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	⊢
	: , : , :
1	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOC
2	reference	89	⊢
3	reference	107	⊢
4	reference	123	⊢
5	instantiation	6, 7, 8	⊢
	: , :
6	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
7	instantiation	19, 9, 10	⊢
	: , : , :
8	instantiation	136, 11	⊢
	: , :
9	instantiation	12, 13, 81, 14, 62, 15*, 16*	⊢
	: , : , :
10	instantiation	17, 32, 44, 18	⊢
	: , : , :
11	instantiation	19, 20, 21	⊢
	: , : , :
12	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
13	instantiation	22, 89, 23	⊢
	: , :
14	instantiation	297, 284, 24	⊢
	: , : , :
15	instantiation	238, 25, 26	⊢
	: , : , :
16	instantiation	154, 27, 28, 29	⊢
	: , : , : , :
17	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_cc_lower_bound
18	instantiation	30, 32, 44, 33	⊢
	: , : , :
19	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
20	instantiation	31, 32, 44, 33	⊢
	: , : , :
21	instantiation	151, 34, 63	⊢
	: , : , :
22	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
23	instantiation	263, 81	⊢
	:
24	instantiation	35, 36, 105	⊢
	: , :
25	instantiation	238, 37, 38	⊢
	: , : , :
26	instantiation	238, 39, 40	⊢
	: , : , :
27	instantiation	238, 41, 42	⊢
	: , : , :
28	instantiation	91	⊢
	:
29	instantiation	115, 43	⊢
	: , :
30	theorem		⊢
	proveit.numbers.number_sets.real_numbers.relax_IntervalCO
31	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
32	instantiation	263, 44	⊢
	:
33	instantiation	45, 89, 214, 126, 46, 47*, 48*, 61*	⊢
	: , : , : , :
34	instantiation	151, 49, 79	⊢
	: , : , :
35	theorem		⊢
	proveit.numbers.addition.add_rational_closure_bin
36	instantiation	127, 129	⊢
	: , :
37	instantiation	257, 114	⊢
	: , : , :
38	instantiation	257, 64	⊢
	: , : , :
39	instantiation	65, 299, 296, 186, 66, 187, 90, 67, 69	⊢
	: , : , : , : , : , :
40	instantiation	50, 90, 67, 51	⊢
	: , : , :
41	instantiation	238, 52, 53	⊢
	: , : , :
42	instantiation	238, 54, 55	⊢
	: , : , :
43	instantiation	257, 56	⊢
	: , : , :
44	instantiation	297, 284, 57	⊢
	: , : , :
45	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rescale_interval_co_membership
46	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval
47	instantiation	154, 58, 59, 60	⊢
	: , : , : , :
48	instantiation	265, 102, 189, 61*	⊢
	: , :
49	instantiation	136, 62	⊢
	: , :
50	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
51	instantiation	91	⊢
	:
52	instantiation	257, 63	⊢
	: , : , :
53	instantiation	257, 64	⊢
	: , : , :
54	instantiation	65, 299, 296, 186, 66, 187, 189, 67, 69	⊢
	: , : , : , : , : , :
55	instantiation	68, 189, 69, 70	⊢
	: , : , :
56	instantiation	238, 71, 72	⊢
	: , : , :
57	instantiation	160, 285, 73, 74	⊢
	: , :
58	instantiation	184, 299, 296, 186, 75, 187, 102, 273, 99	⊢
	: , : , : , : , : , :
59	instantiation	238, 76, 77	⊢
	: , : , :
60	instantiation	256, 99	⊢
	:
61	instantiation	238, 78, 79	⊢
	: , : , :
62	instantiation	80, 81, 82, 83, 84, 85	⊢
	: , : , :
63	instantiation	238, 86, 87	⊢
	: , : , :
64	instantiation	257, 88	⊢
	: , : , :
65	theorem		⊢
	proveit.numbers.addition.disassociation
66	instantiation	241	⊢
	: , :
67	instantiation	297, 286, 89	⊢
	: , : , :
68	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
69	instantiation	269, 90	⊢
	:
70	instantiation	91	⊢
	:
71	instantiation	257, 92	⊢
	: , : , :
72	instantiation	179, 266, 93, 94, 95*	⊢
	: , :
73	instantiation	297, 291, 96	⊢
	: , : , :
74	instantiation	264, 122	⊢
	:
75	instantiation	241	⊢
	: , :
76	instantiation	97, 186, 296, 299, 187, 98, 102, 273, 99	⊢
	: , : , : , : , : , :
77	instantiation	257, 100	⊢
	: , : , :
78	instantiation	101, 102, 189	⊢
	: , :
79	instantiation	138, 266, 212, 194, 157*, 116*	⊢
