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.logic.sets.enumeration.true_for_each_then_true_for_all
2	reference	334	⊢
3	instantiation	279	⊢
	: , :
4	instantiation	7, 125, 143, 147, 6	⊢
	: , : , :
5	instantiation	7, 125, 143, 160, 8	⊢
	: , : , :
6	instantiation	11, 9, 10	⊢
	: , :
7	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOC
8	instantiation	11, 12, 13	⊢
	: , :
9	instantiation	28, 125, 58, 14, 15, 16*, 17*	⊢
	: , : , :
10	instantiation	173, 18	⊢
	: , :
11	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
12	instantiation	35, 19, 20	⊢
	: , : , :
13	instantiation	173, 21	⊢
	: , :
14	instantiation	45, 147, 143	⊢
	: , :
15	instantiation	115, 58, 147, 143, 22, 174	⊢
	: , : , :
16	instantiation	189, 23, 24	⊢
	: , : , :
17	instantiation	276, 25, 26	⊢
	: , : , :
18	instantiation	35, 27, 145	⊢
	: , : , :
19	instantiation	28, 29, 116, 30, 94, 31*, 32*	⊢
	: , : , :
20	instantiation	33, 55, 72, 34	⊢
	: , : , :
21	instantiation	35, 36, 37	⊢
	: , : , :
22	instantiation	38, 58, 144, 44	⊢
	: , : , :
23	instantiation	39, 102	⊢
	:
24	instantiation	40, 102, 41	⊢
	: , :
25	instantiation	100, 224, 334, 337, 225, 42, 121, 227, 102	⊢
	: , : , : , : , : , :
26	instantiation	43, 227, 121, 105	⊢
	: , : , :
27	instantiation	54, 58, 144, 44	⊢
	: , : , :
28	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
29	instantiation	45, 125, 46	⊢
	: , :
30	instantiation	335, 322, 47	⊢
	: , : , :
31	instantiation	276, 48, 49	⊢
	: , : , :
32	instantiation	192, 50, 51, 52	⊢
	: , : , : , :
33	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_cc_lower_bound
34	instantiation	53, 55, 72, 56	⊢
	: , : , :
35	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
36	instantiation	54, 55, 72, 56	⊢
	: , : , :
37	instantiation	189, 57, 98	⊢
	: , : , :
38	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
39	theorem		⊢
	proveit.numbers.addition.elim_zero_right
40	theorem		⊢
	proveit.numbers.addition.commutation
41	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
42	instantiation	279	⊢
	: , :
43	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
44	instantiation	73, 58, 314, 163, 59, 60*, 61*, 62*	⊢
	: , : , : , :
45	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
46	instantiation	301, 116	⊢
	:
47	instantiation	63, 64, 141	⊢
	: , :
48	instantiation	276, 65, 66	⊢
	: , : , :
49	instantiation	276, 67, 68	⊢
	: , : , :
50	instantiation	276, 69, 70	⊢
	: , : , :
51	instantiation	127	⊢
	:
52	instantiation	152, 71	⊢
	: , :
53	theorem		⊢
	proveit.numbers.number_sets.real_numbers.relax_IntervalCO
54	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
55	instantiation	301, 72	⊢
	:
56	instantiation	73, 125, 252, 163, 74, 75*, 76*, 93*	⊢
	: , : , : , :
57	instantiation	189, 77, 114	⊢
	: , : , :
58	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
59	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
60	instantiation	192, 78, 79, 80	⊢
	: , : , : , :
61	instantiation	81, 138	⊢
	:
62	instantiation	315, 138	⊢
	:
63	theorem		⊢
	proveit.numbers.addition.add_rational_closure_bin
64	instantiation	164, 166	⊢
	: , :
65	instantiation	295, 151	⊢
	: , : , :
66	instantiation	295, 99	⊢
	: , : , :
67	instantiation	100, 337, 334, 224, 101, 225, 126, 102, 104	⊢
	: , : , : , : , : , :
68	instantiation	82, 126, 102, 83	⊢
	: , : , :
69	instantiation	276, 84, 85	⊢
	: , : , :
70	instantiation	276, 86, 87	⊢
	: , : , :
71	instantiation	295, 88	⊢
	: , : , :
72	instantiation	335, 322, 89	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rescale_interval_co_membership
74	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval
75	instantiation	192, 90, 91, 92	⊢
	: , : , : , :
76	instantiation	303, 138, 227, 93*	⊢
	: , :
77	instantiation	173, 94	⊢
	: , :
78	instantiation	222, 337, 334, 224, 95, 225, 138, 311, 121	⊢
	: , : , : , : , : , :
79	instantiation	276, 96, 97	⊢
	: , : , :
80	instantiation	294, 121	⊢
	:
81	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
82	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
83	instantiation	127	⊢
	:
84	instantiation	295, 98	⊢
	: , : , :
85	instantiation	295, 99	⊢
	: , : , :
86	instantiation	100, 337, 334, 224, 101, 225, 227, 102, 104	⊢
