Show the Proof

import proveit
# Automation is not needed when only showing a stored proof:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%show_proof

	step type	requirements	statement
0	instantiation	1, 2, 3, 4, 5, 6	,  ⊢
	: , : , :
1	theorem		⊢
	proveit.numbers.division.weak_div_from_numer_bound__pos_denom
2	instantiation	82, 7, 8	,  ⊢
	: , : , :
3	instantiation	289, 173, 9	⊢
	: , : , :
4	reference	280	⊢
5	instantiation	10, 30, 31, 11*	⊢
	: , :
6	instantiation	12, 291, 180, 26, 181, 49, 13, 14, 15*	,  ⊢
	: , : , : , : , : , :
7	instantiation	199, 16, 28	,  ⊢
	: , :
8	instantiation	115, 180, 291, 284, 181, 26, 272, 27, 17	,  ⊢
	: , : , : , : , : , :
9	instantiation	188, 18	⊢
	:
10	theorem		⊢
	proveit.numbers.absolute_value.triangle_inequality
11	instantiation	240, 19, 20	⊢
	: , : , :
12	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_factor_bound
13	instantiation	289, 67, 21	⊢
	: , : , :
14	instantiation	22, 23, 24	,  ⊢
	: , : , :
15	instantiation	25, 291, 180, 26, 181, 272, 27	⊢
	: , : , : , :
16	instantiation	199, 280, 40	⊢
	: , :
17	instantiation	289, 279, 28	,  ⊢
	: , : , :
18	instantiation	29, 30, 31	⊢
	: , :
19	instantiation	257, 291, 32, 33, 34, 35	⊢
	: , : , : , :
20	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
21	instantiation	289, 256, 57	⊢
	: , : , :
22	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
23	instantiation	162, 36	,  ⊢
	:
24	instantiation	37, 38, 111, 39*	,  ⊢
	:
25	theorem		⊢
	proveit.numbers.multiplication.mult_zero_any
26	instantiation	269	⊢
	: , :
27	instantiation	289, 279, 40	⊢
	: , : , :
28	instantiation	92, 185, 253, 41	,  ⊢
	: , : , :
29	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
30	instantiation	289, 279, 253	⊢
	: , : , :
31	instantiation	42, 46	⊢
	:
32	instantiation	269	⊢
	: , :
33	instantiation	269	⊢
	: , :
34	instantiation	43, 44	⊢
	:
35	instantiation	45, 46, 47*	⊢
	:
36	instantiation	48, 49, 50	,  ⊢
	: , :
37	theorem		⊢
	proveit.trigonometry.sine_linear_bound_nonneg
38	instantiation	51, 77, 52	,  ⊢
	:
39	instantiation	240, 53, 54	⊢
	: , : , :
40	instantiation	55, 56, 57	⊢
	: , : , :
41	instantiation	58, 59	,  ⊢
	:
42	theorem		⊢
	proveit.numbers.negation.complex_closure
43	theorem		⊢
	proveit.numbers.absolute_value.abs_non_neg_elim
44	instantiation	60, 284	⊢
	:
45	theorem		⊢
	proveit.numbers.absolute_value.abs_even
46	instantiation	61, 62, 63	⊢
	: , :
47	instantiation	64, 65, 66*	⊢
	:
48	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
49	instantiation	289, 67, 239	⊢
	: , : , :
50	instantiation	68, 69	,  ⊢
	:
51	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_real_is_real_nonneg
52	instantiation	70, 94	,  ⊢
	: , :
53	instantiation	216, 71	⊢
	: , : , :
54	instantiation	240, 72, 73	⊢
	: , : , :
55	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
56	instantiation	74, 75	⊢
	: , :
57	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
58	theorem		⊢
	proveit.trigonometry.sine_pos_interval
59	instantiation	76, 185, 235, 77, 78	,  ⊢
	: , : , :
60	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
61	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
62	instantiation	289, 279, 79	⊢
	: , : , :
63	instantiation	82, 80, 81	⊢
	: , : , :
64	theorem		⊢
	proveit.numbers.absolute_value.complex_unit_length
65	instantiation	82, 83, 84	⊢
	: , : , :
66	instantiation	167, 85	⊢
	: , :
67	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
68	theorem		⊢
	proveit.numbers.absolute_value.abs_nonzero_closure
69	instantiation	86, 189, 87	,  ⊢
	:
70	theorem		⊢
	proveit.numbers.ordering.relax_less
71	instantiation	115, 284, 291, 180, 88, 181, 272, 219, 144	⊢
	: , : , : , : , : , :
72	instantiation	240, 89, 90	⊢
	: , : , :
73	instantiation	91, 107	⊢
	:
74	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
75	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
76	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOO
77	instantiation	92, 185, 129, 127	,  ⊢
	: , : , :
78	instantiation	93, 94, 95	,  ⊢
	: , :
79	instantiation	289, 250, 96	⊢
	: , : , :
80	instantiation	113, 114, 97	⊢
	: , :
81	instantiation	240, 98, 99	⊢
	: , : , :
82	theorem		⊢
	proveit.logic.equality.sub_right_side_into
83	instantiation	199, 134, 159	⊢
	: , :
84	instantiation	115, 180, 291, 284, 181, 135, 272, 219, 139	⊢
	: , : , : , : , : , :
85	instantiation	216, 100	⊢
	: , : , :
86	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
87	instantiation	101, 102	,  ⊢
	: , :
88	instantiation	269	⊢
	: , :
89	instantiation	216, 103	⊢
	: , : , :
90	instantiation	104, 105, 106, 107, 108*	⊢
	: , : , :
91	theorem		⊢
	proveit.numbers.division.frac_one_denom
92	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
93	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
94	instantiation	109, 185, 129, 127	,  ⊢
	: , : , :
95	instantiation	110, 111, 112	,  ⊢
	: , : , :
96	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
97	instantiation	113, 138, 139	⊢
	: , :
98	instantiation	115, 284, 291, 180, 116, 181, 114, 138, 139	⊢
	: , : , : , : , : , :
99	instantiation	115, 180, 291, 181, 135, 116, 272, 219, 138, 139	⊢
	: , : , : , : , : , :
100	instantiation	240, 117, 118	⊢
	: , : , :
101	theorem		⊢
	proveit.numbers.addition.subtraction.nonzero_difference_if_different
102	instantiation	119, 120, 203, 148	,  ⊢
	: , :
103	instantiation	240, 121, 122	⊢
	: , : , :
104	theorem		⊢
	proveit.numbers.division.frac_cancel_left
105	instantiation	289, 247, 123	⊢
	: , : , :
106	instantiation	289, 247, 124	⊢
	: , : , :
107	instantiation	289, 279, 125	⊢
	: , : , :
108	instantiation	271, 219	⊢
	:
109	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
110	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
111	instantiation	126, 185, 129, 127	,  ⊢
	: , : , :
112	instantiation	128, 129, 130, 192, 131, 132*, 133*	⊢
	: , : , :
113	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
114	instantiation	289, 279, 134	⊢
	: , : , :
115	theorem		⊢
	proveit.numbers.multiplication.disassociation
116	instantiation	269	⊢
	: , :
117	instantiation	141, 180, 291, 284, 181, 135, 272, 219, 138, 139	⊢
	: , : , : , : , : , : , :
118	instantiation	142, 284, 136, 180, 137, 181, 138, 272, 219, 139	⊢
	: , : , : , : , : , :
119	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_not_eq_scaledNonzeroInt
120	instantiation	140, 180, 284, 181	⊢
	: , : , : , : , :
121	instantiation	141, 180, 284, 181, 272, 219, 144	⊢
	: , : , : , : , : , : , :
122	instantiation	142, 284, 291, 180, 143, 181, 219, 272, 144	⊢
	: , : , : , : , : , :
123	instantiation	289, 145, 251	⊢
	: , : , :
124	instantiation	289, 265, 146	⊢
	: , : , :
125	instantiation	199, 280, 160	⊢
	: , :
126	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
127	instantiation	147, 215, 148	,  ⊢
	:
128	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
129	instantiation	252, 235, 280, 254	⊢
	: , :
130	instantiation	199, 200, 185	⊢
	: , :
131	instantiation	149, 200, 185, 235, 150, 151	⊢
	: , : , :
132	instantiation	152, 153, 154, 155	⊢
	: , : , : , :
133	instantiation	240, 156, 157	⊢
	: , : , :
134	instantiation	199, 280, 235	⊢
	: , :
135	instantiation	269	⊢
	: , :
136	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
137	instantiation	158	⊢
	: , : , :
138	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
139	instantiation	289, 279, 159	⊢
	: , : , :
140	theorem		⊢
	proveit.logic.sets.enumeration.in_enumerated_set
141	theorem		⊢
	proveit.numbers.multiplication.leftward_commutation
