms_count_summand. (0) = ff_q_fms_count_summand * S ((S (ff_i_fms_count)) * (0)) + (ff_a_fms_count))) /\ ((((exists ff_h_fms_count_partial. ff_h_fms_count_partial + S (ff_r_fms_count) = S ((S (ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_partial. ff_u_fms_count = ff_q_fms_count_partial * S ((S (ff_i_fms_count)) * ff_v_fms_count) + (ff_r_fms_count))) /\ ((((exists ff_h_fms_count_successor. ff_h_fms_count_successor + S (ff_s_fms_count) = S ((S (S ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_successor. ff_u_fms_count = ff_q_fms_count_successor * S ((S (S ff_i_fms_count)) * ff_v_fms_count) + (ff_s_fms_count))) /\ ff_s_fms_count = ff_r_fms_count + ff_a_fms_count)))))) /\ (forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (p)) -> exists ff_bit_fms_count. ((((exists ff_h_fms_count_decoded. ff_h_fms_count_decoded + S (ff_bit_fms_count) = S ((S (ff_i_fms_count)) * (0))) /\ exists ff_q_fms_count_decoded. (0) = ff_q_fms_count_decoded * S ((S (ff_i_fms_count)) * (0)) + (ff_bit_fms_count))) /\ (ff_bit_fms_count = 0 \/ ff_bit_fms_count = 1))))Constructive proof overview
Generated structural guide
The literal zero code is an actual empty characteristic set with an exact zero sum trace at every ambient size.
The unchanged tactic script uses 1 declared prerequisite and contains 33 exact native proof lines.
Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
Direct dependents
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
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 assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro p
02Separate the logical casesL2–2
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L2
split
03Construct an explicit witnessL3–4
04Separate the logical casesL5–5
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L5
split
05Use earlier factsL6–7
06Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
split
07Use earlier factsL9–10
08Fix variables and assumptionsL11–12
09Construct an explicit witnessL13–15
10Separate the logical casesL16–16
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L16
split
11Use earlier factsL17–18
12Separate the logical casesL19–19
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L19
split
13Use earlier factsL20–21
14Separate the logical casesL22–22
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L22
split
15Use earlier factsL23–24
16Calculate and transport equalitiesL25–25
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L25
simp
17Fix variables and assumptionsL26–27
18Construct an explicit witnessL28–28
Supply the displayed value, then prove that it has the required property.
- L28
exists 0
19Separate the logical casesL29–29
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L29
split
20Use earlier factsL30–31
21Separate the logical casesL32–32
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L32
left
22Calculate and transport equalitiesL33–33
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L33
refl
Original exact command ledger · 33 lines
- 0001
intro p - 0002
split - 0003
exists 0 - 0004
exists 0 - 0005
split - 0006
specialize finite_beta_zero_code 0 - 0007
apply finite_beta_zero_code - 0008
split - 0009
specialize finite_beta_zero_code p - 0010
apply finite_beta_zero_code - 0011
intro i - 0012
intro hi - 0013
exists 0 - 0014
exists 0 - 0015
exists 0 - 0016
split - 0017
specialize finite_beta_zero_code i - 0018
apply finite_beta_zero_code - 0019
split - 0020
specialize finite_beta_zero_code i - 0021
apply finite_beta_zero_code - 0022
split - 0023
specialize finite_beta_zero_code S i - 0024
apply finite_beta_zero_code - 0025
simp - 0026
intro i - 0027
intro hi - 0028
exists 0 - 0029
split - 0030
specialize finite_beta_zero_code i - 0031
apply finite_beta_zero_code - 0032
left - 0033
refl