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	reference	156	⊢
2	instantiation	144, 76, 250, 4, 5, 6*, 7*	⊢
	: , : , :
3	instantiation	8, 9, 10	⊢
	: , : , :
4	instantiation	163, 15, 297	⊢
	: , :
5	instantiation	112, 250, 15, 297, 11, 12	⊢
	: , : , :
6	instantiation	13, 258, 237, 128	⊢
	: , : , :
7	instantiation	210, 14	⊢
	: , : , :
8	theorem		⊢
	proveit.logic.equality.sub_right_side_into
9	instantiation	112, 15, 46, 16, 17, 18*	⊢
	: , : , :
10	instantiation	210, 19	⊢
	: , : , :
11	theorem		⊢
	proveit.physics.quantum.QPE._e_value_ge_two
12	instantiation	236, 305	⊢
	:
13	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
14	instantiation	231, 20, 21	⊢
	: , : , :
15	instantiation	163, 43, 76	⊢
	: , :
16	instantiation	144, 76, 43, 78, 22	⊢
	: , : , :
17	instantiation	144, 76, 297, 78, 23, 24*	⊢
	: , : , :
18	instantiation	122, 25, 26, 27	⊢
	: , : , : , :
19	instantiation	210, 28	⊢
	: , : , :
20	instantiation	177, 178, 276, 305, 179, 29, 30, 51, 258	⊢
	: , : , : , : , : , :
21	instantiation	154, 258, 30, 36	⊢
	: , : , :
22	instantiation	31, 250, 113, 72, 277, 32	⊢
	: , : , :
23	instantiation	33, 34	⊢
	:
24	instantiation	35, 258, 36	⊢
	: , :
25	instantiation	177, 178, 276, 305, 179, 38, 70, 51, 37	⊢
	: , : , : , : , : , :
26	instantiation	177, 276, 178, 38, 179, 70, 51	⊢
	: , : , : , : , : , :
27	instantiation	122, 39, 40, 41	⊢
	: , : , : , :
28	instantiation	42, 51, 155	⊢
	: , :
29	instantiation	256	⊢
	: , :
30	instantiation	303, 265, 43	⊢
	: , : , :
31	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
32	instantiation	173, 165, 203, 164, 44	⊢
	: , : , :
33	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
34	instantiation	45, 276, 85	⊢
	: , :
35	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
36	instantiation	205	⊢
	:
37	instantiation	303, 265, 46	⊢
	: , : , :
38	instantiation	256	⊢
	: , :
39	instantiation	92, 305, 70, 51	⊢
	: , : , : , : , : , : , :
40	instantiation	93, 178, 276, 179, 47, 49, 70, 51, 48*	⊢
	: , : , : , : , : , :
41	instantiation	93, 305, 276, 178, 49, 179, 50, 51, 52*	⊢
	: , : , : , : , : , :
42	theorem		⊢
	proveit.numbers.addition.commutation
43	instantiation	84, 250, 53	⊢
	: , :
44	instantiation	54, 250, 226, 105	⊢
	: , :
45	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
46	instantiation	163, 78, 76	⊢
	: , :
47	instantiation	256	⊢
	: , :
48	instantiation	55, 56, 57*	⊢
	: , :
49	instantiation	256	⊢
	: , :
50	instantiation	58, 237, 59	⊢
	: , :
51	instantiation	150, 258	⊢
	:
52	instantiation	60, 258, 237, 128	⊢
	: , : , :
53	instantiation	89, 61, 62	⊢
	:
54	theorem		⊢
	proveit.numbers.negation.negated_weak_bound
55	theorem		⊢
	proveit.logic.equality.equals_reversal
56	instantiation	63, 178, 276, 305, 179, 64, 258, 70, 65*	⊢
	: , : , : , : , : , :
57	instantiation	231, 66, 67	⊢
	: , : , :
58	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
59	instantiation	303, 265, 68	⊢
	: , : , :
60	theorem		⊢
	proveit.numbers.addition.subtraction.negated_add
61	instantiation	96, 97, 69	⊢
	: , :
62	instantiation	99, 106	⊢
	: , :
63	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
64	instantiation	256	⊢
	: , :
65	instantiation	224, 70	⊢
	:
66	instantiation	210, 128	⊢
	: , : , :
67	instantiation	71, 237, 297, 72, 73, 74*, 75*	⊢
	: , : , :
68	instantiation	163, 76, 165	⊢
	: , :
69	instantiation	303, 103, 77	⊢
	: , : , :
70	instantiation	303, 265, 78	⊢
	: , : , :
71	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
72	instantiation	303, 299, 79	⊢
	: , : , :
73	instantiation	80, 293	⊢
	:
74	instantiation	81, 237	⊢
	:
75	instantiation	231, 82, 83	⊢
	: , : , :
76	instantiation	225, 297	⊢
	:
77	instantiation	107, 242	⊢
	:
78	instantiation	84, 250, 85	⊢
	: , :
79	instantiation	303, 301, 90	⊢
	: , : , :
80	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
81	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