142	theorem		⊢
	proveit.numbers.multiplication.association
143	instantiation	269	⊢
	: , :
144	instantiation	289, 279, 160	⊢
	: , : , :
145	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
146	instantiation	289, 276, 161	⊢
	: , : , :
147	theorem		⊢
	proveit.physics.quantum.QPE._scaled_abs_delta_b_floor_diff_interval
148	assumption		⊢
149	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
150	instantiation	162, 251	⊢
	:
151	instantiation	163, 222	⊢
	:
152	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
153	instantiation	240, 164, 165	⊢
	: , : , :
154	instantiation	166	⊢
	:
155	instantiation	167, 191	⊢
	: , :
156	instantiation	216, 191	⊢
	: , : , :
157	instantiation	167, 168, 169*	⊢
	: , :
158	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
159	instantiation	170, 171, 172	⊢
	: , :
160	instantiation	289, 173, 174	⊢
	: , : , :
161	instantiation	289, 282, 255	⊢
	: , : , :
162	theorem		⊢
	proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos
163	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
164	instantiation	240, 175, 176	⊢
	: , : , :
165	instantiation	177, 178	⊢
	:
166	axiom		⊢
	proveit.logic.equality.equals_reflexivity
167	theorem		⊢
	proveit.logic.equality.equals_reversal
168	instantiation	179, 180, 291, 284, 181, 182, 220, 219	⊢
	: , : , : , : , : , :
169	instantiation	240, 183, 184	⊢
	: , : , :
170	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
171	instantiation	204, 185, 253, 186	⊢
	: , : , :
172	instantiation	213, 187	⊢
	:
173	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_nonneg_within_real
174	instantiation	188, 189	⊢
	:
175	instantiation	216, 190	⊢
	: , : , :
176	instantiation	216, 191	⊢
	: , : , :
177	theorem		⊢
	proveit.numbers.addition.elim_zero_left
178	instantiation	289, 279, 192	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
180	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
181	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
182	instantiation	269	⊢
	: , :
183	instantiation	216, 193	⊢
	: , : , :
184	instantiation	270, 219	⊢
	:
185	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
186	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
187	instantiation	289, 285, 194	⊢
	: , : , :
188	theorem		⊢
	proveit.numbers.absolute_value.abs_complex_closure
189	instantiation	289, 279, 195	⊢
	: , : , :
190	instantiation	196, 220	⊢
	:
191	instantiation	197, 219, 272, 254, 198*	⊢
	: , :
192	instantiation	199, 200, 235	⊢
	: , :
193	instantiation	201, 278, 288, 202*	⊢
	: , : , : , :
194	instantiation	289, 287, 203	⊢
	: , : , :
195	instantiation	204, 205, 236, 206	⊢
	: , : , :
196	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
197	theorem		⊢
	proveit.numbers.division.div_as_mult
198	instantiation	240, 207, 208	⊢
	: , : , :
199	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
200	instantiation	289, 285, 209	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
202	instantiation	240, 210, 211	⊢
	: , : , :
203	instantiation	289, 212, 215	⊢
	: , : , :
204	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
205	instantiation	213, 236	⊢
	:
206	instantiation	214, 215	⊢
	:
207	instantiation	216, 217	⊢
	: , : , :
208	instantiation	218, 219, 220	⊢
	: , :
209	instantiation	289, 221, 222	⊢
	: , : , :
210	instantiation	257, 291, 223, 224, 225, 226	⊢
	: , : , : , :
211	instantiation	227, 228, 229	⊢
	:
212	instantiation	230, 231, 264	⊢
	: , :
213	theorem		⊢
	proveit.numbers.negation.real_closure
214	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_floor_diff_in_interval
215	assumption		⊢
216	axiom		⊢
	proveit.logic.equality.substitution
217	instantiation	232, 233, 255, 234*	⊢
	: , :
218	theorem		⊢
	proveit.numbers.multiplication.commutation
219	instantiation	289, 279, 235	⊢
	: , : , :
220	instantiation	289, 279, 236	⊢
	: , : , :
221	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
222	instantiation	237, 238, 239	⊢
	: , :
223	instantiation	269	⊢
	: , :
224	instantiation	269	⊢
	: , :
225	instantiation	240, 241, 242	⊢
	: , : , :
226	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
227	theorem		⊢
	proveit.numbers.division.frac_cancel_complete
228	instantiation	289, 279, 243	⊢
	: , : , :
229	instantiation	268, 244	⊢
	:
230	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
231	instantiation	245, 246, 278	⊢
	: , :
232	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
233	instantiation	289, 247, 248	⊢
	: , : , :
234	instantiation	249, 272	⊢
	:
235	instantiation	289, 250, 251	⊢
	: , : , :
236	instantiation	252, 253, 280, 254	⊢
	: , :
237	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
238	instantiation	289, 256, 255	⊢
	: , : , :
239	instantiation	289, 256, 283	⊢
	: , : , :
240	axiom		⊢
	proveit.logic.equality.equals_transitivity
241	instantiation	257, 291, 258, 259, 260, 261	⊢
	: , : , : , :
242	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
243	instantiation	289, 285, 262	⊢
	: , : , :
244	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
245	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
246	instantiation	263, 264	⊢
	:
247	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
248	instantiation	289, 265, 266	⊢
	: , : , :
249	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
250	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
251	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
252	theorem		⊢
	proveit.numbers.division.div_real_closure
253	instantiation	289, 285, 267	⊢
	: , : , :
254	instantiation	268, 283	⊢
	:
255	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
256	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
257	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
258	instantiation	269	⊢
	: , :
259	instantiation	269	⊢
	: , :
260	instantiation	270, 272	⊢
	:
261	instantiation	271, 272	⊢
	:
262	instantiation	289, 287, 273	⊢
	: , : , :
263	theorem		⊢
	proveit.numbers.negation.int_closure
264	instantiation	289, 274, 275	⊢
	: , : , :
265	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
266	instantiation	289, 276, 277	⊢
	: , : , :
267	instantiation	289, 287, 278	⊢
	: , : , :
268	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
269	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
270	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
271	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
272	instantiation	289, 279, 280	⊢
	: , : , :
273	instantiation	289, 290, 281	⊢
	: , : , :
274	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
275	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
276	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
277	instantiation	289, 282, 283	⊢
	: , : , :
278	instantiation	289, 290, 284	⊢
	: , : , :
279	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
280	instantiation	289, 285, 286	⊢
	: , : , :
281	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
282	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
283	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
284	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
285	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
286	instantiation	289, 287, 288	⊢
	: , : , :
287	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
288	instantiation	289, 290, 291	⊢
	: , : , :
289	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
290	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
291	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements