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
∀ n. ∀ m. ∀ k. ∀ z. n = m → Choose(n,k,z) → Choose(m,k,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
2 occurrences
In local proof propositions
none
0 occurrences
Exact expanded native-PA statement
forall n m k z. n = m -> (((exists bcf_lt_gap_bcuet_source_out_of_range. bcf_lt_gap_bcuet_source_out_of_range + S (n) = k) /\ z = 0) \/ ((exists bcf_le_gap_bcuet_source_in_range. bcf_le_gap_bcuet_source_in_range + (k) = n) /\ (exists bcf_row_code_code_bcuet_source bcf_row_code_scale_bcuet_source bcf_row_scale_code_bcuet_source bcf_row_scale_scale_bcuet_source bcf_row_code_bcuet_source bcf_row_scale_bcuet_source. ((forall bcf_row_index_bcuet_source_table. (exists bcf_lt_gap_bcuet_source_table_row_bound. bcf_lt_gap_bcuet_source_table_row_bound + S (bcf_row_index_bcuet_source_table) = S (n)) -> exists bcf_row_code_bcuet_source_table bcf_row_scale_bcuet_source_table. ((((exists bcf_height_bcuet_source_table_decoded_row_code. bcf_height_bcuet_source_table_decoded_row_code + S (bcf_row_code_bcuet_source_table) = S ((S (bcf_row_index_bcuet_source_table)) * bcf_row_code_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_table_decoded_row_code. bcf_row_code_code_bcuet_source = bcf_quotient_bcuet_source_table_decoded_row_code * S ((S (bcf_row_index_bcuet_source_table)) * bcf_row_code_scale_bcuet_source) + (bcf_row_code_bcuet_source_table))) /\ ((((exists bcf_height_bcuet_source_table_decoded_row_scale. bcf_height_bcuet_source_table_decoded_row_scale + S (bcf_row_scale_bcuet_source_table) = S ((S (bcf_row_index_bcuet_source_table)) * bcf_row_scale_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_table_decoded_row_scale. bcf_row_scale_code_bcuet_source = bcf_quotient_bcuet_source_table_decoded_row_scale * S ((S (bcf_row_index_bcuet_source_table)) * bcf_row_scale_scale_bcuet_source) + (bcf_row_scale_bcuet_source_table))) /\ ((bcf_row_index_bcuet_source_table = 0 /\ (forall bcf_index_bcuet_source_table_zero_row. (exists bcf_lt_gap_bcuet_source_table_zero_row_bound. bcf_lt_gap_bcuet_source_table_zero_row_bound + S (bcf_index_bcuet_source_table_zero_row) = S (n)) -> exists bcf_value_bcuet_source_table_zero_row. ((((exists bcf_height_bcuet_source_table_zero_row_entry. bcf_height_bcuet_source_table_zero_row_entry + S (bcf_value_bcuet_source_table_zero_row) = S ((S (bcf_index_bcuet_source_table_zero_row)) * bcf_row_scale_bcuet_source_table)) /\ exists bcf_quotient_bcuet_source_table_zero_row_entry. bcf_row_code_bcuet_source_table = bcf_quotient_bcuet_source_table_zero_row_entry * S ((S (bcf_index_bcuet_source_table_zero_row)) * bcf_row_scale_bcuet_source_table) + (bcf_value_bcuet_source_table_zero_row))) /\ ((bcf_index_bcuet_source_table_zero_row = 0 /\ bcf_value_bcuet_source_table_zero_row = 1) \/ exists bcf_predecessor_bcuet_source_table_zero_row. bcf_index_bcuet_source_table_zero_row = S bcf_predecessor_bcuet_source_table_zero_row /\ bcf_value_bcuet_source_table_zero_row = 0)))) \/ exists bcf_predecessor_bcuet_source_table bcf_previous_code_bcuet_source_table bcf_previous_scale_bcuet_source_table. bcf_row_index_bcuet_source_table = S bcf_predecessor_bcuet_source_table /\ ((((exists bcf_height_bcuet_source_table_decoded_previous_code. bcf_height_bcuet_source_table_decoded_previous_code + S (bcf_previous_code_bcuet_source_table) = S ((S (bcf_predecessor_bcuet_source_table)) * bcf_row_code_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_table_decoded_previous_code. bcf_row_code_code_bcuet_source = bcf_quotient_bcuet_source_table_decoded_previous_code * S ((S (bcf_predecessor_bcuet_source_table)) * bcf_row_code_scale_bcuet_source) + (bcf_previous_code_bcuet_source_table))) /\ ((((exists bcf_height_bcuet_source_table_decoded_previous_scale. bcf_height_bcuet_source_table_decoded_previous_scale + S (bcf_previous_scale_bcuet_source_table) = S ((S (bcf_predecessor_bcuet_source_table)) * bcf_row_scale_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_table_decoded_previous_scale. bcf_row_scale_code_bcuet_source = bcf_quotient_bcuet_source_table_decoded_previous_scale * S ((S (bcf_predecessor_bcuet_source_table)) * bcf_row_scale_scale_bcuet_source) + (bcf_previous_scale_bcuet_source_table))) /\ (forall bcf_index_bcuet_source_table_row_step. (exists bcf_lt_gap_bcuet_source_table_row_step_bound. bcf_lt_gap_bcuet_source_table_row_step_bound + S (bcf_index_bcuet_source_table_row_step) = S (n)) -> exists bcf_value_bcuet_source_table_row_step. ((((exists bcf_height_bcuet_source_table_row_step_entry. bcf_height_bcuet_source_table_row_step_entry + S (bcf_value_bcuet_source_table_row_step) = S ((S (bcf_index_bcuet_source_table_row_step)) * bcf_row_scale_bcuet_source_table)) /\ exists bcf_quotient_bcuet_source_table_row_step_entry. bcf_row_code_bcuet_source_table = bcf_quotient_bcuet_source_table_row_step_entry * S ((S (bcf_index_bcuet_source_table_row_step)) * bcf_row_scale_bcuet_source_table) + (bcf_value_bcuet_source_table_row_step))) /\ ((bcf_index_bcuet_source_table_row_step = 0 /\ bcf_value_bcuet_source_table_row_step = 1) \/ exists bcf_predecessor_bcuet_source_table_row_step bcf_left_bcuet_source_table_row_step bcf_right_bcuet_source_table_row_step. bcf_index_bcuet_source_table_row_step = S bcf_predecessor_bcuet_source_table_row_step /\ ((((exists bcf_height_bcuet_source_table_row_step_previous_left. bcf_height_bcuet_source_table_row_step_previous_left + S (bcf_left_bcuet_source_table_row_step) = S ((S (bcf_predecessor_bcuet_source_table_row_step)) * bcf_previous_scale_bcuet_source_table)) /\ exists bcf_quotient_bcuet_source_table_row_step_previous_left. bcf_previous_code_bcuet_source_table = bcf_quotient_bcuet_source_table_row_step_previous_left * S ((S (bcf_predecessor_bcuet_source_table_row_step)) * bcf_previous_scale_bcuet_source_table) + (bcf_left_bcuet_source_table_row_step))) /\ ((((exists bcf_height_bcuet_source_table_row_step_previous_right. bcf_height_bcuet_source_table_row_step_previous_right + S (bcf_right_bcuet_source_table_row_step) = S ((S (S (bcf_predecessor_bcuet_source_table_row_step))) * bcf_previous_scale_bcuet_source_table)) /\ exists bcf_quotient_bcuet_source_table_row_step_previous_right. bcf_previous_code_bcuet_source_table = bcf_quotient_bcuet_source_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcuet_source_table_row_step))) * bcf_previous_scale_bcuet_source_table) + (bcf_right_bcuet_source_table_row_step))) /\ bcf_value_bcuet_source_table_row_step = bcf_left_bcuet_source_table_row_step + bcf_right_bcuet_source_table_row_step))))))))))) /\ ((((exists bcf_height_bcuet_source_decoded_row_code. bcf_height_bcuet_source_decoded_row_code + S (bcf_row_code_bcuet_source) = S ((S (n)) * bcf_row_code_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_decoded_row_code. bcf_row_code_code_bcuet_source = bcf_quotient_bcuet_source_decoded_row_code * S ((S (n)) * bcf_row_code_scale_bcuet_source) + (bcf_row_code_bcuet_source))) /\ ((((exists bcf_height_bcuet_source_decoded_row_scale. bcf_height_bcuet_source_decoded_row_scale + S (bcf_row_scale_bcuet_source) = S ((S (n)) * bcf_row_scale_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_decoded_row_scale. bcf_row_scale_code_bcuet_source = bcf_quotient_bcuet_source_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_bcuet_source) + (bcf_row_scale_bcuet_source))) /\ (((exists bcf_height_bcuet_source_decoded_value. bcf_height_bcuet_source_decoded_value + S (z) = S ((S (k)) * bcf_row_scale_bcuet_source)) /\ exists bcf_quotient_bcuet_source_decoded_value. bcf_row_code_bcuet_source = bcf_quotient_bcuet_source_decoded_value * S ((S (k)) * bcf_row_scale_bcuet_source) + (z))))))))) -> (((exists bcf_lt_gap_bcuet_target_out_of_range. bcf_lt_gap_bcuet_target_out_of_range + S (m) = k) /\ z = 0) \/ ((exists bcf_le_gap_bcuet_target_in_range. bcf_le_gap_bcuet_target_in_range + (k) = m) /\ (exists bcf_row_code_code_bcuet_target