	: , : , : , :
80	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
81	instantiation	297, 284, 103	⊢
	: , : , :
82	instantiation	104, 107	⊢
	: , :
83	instantiation	297, 284, 105	⊢
	: , : , :
84	instantiation	106, 107, 108, 109, 110	⊢
	: , : , :
85	instantiation	169, 128	⊢
	:
86	instantiation	257, 111	⊢
	: , : , :
87	instantiation	112, 292, 283, 113*	⊢
	: , : , : , :
88	instantiation	257, 114	⊢
	: , : , :
89	instantiation	263, 214	⊢
	:
90	instantiation	117, 189, 230	⊢
	: , :
91	axiom		⊢
	proveit.logic.equality.equals_reflexivity
92	instantiation	115, 116	⊢
	: , :
93	instantiation	117, 250, 273	⊢
	: , :
94	instantiation	118, 296, 119, 212, 194	⊢
	: , :
95	instantiation	238, 120, 121	⊢
	: , : , :
96	instantiation	297, 220, 122	⊢
	: , : , :
97	theorem		⊢
	proveit.numbers.multiplication.association
98	instantiation	241	⊢
	: , :
99	instantiation	297, 286, 123	⊢
	: , : , :
100	instantiation	151, 124, 125	⊢
	: , : , :
101	theorem		⊢
	proveit.numbers.multiplication.commutation
102	instantiation	297, 286, 126	⊢
	: , : , :
103	instantiation	127, 129, 130	⊢
	: , :
104	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
105	instantiation	297, 159, 128	⊢
	: , : , :
106	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
107	instantiation	297, 284, 129	⊢
	: , : , :
108	instantiation	297, 284, 130	⊢
	: , : , :
109	instantiation	131, 132, 133, 134, 135	⊢
	: , : , :
110	instantiation	136, 137	⊢
	: , :
111	instantiation	138, 266, 212, 157*, 139*	⊢
	: , : , : , :
112	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
113	instantiation	238, 140, 141	⊢
	: , : , :
114	instantiation	257, 142	⊢
	: , : , :
115	theorem		⊢
	proveit.logic.equality.equals_reversal
116	instantiation	143, 250, 272, 295, 213*	⊢
	: , : , :
117	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
118	theorem		⊢
	proveit.numbers.multiplication.mult_not_eq_zero
119	instantiation	241	⊢
	: , :
120	instantiation	257, 144	⊢
	: , : , :
121	instantiation	238, 145, 146	⊢
	: , : , :
122	instantiation	147, 296, 148	⊢
	: , :
123	instantiation	149, 150	⊢
	:
124	instantiation	151, 152, 153	⊢
	: , : , :
125	instantiation	154, 155, 156, 157	⊢
	: , : , : , :
126	instantiation	234, 276, 282, 182	⊢
	: , :
127	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
128	instantiation	202, 203, 158	⊢
	: , :
129	instantiation	297, 159, 170	⊢
	: , : , :
130	instantiation	160, 285, 161, 182	⊢
	: , :
131	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
132	instantiation	297, 162, 163	⊢
	: , : , :
133	instantiation	297, 164, 204	⊢
	: , : , :
134	instantiation	297, 164, 165	⊢
	: , : , :
135	instantiation	166, 268, 276, 287, 167, 168, 213*	⊢
	: , : , :
136	theorem		⊢
	proveit.numbers.ordering.relax_less
137	instantiation	169, 170	⊢
	:
138	theorem		⊢
	proveit.numbers.division.prod_of_fracs
139	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
140	instantiation	222, 296, 171, 172, 173, 174	⊢
	: , : , : , :
141	instantiation	175, 176, 212, 266, 177*, 178*	⊢
	: , : , :
142	instantiation	179, 266, 273, 182, 180*	⊢
	: , :
143	theorem		⊢
	proveit.numbers.exponentiation.product_of_posnat_powers
144	instantiation	181, 250, 273, 246, 247, 182, 183*, 229*	⊢
	: , : , :
145	instantiation	184, 299, 296, 186, 188, 187, 266, 189, 230	⊢
	: , : , : , : , : , :
146	instantiation	185, 186, 296, 187, 188, 189, 230	⊢
	: , : , : , :
147	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
148	instantiation	190, 299, 191	⊢
	: , :
149	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
150	theorem		⊢
	proveit.physics.quantum.QPE._best_round_is_int
151	theorem		⊢
	proveit.logic.equality.sub_right_side_into
152	instantiation	192, 266, 193, 194	⊢