82	instantiation	177, 305, 276, 178, 86, 179, 258, 155, 136	⊢
	: , : , : , : , : , :
83	instantiation	231, 87, 88	⊢
	: , : , :
84	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
85	instantiation	89, 90, 91	⊢
	:
86	instantiation	256	⊢
	: , :
87	instantiation	92, 305, 178, 179, 258, 155, 136	⊢
	: , : , : , : , : , : , :
88	instantiation	93, 178, 276, 305, 179, 94, 258, 136, 155, 95*	⊢
	: , : , : , : , : , :
89	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
90	instantiation	96, 97, 98	⊢
	: , :
91	instantiation	99, 100	⊢
	: , :
92	theorem		⊢
	proveit.numbers.addition.leftward_commutation
93	theorem		⊢
	proveit.numbers.addition.association
94	instantiation	256	⊢
	: , :
95	instantiation	101, 258, 237, 128	⊢
	: , : , :
96	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
97	instantiation	303, 102, 186	⊢
	: , : , :
98	instantiation	303, 103, 104	⊢
	: , : , :
99	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
100	instantiation	129, 105, 106	⊢
	: , : , :
101	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
102	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
103	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
104	instantiation	107, 293	⊢
	:
105	axiom		⊢
	proveit.physics.quantum.QPE._n_ge_two
106	instantiation	144, 148, 250, 108, 109, 110*, 111*	⊢
	: , : , :
107	theorem		⊢
	proveit.numbers.negation.int_neg_closure
108	instantiation	163, 113, 250	⊢
	: , :
109	instantiation	112, 250, 113, 114, 220	⊢
	: , : , :
110	instantiation	231, 115, 116	⊢
	: , : , :
111	instantiation	231, 117, 118	⊢
	: , : , :
112	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
113	instantiation	163, 165, 203	⊢
	: , :
114	instantiation	129, 119, 120	⊢
	: , : , :
115	instantiation	177, 305, 276, 178, 135, 179, 237, 183, 136	⊢
	: , : , : , : , : , :
116	instantiation	182, 237, 183, 153	⊢
	: , : , :
117	instantiation	210, 121	⊢
	: , : , :
118	instantiation	122, 123, 124, 125	⊢
	: , : , : , :
119	instantiation	126, 302, 142, 127, 128*	⊢
	: , :
120	instantiation	129, 130, 131	⊢
	: , : , :
121	instantiation	177, 178, 276, 305, 179, 152, 155, 181, 237	⊢
	: , : , : , : , : , :
122	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
123	instantiation	177, 178, 133, 305, 179, 134, 155, 181, 237, 132	⊢
	: , : , : , : , : , :
124	instantiation	177, 133, 276, 178, 134, 135, 179, 155, 181, 237, 183, 136	⊢
	: , : , : , : , : , :
125	instantiation	231, 137, 138	⊢
	: , : , :
126	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
127	instantiation	139, 285, 159, 140	⊢
	: , :
128	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
129	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
130	instantiation	141, 142, 254, 143	⊢
	: , :
131	instantiation	144, 203, 145, 165, 146, 147*	⊢
	: , : , :
132	instantiation	303, 265, 148	⊢
	: , : , :
133	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
134	instantiation	149	⊢
	: , : , :
135	instantiation	256	⊢
	: , :
136	instantiation	150, 237	⊢
	:
137	instantiation	151, 276, 305, 178, 152, 179, 155, 181, 237, 183, 153	⊢
	: , : , : , : , : , : , : , :
138	instantiation	154, 183, 155, 185	⊢
	: , : , :
139	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
140	instantiation	156, 277, 157	⊢
	: , :
141	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
142	instantiation	260, 285, 159, 262	⊢
	: , :
143	instantiation	158, 285, 159, 261, 160, 277	⊢
	: , : , :
144	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
145	instantiation	163, 226, 204	⊢
	: , :
146	axiom		⊢
	proveit.physics.quantum.QPE._t_req
147	instantiation	231, 161, 162	⊢
	: , : , :
148	instantiation	163, 226, 164	⊢
	: , :
149	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
150	theorem		⊢
	proveit.numbers.negation.complex_closure
151	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
152	instantiation	256	⊢
	: , :
153	instantiation	205	⊢
	:
154	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
155	instantiation	303, 265, 165	⊢
	: , : , :
156	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
157	instantiation	166, 250, 167, 168, 169, 170*, 171*	⊢
	: , : , :
158	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
159	instantiation	303, 287, 172	⊢
	: , : , :
160	instantiation	173, 250, 244, 174, 175, 176*	⊢
	: , : , :
161	instantiation	177, 178, 276, 305, 179, 180, 183, 184, 181	⊢
	: , : , : , : , : , :
162	instantiation	182, 183, 184, 185	⊢
	: , : , :
163	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
164	instantiation	225, 250	⊢
	:
165	instantiation	240, 241, 186	⊢
	: , : , :
166	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
167	instantiation	187, 244, 296	⊢
	: , :
168	instantiation	303, 299, 188	⊢
	: , : , :
169	instantiation	189, 244, 296, 297, 190, 191	⊢
	: , : , :
170	instantiation	231, 192, 193	⊢
	: , : , :
171	instantiation	231, 194, 195	⊢
	: , : , :
172	instantiation	271, 196, 288	⊢
	: , :
173	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
174	instantiation	303, 197, 268	⊢
	: , : , :
175	instantiation	198, 199, 283, 285, 200	⊢
	: , : , :
176	instantiation	212, 266, 302, 213*, 201*, 202*	⊢
	: , : , : , :
177	theorem		⊢
	proveit.numbers.addition.disassociation
178	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
179	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
180	instantiation	256	⊢
	: , :
181	instantiation	303, 265, 203	⊢
	: , : , :
182	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
183	instantiation	303, 265, 226	⊢
	: , : , :
184	instantiation	303, 265, 204	⊢
	: , : , :
185	instantiation	205	⊢
	:
186	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
187	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
188	instantiation	206, 255, 300	⊢
	: , :
189	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
190	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
191	instantiation	207, 264	⊢
	:
192	instantiation	210, 208	⊢
	: , : , :
193	instantiation	209, 237	⊢
	:
194	instantiation	210, 211	⊢
	: , : , :
195	instantiation	212, 302, 266, 213*, 214*, 221*	⊢
	: , : , : , :
196	instantiation	303, 292, 215	⊢
	: , : , :
197	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
198	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
199	instantiation	303, 216, 217	⊢
	: , : , :
200	instantiation	218, 250, 290, 297, 219, 220, 221*	⊢
	: , : , :
201	instantiation	231, 222, 223	⊢
	: , : , :
202	instantiation	224, 237	⊢
	:
203	instantiation	225, 226	⊢
	:
204	instantiation	303, 299, 227	⊢
	: , : , :
205	axiom		⊢
	proveit.logic.equality.equals_reflexivity
206	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
207	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
208	instantiation	228, 229	⊢
	:
209	theorem		⊢
	proveit.numbers.addition.elim_zero_left
210	axiom		⊢
	proveit.logic.equality.substitution
211	instantiation	257, 229	⊢
	:
212	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
213	instantiation	230, 237	⊢
	:
214	instantiation	231, 232, 233	⊢
	: , : , :
215	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
216	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
217	instantiation	303, 234, 305	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
219	instantiation	235, 296, 297, 298	⊢
	: , : , :
220	instantiation	236, 276	⊢
	:
221	instantiation	257, 237	⊢
	:
222	instantiation	245, 276, 238, 239, 249, 248	⊢
	: , : , : , :
223	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
224	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
225	theorem		⊢
	proveit.numbers.negation.real_closure
226	instantiation	240, 241, 242	⊢
	: , : , :
227	instantiation	303, 301, 243	⊢
	: , : , :
228	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
229	instantiation	303, 265, 244	⊢
	: , : , :
230	theorem		⊢
	proveit.numbers.division.frac_one_denom
231	axiom		⊢
	proveit.logic.equality.equals_transitivity
232	instantiation	245, 276, 246, 247, 248, 249	⊢
	: , : , : , :
233	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
234	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
235	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
236	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
237	instantiation	303, 265, 250	⊢
	: , : , :
238	instantiation	256	⊢
	: , :
239	instantiation	256	⊢
	: , :
240	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
241	instantiation	251, 252	⊢
	: , :
242	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
243	instantiation	253, 254	⊢
	:
244	instantiation	303, 299, 255	⊢
	: , : , :
245	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
246	instantiation	256	⊢
	: , :
247	instantiation	256	⊢
	: , :
248	instantiation	257, 258	⊢
	:
249	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
250	instantiation	303, 299, 259	⊢
	: , : , :
251	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
252	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
253	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
254	instantiation	260, 285, 261, 262	⊢
	: , :
255	instantiation	303, 263, 264	⊢
	: , : , :
256	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
257	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
258	instantiation	303, 265, 297	⊢
	: , : , :
259	instantiation	303, 301, 266	⊢
	: , : , :
260	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
261	instantiation	267, 285, 268	⊢
	: , :
262	instantiation	269, 270	⊢
	: , :
263	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
264	instantiation	271, 278, 288	⊢
	: , :
265	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
266	instantiation	303, 304, 276	⊢
	: , : , :
267	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
268	instantiation	272, 273, 283, 274	⊢
	: , :
269	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
270	instantiation	275, 305, 276, 277	⊢
	: , :
271	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
272	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
273	instantiation	303, 287, 278	⊢
	: , : , :
274	instantiation	279, 280	⊢
	:
275	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
276	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
277	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
278	instantiation	303, 292, 281	⊢
	: , : , :
279	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
280	instantiation	303, 282, 283	⊢
	: , : , :
281	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
282	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
283	instantiation	284, 285, 286	⊢
	: , :
284	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
285	instantiation	303, 287, 288	⊢
	: , : , :
286	instantiation	289, 290, 291	⊢
	:
287	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
288	instantiation	303, 292, 293	⊢
	: , : , :
289	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
290	instantiation	294, 296, 297, 298	⊢
	: , : , :
291	instantiation	295, 296, 297, 298	⊢
	: , : , :
292	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
293	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
294	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
295	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
296	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
297	instantiation	303, 299, 300	⊢
	: , : , :
298	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
299	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
300	instantiation	303, 301, 302	⊢
	: , : , :
301	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
302	instantiation	303, 304, 305	⊢
	: , : , :
303	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
304	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
305	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements