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.
Statement with defined notation
∀ q. ∀ b. ∀ c. ∀ l. (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ (y = 1 ∧ Lt(x,q) ∨ y = 0 ∧ Lt(q,S x))) → (∃ x. x = 1 ∧ Lt(l,q) ∨ x = 0 ∧ Lt(q,S l)) → ∃ x. ∃ y. ∀ z. Lt(z,S l) → ∃ n. BetaAt(x,y,z,n) ∧ (n = 1 ∧ Lt(z,q) ∨ n = 0 ∧ Lt(q,S z))Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
10 occurrences
In local proof propositions
4 occurrences
Exact expanded native-PA statement
forall q b c l. (forall eis_index_initial_segment_extend_before. (exists eis_lt_gap_initial_segment_extend_before_bound. eis_lt_gap_initial_segment_extend_before_bound + S (eis_index_initial_segment_extend_before) = l) -> exists eis_bit_initial_segment_extend_before. ((((exists ff_h_eis_initial_segment_extend_before_decoded. ff_h_eis_initial_segment_extend_before_decoded + S (eis_bit_initial_segment_extend_before) = S ((S (eis_index_initial_segment_extend_before)) * c)) /\ exists ff_q_eis_initial_segment_extend_before_decoded. b = ff_q_eis_initial_segment_extend_before_decoded * S ((S (eis_index_initial_segment_extend_before)) * c) + (eis_bit_initial_segment_extend_before))) /\ (((eis_bit_initial_segment_extend_before = 1 /\ (exists eis_le_gap_initial_segment_extend_before_choice_inside. eis_le_gap_initial_segment_extend_before_choice_inside + (S eis_index_initial_segment_extend_before) = q)) \/ (eis_bit_initial_segment_extend_before = 0 /\ (exists eis_lt_gap_initial_segment_extend_before_choice_outside. eis_lt_gap_initial_segment_extend_before_choice_outside + S (q) = S eis_index_initial_segment_extend_before)))))) -> (exists bit. (((bit = 1 /\ (exists eis_le_gap_initial_segment_extend_last_inside. eis_le_gap_initial_segment_extend_last_inside + (S l) = q)) \/ (bit = 0 /\ (exists eis_lt_gap_initial_segment_extend_last_outside. eis_lt_gap_initial_segment_extend_last_outside + S (q) = S l))))) -> exists z d. (forall eis_index_initial_segment_extend_after. (exists eis_lt_gap_initial_segment_extend_after_bound. eis_lt_gap_initial_segment_extend_after_bound + S (eis_index_initial_segment_extend_after) = S l) -> exists eis_bit_initial_segment_extend_after. ((((exists ff_h_eis_initial_segment_extend_after_decoded. ff_h_eis_initial_segment_extend_after_decoded + S (eis_bit_initial_segment_extend_after) = S ((S (eis_index_initial_segment_extend_after)) * d)) /\ exists ff_q_eis_initial_segment_extend_after_decoded. z = ff_q_eis_initial_segment_extend_after_decoded * S ((S (eis_index_initial_segment_extend_after)) * d) + (eis_bit_initial_segment_extend_after))) /\ (((eis_bit_initial_segment_extend_after = 1 /\ (exists eis_le_gap_initial_segment_extend_after_choice_inside. eis_le_gap_initial_segment_extend_after_choice_inside + (S eis_index_initial_segment_extend_after) = q)) \/ (eis_bit_initial_segment_extend_after = 0 /\ (exists eis_lt_gap_initial_segment_extend_after_choice_outside. eis_lt_gap_initial_segment_extend_after_choice_outside + S (q) = S eis_index_initial_segment_extend_after))))))Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
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 (2)
01Fix variables and assumptionsL1–6
02Separate the logical casesL7–7
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L7
cases hchoice
03Use earlier factsL8–11
04Separate the logical casesL12–14
05Construct an explicit witnessL15–16
06Fix variables and assumptionsL17–18
07Establish hsplitL19–23
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite lt succ eq or lt.
08Separate the logical casesL24–24
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L24
cases hsplit
09Construct an explicit witnessL25–25
Supply the displayed value, then prove that it has the required property.
- L25
exists x
10Separate the logical casesL26–26
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L26
split
11Calculate and transport equalitiesL27–28
12Use earlier factsL29–29
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L29
exact beta_prefix_extend_witness_witness_left
13Calculate and transport equalitiesL30–31
14Use earlier factsL32–32
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L32
exact hchoice_witness
15Establish holdL33–36
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprefix.
- L33
have hold : ∃ oldbit. BetaAt(b,c,j,oldbit) ∧ (oldbit = 1 ∧ Lt(j,q) ∨ oldbit = 0 ∧ Lt(q,S j))Definitions: BetaAt(b,c,j,oldbit)Lt(j,q)Lt(q,S j)Original native command in the exact edition - L34
specialize hprefix j - L35
apply hprefix - L36
exact hsplit_right
16Separate the logical casesL37–38
17Construct an explicit witnessL39–39
Supply the displayed value, then prove that it has the required property.
- L39
exists x3
18Separate the logical casesL40–40
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L40
split
19Use earlier factsL41–46
Instantiate or apply named facts and discharge the corresponding proof obligations.
Original defined command ledger · 46 lines
- 0001
intro q - 0002
intro b - 0003
intro c - 0004
intro l - 0005
intro hprefix - 0006
intro hchoice - 0007
cases hchoice - 0008
specialize beta_prefix_extend l - 0009
specialize beta_prefix_extend b - 0010
specialize beta_prefix_extend c - 0011
specialize beta_prefix_extend x - 0012
cases beta_prefix_extend - 0013
cases beta_prefix_extend_witness - 0014
cases beta_prefix_extend_witness_witness - 0015
exists x1 - 0016
exists x2 - 0017
intro j - 0018
intro hj - 0019
have hsplit : j = l ∨ Lt(j,l)Exact native replay line
have hsplit : j = l \/ exists gap. gap + S j = l - 0020
specialize finite_lt_succ_eq_or_lt l - 0021
specialize finite_lt_succ_eq_or_lt j - 0022
apply finite_lt_succ_eq_or_lt - 0023
exact hj - 0024
cases hsplit - 0025
exists x - 0026
split - 0027
rewrite hsplit_left - 0028
rewrite hsplit_left - 0029
exact beta_prefix_extend_witness_witness_left - 0030
rewrite hsplit_left - 0031
rewrite hsplit_left - 0032
exact hchoice_witness - 0033
have hold : ∃ oldbit. BetaAt(b,c,j,oldbit) ∧ (oldbit = 1 ∧ Lt(j,q) ∨ oldbit = 0 ∧ Lt(q,S j))Exact native replay line
have hold : exists oldbit. ((((exists ff_h_initial_segment_old. ff_h_initial_segment_old + S (oldbit) = S ((S (j)) * c)) /\ exists ff_q_initial_segment_old. b = ff_q_initial_segment_old * S ((S (j)) * c) + (oldbit))) /\ (((oldbit = 1 /\ (exists eis_le_gap_initial_segment_extend_old_choice_inside. eis_le_gap_initial_segment_extend_old_choice_inside + (S j) = q)) \/ (oldbit = 0 /\ (exists eis_lt_gap_initial_segment_extend_old_choice_outside. eis_lt_gap_initial_segment_extend_old_choice_outside + S (q) = S j))))) - 0034
specialize hprefix j - 0035
apply hprefix - 0036
exact hsplit_right - 0037
cases hold - 0038
cases hold_witness - 0039
exists x3 - 0040
split - 0041
specialize beta_prefix_extend_witness_witness_right j - 0042
specialize beta_prefix_extend_witness_witness_right x3 - 0043
apply beta_prefix_extend_witness_witness_right - 0044
exact hsplit_right - 0045
exact hold_witness_left - 0046
exact hold_witness_right