	: , : , : , : , :
153	instantiation	238, 195, 196	⊢
	: , : , :
154	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
155	instantiation	257, 197	⊢
	: , : , :
156	instantiation	257, 197	⊢
	: , : , :
157	instantiation	277, 266	⊢
	:
158	instantiation	297, 221, 198	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
160	theorem		⊢
	proveit.numbers.division.div_rational_closure
161	instantiation	297, 291, 199	⊢
	: , : , :
162	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
163	instantiation	297, 200, 299	⊢
	: , : , :
164	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
165	instantiation	297, 221, 289	⊢
	: , : , :
166	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
167	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
168	instantiation	201, 295	⊢
	:
169	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
170	instantiation	202, 203, 204	⊢
	: , :
171	instantiation	241	⊢
	: , :
172	instantiation	241	⊢
	: , :
173	instantiation	238, 205, 206	⊢
	: , : , :
174	theorem		⊢
	proveit.numbers.numerals.decimals.mult_4_4
175	theorem		⊢
	proveit.numbers.division.frac_cancel_left
176	instantiation	297, 231, 207	⊢
	: , : , :
177	instantiation	277, 208	⊢
	:
178	theorem		⊢
	proveit.numbers.numerals.decimals.mult_8_2
179	theorem		⊢
	proveit.numbers.division.div_as_mult
180	instantiation	238, 209, 210	⊢
	: , : , :
181	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
182	instantiation	264, 289	⊢
	:
183	instantiation	211, 212, 272, 213*	⊢
	: , :
184	theorem		⊢
	proveit.numbers.multiplication.disassociation
185	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
186	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
187	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
188	instantiation	241	⊢
	: , :
189	instantiation	297, 286, 214	⊢
	: , : , :
190	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
191	instantiation	297, 215, 295	⊢
	: , : , :
192	theorem		⊢
	proveit.numbers.division.mult_frac_cancel_denom_left
193	instantiation	297, 231, 216	⊢
	: , : , :
194	instantiation	297, 231, 217	⊢
	: , : , :
195	instantiation	257, 218	⊢
	: , : , :
196	instantiation	257, 219	⊢
	: , : , :
197	instantiation	259, 266	⊢
	:
198	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
199	instantiation	297, 220, 289	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
201	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
202	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
203	instantiation	297, 221, 272	⊢
	: , : , :
204	instantiation	297, 221, 281	⊢
	: , : , :
205	instantiation	222, 296, 223, 224, 225, 226	⊢
	: , : , : , :
206	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_4
207	instantiation	297, 252, 227	⊢
	: , : , :
208	instantiation	297, 286, 228	⊢
	: , : , :
209	instantiation	257, 229	⊢
	: , : , :
210	instantiation	256, 230	⊢
	:
211	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
212	instantiation	297, 231, 232	⊢
	: , : , :
213	instantiation	233, 250	⊢
	:
214	instantiation	234, 276, 268, 247	⊢
	: , :
215	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
216	instantiation	297, 252, 235	⊢
	: , : , :
217	instantiation	297, 252, 236	⊢
	: , : , :
218	instantiation	257, 237	⊢
	: , : , :
219	instantiation	238, 239, 240	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
221	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
222	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
223	instantiation	241	⊢
	: , :
224	instantiation	241	⊢
	: , :
225	instantiation	256, 242	⊢
	:
226	instantiation	277, 242	⊢
	:
227	instantiation	297, 270, 243	⊢
	: , : , :
228	instantiation	297, 284, 244	⊢
	: , : , :
229	instantiation	245, 250, 287, 246, 247, 248*	⊢
	: , : , :
230	instantiation	249, 250, 251	⊢
	: , :
231	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
232	instantiation	297, 252, 253	⊢
	: , : , :
233	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
234	theorem		⊢
	proveit.numbers.division.div_real_closure
235	instantiation	297, 270, 254	⊢
	: , : , :
236	instantiation	297, 270, 255	⊢
	: , : , :
237	instantiation	256, 273	⊢
	:
238	axiom		⊢
	proveit.logic.equality.equals_transitivity
239	instantiation	257, 258	⊢
	: , : , :
240	instantiation	259, 273	⊢
	:
241	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
242	instantiation	297, 286, 260	⊢
	: , : , :
243	instantiation	297, 280, 261	⊢
	: , : , :
244	instantiation	297, 291, 262	⊢
	: , : , :
245	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_real_power
246	instantiation	263, 276	⊢
	:
247	instantiation	264, 281	⊢
	:
248	instantiation	265, 278, 266, 267*	⊢
	: , :
249	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
250	instantiation	297, 286, 268	⊢
	: , : , :
251	instantiation	269, 278	⊢
	:
252	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
253	instantiation	297, 270, 271	⊢
	: , : , :
254	instantiation	297, 280, 272	⊢
	: , : , :
255	instantiation	297, 280, 289	⊢
	: , : , :
256	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
257	axiom		⊢
	proveit.logic.equality.substitution
258	instantiation	277, 273	⊢
	:
259	theorem		⊢
	proveit.numbers.division.frac_one_denom
260	instantiation	297, 284, 274	⊢
	: , : , :
261	theorem		⊢
	proveit.numbers.numerals.decimals.posnat8
262	instantiation	297, 298, 275	⊢
	: , : , :
263	theorem		⊢
	proveit.numbers.negation.real_closure
264	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
265	theorem		⊢
	proveit.numbers.multiplication.mult_neg_right
266	instantiation	297, 286, 276	⊢
	: , : , :
267	instantiation	277, 278	⊢
	:
268	instantiation	297, 284, 279	⊢
	: , : , :
269	theorem		⊢
	proveit.numbers.negation.complex_closure
270	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
271	instantiation	297, 280, 281	⊢
	: , : , :
272	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
273	instantiation	297, 286, 282	⊢
	: , : , :
274	instantiation	297, 291, 283	⊢
	: , : , :
275	theorem		⊢
	proveit.numbers.numerals.decimals.nat8
276	instantiation	297, 284, 285	⊢
	: , : , :
277	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
278	instantiation	297, 286, 287	⊢
	: , : , :
279	instantiation	297, 291, 288	⊢
	: , : , :
280	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
281	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
282	instantiation	293, 294, 289	⊢
	: , : , :
283	instantiation	297, 298, 290	⊢
	: , : , :
284	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
285	instantiation	297, 291, 292	⊢
	: , : , :
286	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
287	instantiation	293, 294, 295	⊢
	: , : , :
288	instantiation	297, 298, 296	⊢
	: , : , :
289	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
290	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
291	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
292	instantiation	297, 298, 299	⊢
	: , : , :
293	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
294	instantiation	300, 301	⊢
	: , :
295	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
296	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
297	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
298	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
299	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
300	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
301	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
*equality replacement requirements