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. ∀ k. (∀ x. Lt(x,k) → ∃ y. BetaAt(b,c,x,y) ∧ (y = 1 ∧ Lt(x,q) ∨ y = 0 ∧ Lt(q,S x))) → AllBits(b,c,k)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
5 occurrences
In local proof propositions
3 occurrences
Exact expanded native-PA statement
forall q b c k. (forall eis_index_initial_segment_bits_source. (exists eis_lt_gap_initial_segment_bits_source_bound. eis_lt_gap_initial_segment_bits_source_bound + S (eis_index_initial_segment_bits_source) = k) -> exists eis_bit_initial_segment_bits_source. ((((exists ff_h_eis_initial_segment_bits_source_decoded. ff_h_eis_initial_segment_bits_source_decoded + S (eis_bit_initial_segment_bits_source) = S ((S (eis_index_initial_segment_bits_source)) * c)) /\ exists ff_q_eis_initial_segment_bits_source_decoded. b = ff_q_eis_initial_segment_bits_source_decoded * S ((S (eis_index_initial_segment_bits_source)) * c) + (eis_bit_initial_segment_bits_source))) /\ (((eis_bit_initial_segment_bits_source = 1 /\ (exists eis_le_gap_initial_segment_bits_source_choice_inside. eis_le_gap_initial_segment_bits_source_choice_inside + (S eis_index_initial_segment_bits_source) = q)) \/ (eis_bit_initial_segment_bits_source = 0 /\ (exists eis_lt_gap_initial_segment_bits_source_choice_outside. eis_lt_gap_initial_segment_bits_source_choice_outside + S (q) = S eis_index_initial_segment_bits_source)))))) -> (forall ff_i_initial_segment_bits_result. (exists ff_lt_initial_segment_bits_result_bound. ff_lt_initial_segment_bits_result_bound + S ff_i_initial_segment_bits_result = k) -> exists ff_bit_initial_segment_bits_result. ((((exists ff_h_initial_segment_bits_result_decoded. ff_h_initial_segment_bits_result_decoded + S (ff_bit_initial_segment_bits_result) = S ((S (ff_i_initial_segment_bits_result)) * c)) /\ exists ff_q_initial_segment_bits_result_decoded. b = ff_q_initial_segment_bits_result_decoded * S ((S (ff_i_initial_segment_bits_result)) * c) + (ff_bit_initial_segment_bits_result))) /\ (ff_bit_initial_segment_bits_result = 0 \/ ff_bit_initial_segment_bits_result = 1)))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.
01Fix variables and assumptionsL1–7
02Use earlier factsL8–8
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L8
specialize hprefix j
03Establish hstoredL9–11
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprefix.
- L9
have hstored : ∃ bit. BetaAt(b,c,j,bit) ∧ (bit = 1 ∧ Lt(j,q) ∨ bit = 0 ∧ Lt(q,S j))Definitions: BetaAt(b,c,j,bit)Lt(j,q)Lt(q,S j)Original native command in the exact edition - L10
apply hprefix - L11
exact hj
04Separate the logical casesL12–13
05Construct an explicit witnessL14–14
Supply the displayed value, then prove that it has the required property.
- L14
exists x
06Separate the logical casesL15–15
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L15
split
07Use earlier factsL16–16
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L16
exact hstored_witness_left
08Separate the logical casesL17–19
09Use earlier factsL20–20
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L20
exact hstored_witness_right_left_left
10Separate the logical casesL21–22
11Use earlier factsL23–23
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L23
exact hstored_witness_right_right_left
Original defined command ledger · 23 lines
- 0001
intro q - 0002
intro b - 0003
intro c - 0004
intro k - 0005
intro hprefix - 0006
intro j - 0007
intro hj - 0008
specialize hprefix j - 0009
have hstored : ∃ bit. BetaAt(b,c,j,bit) ∧ (bit = 1 ∧ Lt(j,q) ∨ bit = 0 ∧ Lt(q,S j))Exact native replay line
have hstored : exists bit. ((((exists ff_h_initial_segment_bits_stored. ff_h_initial_segment_bits_stored + S (bit) = S ((S (j)) * c)) /\ exists ff_q_initial_segment_bits_stored. b = ff_q_initial_segment_bits_stored * S ((S (j)) * c) + (bit))) /\ (((bit = 1 /\ (exists eis_le_gap_initial_segment_bits_choice_inside. eis_le_gap_initial_segment_bits_choice_inside + (S j) = q)) \/ (bit = 0 /\ (exists eis_lt_gap_initial_segment_bits_choice_outside. eis_lt_gap_initial_segment_bits_choice_outside + S (q) = S j))))) - 0010
apply hprefix - 0011
exact hj - 0012
cases hstored - 0013
cases hstored_witness - 0014
exists x - 0015
split - 0016
exact hstored_witness_left - 0017
cases hstored_witness_right - 0018
cases hstored_witness_right_left - 0019
right - 0020
exact hstored_witness_right_left_left - 0021
cases hstored_witness_right_right - 0022
left - 0023
exact hstored_witness_right_right_left