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