	: , : , : , : , : , :
87	instantiation	103, 227, 104, 105	⊢
	: , : , :
88	instantiation	276, 106, 107	⊢
	: , : , :
89	instantiation	198, 323, 108, 109	⊢
	: , :
90	instantiation	222, 337, 334, 224, 110, 225, 138, 311, 135	⊢
	: , : , : , : , : , :
91	instantiation	276, 111, 112	⊢
	: , : , :
92	instantiation	294, 135	⊢
	:
93	instantiation	276, 113, 114	⊢
	: , : , :
94	instantiation	115, 116, 117, 118, 119, 120	⊢
	: , : , :
95	instantiation	279	⊢
	: , :
96	instantiation	133, 224, 334, 337, 225, 134, 138, 311, 121	⊢
	: , : , : , : , : , :
97	instantiation	295, 136	⊢
	: , : , :
98	instantiation	276, 122, 123	⊢
	: , : , :
99	instantiation	295, 124	⊢
	: , : , :
100	theorem		⊢
	proveit.numbers.addition.disassociation
101	instantiation	279	⊢
	: , :
102	instantiation	335, 324, 125	⊢
	: , : , :
103	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
104	instantiation	307, 126	⊢
	:
105	instantiation	127	⊢
	:
106	instantiation	295, 128	⊢
	: , : , :
107	instantiation	217, 304, 129, 130, 131*	⊢
	: , :
108	instantiation	335, 329, 132	⊢
	: , : , :
109	instantiation	302, 159	⊢
	:
110	instantiation	279	⊢
	: , :
111	instantiation	133, 224, 334, 337, 225, 134, 138, 311, 135	⊢
	: , : , : , : , : , :
112	instantiation	295, 136	⊢
	: , : , :
113	instantiation	137, 138, 227	⊢
	: , :
114	instantiation	176, 304, 250, 232, 195*, 153*	⊢
	: , : , : , :
115	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
116	instantiation	335, 322, 139	⊢
	: , : , :
117	instantiation	140, 143	⊢
	: , :
118	instantiation	335, 322, 141	⊢
	: , : , :
119	instantiation	142, 143, 144, 145, 146	⊢
	: , : , :
120	instantiation	207, 165	⊢
	:
121	instantiation	335, 324, 147	⊢
	: , : , :
122	instantiation	295, 148	⊢
	: , : , :
123	instantiation	149, 330, 321, 150*	⊢
	: , : , : , :
124	instantiation	295, 151	⊢
	: , : , :
125	instantiation	301, 252	⊢
	:
126	instantiation	154, 227, 268	⊢
	: , :
127	axiom		⊢
	proveit.logic.equality.equals_reflexivity
128	instantiation	152, 153	⊢
	: , :
129	instantiation	154, 288, 311	⊢
	: , :
130	instantiation	155, 334, 156, 250, 232	⊢
	: , :
131	instantiation	276, 157, 158	⊢
	: , : , :
132	instantiation	335, 258, 159	⊢
	: , : , :
133	theorem		⊢
	proveit.numbers.multiplication.association
134	instantiation	279	⊢
	: , :
135	instantiation	335, 324, 160	⊢
	: , : , :
136	instantiation	189, 161, 162	⊢
	: , : , :
137	theorem		⊢
	proveit.numbers.multiplication.commutation
138	instantiation	335, 324, 163	⊢
	: , : , :
139	instantiation	164, 166, 167	⊢
	: , :
140	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
141	instantiation	335, 197, 165	⊢
	: , : , :
142	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
143	instantiation	335, 322, 166	⊢
	: , : , :
144	instantiation	335, 322, 167	⊢
	: , : , :
145	instantiation	168, 169, 170, 171, 172	⊢
	: , : , :
146	instantiation	173, 174	⊢
	: , :
147	instantiation	187, 175	⊢
	:
148	instantiation	176, 304, 250, 195*, 177*	⊢
	: , : , : , :
149	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
150	instantiation	276, 178, 179	⊢
	: , : , :
151	instantiation	295, 180	⊢
	: , : , :
152	theorem		⊢
	proveit.logic.equality.equals_reversal
153	instantiation	181, 288, 310, 333, 251*	⊢
	: , : , :
154	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
155	theorem		⊢
	proveit.numbers.multiplication.mult_not_eq_zero
156	instantiation	279	⊢
	: , :
157	instantiation	295, 182	⊢
	: , : , :
158	instantiation	276, 183, 184	⊢
	: , : , :
159	instantiation	185, 334, 186	⊢
	: , :
160	instantiation	187, 188	⊢
	:
161	instantiation	189, 190, 191	⊢
	: , : , :
162	instantiation	192, 193, 194, 195	⊢
	: , : , : , :
163	instantiation	272, 314, 320, 220	⊢
	: , :
164	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
165	instantiation	240, 241, 196	⊢
	: , :
166	instantiation	335, 197, 208	⊢
	: , : , :
167	instantiation	198, 323, 199, 220	⊢
	: , :
168	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
169	instantiation	335, 200, 201	⊢
	: , : , :
170	instantiation	335, 202, 242	⊢
	: , : , :
171	instantiation	335, 202, 203	⊢
	: , : , :
172	instantiation	204, 306, 314, 325, 205, 206, 251*	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.ordering.relax_less
174	instantiation	207, 208	⊢
	:
175	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
176	theorem		⊢
	proveit.numbers.division.prod_of_fracs
177	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
178	instantiation	260, 334, 209, 210, 211, 212	⊢
	: , : , : , :
179	instantiation	213, 214, 250, 304, 215*, 216*	⊢
	: , : , :
180	instantiation	217, 304, 311, 220, 218*	⊢
	: , :
181	theorem		⊢
	proveit.numbers.exponentiation.product_of_posnat_powers
182	instantiation	219, 288, 311, 284, 285, 220, 221*, 267*	⊢
	: , : , :
183	instantiation	222, 337, 334, 224, 226, 225, 304, 227, 268	⊢
	: , : , : , : , : , :
184	instantiation	223, 224, 334, 225, 226, 227, 268	⊢
	: , : , : , :
185	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
186	instantiation	228, 337, 229	⊢
	: , :
187	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
188	theorem		⊢
	proveit.physics.quantum.QPE._best_round_is_int
189	theorem		⊢
	proveit.logic.equality.sub_right_side_into
190	instantiation	230, 304, 231, 232	⊢
	: , : , : , : , :
191	instantiation	276, 233, 234	⊢
	: , : , :
192	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
193	instantiation	295, 235	⊢
	: , : , :
194	instantiation	295, 235	⊢
	: , : , :
195	instantiation	315, 304	⊢
	:
196	instantiation	335, 259, 236	⊢
	: , : , :
197	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
198	theorem		⊢
	proveit.numbers.division.div_rational_closure
199	instantiation	335, 329, 237	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
201	instantiation	335, 238, 337	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
203	instantiation	335, 259, 327	⊢
	: , : , :
204	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
205	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
206	instantiation	239, 333	⊢
	:
207	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
208	instantiation	240, 241, 242	⊢
	: , :
209	instantiation	279	⊢
	: , :
210	instantiation	279	⊢
	: , :
211	instantiation	276, 243, 244	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.numerals.decimals.mult_4_4
213	theorem		⊢
	proveit.numbers.division.frac_cancel_left
214	instantiation	335, 269, 245	⊢
	: , : , :
215	instantiation	315, 246	⊢
	:
216	theorem		⊢
	proveit.numbers.numerals.decimals.mult_8_2
217	theorem		⊢
	proveit.numbers.division.div_as_mult
218	instantiation	276, 247, 248	⊢
	: , : , :
219	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
220	instantiation	302, 327	⊢
	:
221	instantiation	249, 250, 310, 251*	⊢
	: , :
222	theorem		⊢
	proveit.numbers.multiplication.disassociation
223	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
224	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
225	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
226	instantiation	279	⊢
	: , :
227	instantiation	335, 324, 252	⊢
	: , : , :
228	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
229	instantiation	335, 253, 333	⊢
	: , : , :
230	theorem		⊢
	proveit.numbers.division.mult_frac_cancel_denom_left
231	instantiation	335, 269, 254	⊢
	: , : , :
232	instantiation	335, 269, 255	⊢
	: , : , :
233	instantiation	295, 256	⊢
	: , : , :
234	instantiation	295, 257	⊢
	: , : , :
235	instantiation	297, 304	⊢
	:
236	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
237	instantiation	335, 258, 327	⊢
	: , : , :
238	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
239	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
240	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
241	instantiation	335, 259, 310	⊢
	: , : , :
242	instantiation	335, 259, 319	⊢
	: , : , :
243	instantiation	260, 334, 261, 262, 263, 264	⊢
	: , : , : , :
244	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_4
245	instantiation	335, 290, 265	⊢
	: , : , :
246	instantiation	335, 324, 266	⊢
	: , : , :
247	instantiation	295, 267	⊢
	: , : , :
248	instantiation	294, 268	⊢
	:
249	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
250	instantiation	335, 269, 270	⊢
	: , : , :
251	instantiation	271, 288	⊢
	:
252	instantiation	272, 314, 306, 285	⊢
	: , :
253	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
254	instantiation	335, 290, 273	⊢
	: , : , :
255	instantiation	335, 290, 274	⊢
	: , : , :
256	instantiation	295, 275	⊢
	: , : , :
257	instantiation	276, 277, 278	⊢
	: , : , :
258	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
259	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
260	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
261	instantiation	279	⊢
	: , :
262	instantiation	279	⊢
	: , :
263	instantiation	294, 280	⊢
	:
264	instantiation	315, 280	⊢
	:
265	instantiation	335, 308, 281	⊢
	: , : , :
266	instantiation	335, 322, 282	⊢
	: , : , :
267	instantiation	283, 288, 325, 284, 285, 286*	⊢
	: , : , :
268	instantiation	287, 288, 289	⊢
	: , :
269	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
270	instantiation	335, 290, 291	⊢
	: , : , :
271	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
272	theorem		⊢
	proveit.numbers.division.div_real_closure
273	instantiation	335, 308, 292	⊢
	: , : , :
274	instantiation	335, 308, 293	⊢
	: , : , :
275	instantiation	294, 311	⊢
	:
276	axiom		⊢
	proveit.logic.equality.equals_transitivity
277	instantiation	295, 296	⊢
	: , : , :
278	instantiation	297, 311	⊢
	:
279	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
280	instantiation	335, 324, 298	⊢
	: , : , :
281	instantiation	335, 318, 299	⊢
	: , : , :
282	instantiation	335, 329, 300	⊢
	: , : , :
283	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_real_power
284	instantiation	301, 314	⊢
	:
285	instantiation	302, 319	⊢
	:
286	instantiation	303, 316, 304, 305*	⊢
	: , :
287	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
288	instantiation	335, 324, 306	⊢
	: , : , :
289	instantiation	307, 316	⊢
	:
290	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
291	instantiation	335, 308, 309	⊢
	: , : , :
292	instantiation	335, 318, 310	⊢
	: , : , :
293	instantiation	335, 318, 327	⊢
	: , : , :
294	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
295	axiom		⊢
	proveit.logic.equality.substitution
296	instantiation	315, 311	⊢
	:
297	theorem		⊢
	proveit.numbers.division.frac_one_denom
298	instantiation	335, 322, 312	⊢
	: , : , :
299	theorem		⊢
	proveit.numbers.numerals.decimals.posnat8
300	instantiation	335, 336, 313	⊢
	: , : , :
301	theorem		⊢
	proveit.numbers.negation.real_closure
302	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
303	theorem		⊢
	proveit.numbers.multiplication.mult_neg_right
304	instantiation	335, 324, 314	⊢
	: , : , :
305	instantiation	315, 316	⊢
	:
306	instantiation	335, 322, 317	⊢
	: , : , :
307	theorem		⊢
	proveit.numbers.negation.complex_closure
308	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
309	instantiation	335, 318, 319	⊢
	: , : , :
310	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
311	instantiation	335, 324, 320	⊢
	: , : , :
312	instantiation	335, 329, 321	⊢
	: , : , :
313	theorem		⊢
	proveit.numbers.numerals.decimals.nat8
314	instantiation	335, 322, 323	⊢
	: , : , :
315	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
316	instantiation	335, 324, 325	⊢
	: , : , :
317	instantiation	335, 329, 326	⊢
	: , : , :
318	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
319	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
320	instantiation	331, 332, 327	⊢
	: , : , :
321	instantiation	335, 336, 328	⊢
	: , : , :
322	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
323	instantiation	335, 329, 330	⊢
	: , : , :
324	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
325	instantiation	331, 332, 333	⊢
	: , : , :
326	instantiation	335, 336, 334	⊢
	: , : , :
327	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
328	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
329	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
330	instantiation	335, 336, 337	⊢
	: , : , :
331	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
332	instantiation	338, 339	⊢
	: , :
333	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
334	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
335	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
336	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
337	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
338	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
339	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
*equality replacement requirements