bcf_row_code_scale_bcuet_target bcf_row_scale_code_bcuet_target bcf_row_scale_scale_bcuet_target bcf_row_code_bcuet_target bcf_row_scale_bcuet_target. ((forall bcf_row_index_bcuet_target_table. (exists bcf_lt_gap_bcuet_target_table_row_bound. bcf_lt_gap_bcuet_target_table_row_bound + S (bcf_row_index_bcuet_target_table) = S (m)) -> exists bcf_row_code_bcuet_target_table bcf_row_scale_bcuet_target_table. ((((exists bcf_height_bcuet_target_table_decoded_row_code. bcf_height_bcuet_target_table_decoded_row_code + S (bcf_row_code_bcuet_target_table) = S ((S (bcf_row_index_bcuet_target_table)) * bcf_row_code_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_table_decoded_row_code. bcf_row_code_code_bcuet_target = bcf_quotient_bcuet_target_table_decoded_row_code * S ((S (bcf_row_index_bcuet_target_table)) * bcf_row_code_scale_bcuet_target) + (bcf_row_code_bcuet_target_table))) /\ ((((exists bcf_height_bcuet_target_table_decoded_row_scale. bcf_height_bcuet_target_table_decoded_row_scale + S (bcf_row_scale_bcuet_target_table) = S ((S (bcf_row_index_bcuet_target_table)) * bcf_row_scale_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_table_decoded_row_scale. bcf_row_scale_code_bcuet_target = bcf_quotient_bcuet_target_table_decoded_row_scale * S ((S (bcf_row_index_bcuet_target_table)) * bcf_row_scale_scale_bcuet_target) + (bcf_row_scale_bcuet_target_table))) /\ ((bcf_row_index_bcuet_target_table = 0 /\ (forall bcf_index_bcuet_target_table_zero_row. (exists bcf_lt_gap_bcuet_target_table_zero_row_bound. bcf_lt_gap_bcuet_target_table_zero_row_bound + S (bcf_index_bcuet_target_table_zero_row) = S (m)) -> exists bcf_value_bcuet_target_table_zero_row. ((((exists bcf_height_bcuet_target_table_zero_row_entry. bcf_height_bcuet_target_table_zero_row_entry + S (bcf_value_bcuet_target_table_zero_row) = S ((S (bcf_index_bcuet_target_table_zero_row)) * bcf_row_scale_bcuet_target_table)) /\ exists bcf_quotient_bcuet_target_table_zero_row_entry. bcf_row_code_bcuet_target_table = bcf_quotient_bcuet_target_table_zero_row_entry * S ((S (bcf_index_bcuet_target_table_zero_row)) * bcf_row_scale_bcuet_target_table) + (bcf_value_bcuet_target_table_zero_row))) /\ ((bcf_index_bcuet_target_table_zero_row = 0 /\ bcf_value_bcuet_target_table_zero_row = 1) \/ exists bcf_predecessor_bcuet_target_table_zero_row. bcf_index_bcuet_target_table_zero_row = S bcf_predecessor_bcuet_target_table_zero_row /\ bcf_value_bcuet_target_table_zero_row = 0)))) \/ exists bcf_predecessor_bcuet_target_table bcf_previous_code_bcuet_target_table bcf_previous_scale_bcuet_target_table. bcf_row_index_bcuet_target_table = S bcf_predecessor_bcuet_target_table /\ ((((exists bcf_height_bcuet_target_table_decoded_previous_code. bcf_height_bcuet_target_table_decoded_previous_code + S (bcf_previous_code_bcuet_target_table) = S ((S (bcf_predecessor_bcuet_target_table)) * bcf_row_code_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_table_decoded_previous_code. bcf_row_code_code_bcuet_target = bcf_quotient_bcuet_target_table_decoded_previous_code * S ((S (bcf_predecessor_bcuet_target_table)) * bcf_row_code_scale_bcuet_target) + (bcf_previous_code_bcuet_target_table))) /\ ((((exists bcf_height_bcuet_target_table_decoded_previous_scale. bcf_height_bcuet_target_table_decoded_previous_scale + S (bcf_previous_scale_bcuet_target_table) = S ((S (bcf_predecessor_bcuet_target_table)) * bcf_row_scale_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_table_decoded_previous_scale. bcf_row_scale_code_bcuet_target = bcf_quotient_bcuet_target_table_decoded_previous_scale * S ((S (bcf_predecessor_bcuet_target_table)) * bcf_row_scale_scale_bcuet_target) + (bcf_previous_scale_bcuet_target_table))) /\ (forall bcf_index_bcuet_target_table_row_step. (exists bcf_lt_gap_bcuet_target_table_row_step_bound. bcf_lt_gap_bcuet_target_table_row_step_bound + S (bcf_index_bcuet_target_table_row_step) = S (m)) -> exists bcf_value_bcuet_target_table_row_step. ((((exists bcf_height_bcuet_target_table_row_step_entry. bcf_height_bcuet_target_table_row_step_entry + S (bcf_value_bcuet_target_table_row_step) = S ((S (bcf_index_bcuet_target_table_row_step)) * bcf_row_scale_bcuet_target_table)) /\ exists bcf_quotient_bcuet_target_table_row_step_entry. bcf_row_code_bcuet_target_table = bcf_quotient_bcuet_target_table_row_step_entry * S ((S (bcf_index_bcuet_target_table_row_step)) * bcf_row_scale_bcuet_target_table) + (bcf_value_bcuet_target_table_row_step))) /\ ((bcf_index_bcuet_target_table_row_step = 0 /\ bcf_value_bcuet_target_table_row_step = 1) \/ exists bcf_predecessor_bcuet_target_table_row_step bcf_left_bcuet_target_table_row_step bcf_right_bcuet_target_table_row_step. bcf_index_bcuet_target_table_row_step = S bcf_predecessor_bcuet_target_table_row_step /\ ((((exists bcf_height_bcuet_target_table_row_step_previous_left. bcf_height_bcuet_target_table_row_step_previous_left + S (bcf_left_bcuet_target_table_row_step) = S ((S (bcf_predecessor_bcuet_target_table_row_step)) * bcf_previous_scale_bcuet_target_table)) /\ exists bcf_quotient_bcuet_target_table_row_step_previous_left. bcf_previous_code_bcuet_target_table = bcf_quotient_bcuet_target_table_row_step_previous_left * S ((S (bcf_predecessor_bcuet_target_table_row_step)) * bcf_previous_scale_bcuet_target_table) + (bcf_left_bcuet_target_table_row_step))) /\ ((((exists bcf_height_bcuet_target_table_row_step_previous_right. bcf_height_bcuet_target_table_row_step_previous_right + S (bcf_right_bcuet_target_table_row_step) = S ((S (S (bcf_predecessor_bcuet_target_table_row_step))) * bcf_previous_scale_bcuet_target_table)) /\ exists bcf_quotient_bcuet_target_table_row_step_previous_right. bcf_previous_code_bcuet_target_table = bcf_quotient_bcuet_target_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcuet_target_table_row_step))) * bcf_previous_scale_bcuet_target_table) + (bcf_right_bcuet_target_table_row_step))) /\ bcf_value_bcuet_target_table_row_step = bcf_left_bcuet_target_table_row_step + bcf_right_bcuet_target_table_row_step))))))))))) /\ ((((exists bcf_height_bcuet_target_decoded_row_code. bcf_height_bcuet_target_decoded_row_code + S (bcf_row_code_bcuet_target) = S ((S (m)) * bcf_row_code_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_decoded_row_code. bcf_row_code_code_bcuet_target = bcf_quotient_bcuet_target_decoded_row_code * S ((S (m)) * bcf_row_code_scale_bcuet_target) + (bcf_row_code_bcuet_target))) /\ ((((exists bcf_height_bcuet_target_decoded_row_scale. bcf_height_bcuet_target_decoded_row_scale + S (bcf_row_scale_bcuet_target) = S ((S (m)) * bcf_row_scale_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_decoded_row_scale. bcf_row_scale_code_bcuet_target = bcf_quotient_bcuet_target_decoded_row_scale * S ((S (m)) * bcf_row_scale_scale_bcuet_target) + (bcf_row_scale_bcuet_target))) /\ (((exists bcf_height_bcuet_target_decoded_value. bcf_height_bcuet_target_decoded_value + S (z) = S ((S (k)) * bcf_row_scale_bcuet_target)) /\ exists bcf_quotient_bcuet_target_decoded_value. bcf_row_code_bcuet_target = bcf_quotient_bcuet_target_decoded_value * S ((S (k)) * bcf_row_scale_bcuet_target) + (z)))))))))Proof neighborhood
Direct theorem prerequisites
none
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 assumptionsL1–6
02Calculate and transport equalitiesL7–15
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
03Use earlier factsL16–16
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L16
exact hchoose
Original defined command ledger · 16 lines
- 0001
intro n - 0002
intro m - 0003
intro k - 0004
intro z - 0005
intro heq - 0006
intro hchoose - 0007
rewrite heq at hchoose - 0008
rewrite heq at hchoose - 0009
rewrite heq at hchoose - 0010
rewrite heq at hchoose - 0011
rewrite heq at hchoose - 0012
rewrite heq at hchoose - 0013
rewrite heq at hchoose - 0014
rewrite heq at hchoose - 0015
rewrite heq at hchoose - 0016
exact hchoose