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_eq_less_eq
2	instantiation	102, 8, 4, 7, 5	⊢
	: , : , :
3	instantiation	6, 7, 8, 9, 10	⊢
	: , : , :
4	modus ponens	11, 12	⊢
5	modus ponens	13, 14	⊢
6	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
7	modus ponens	15, 16	⊢
8	modus ponens	17, 18	⊢
9	modus ponens	19, 20	⊢
10	modus ponens	21, 22	⊢
11	instantiation	27	⊢
	: , : , :
12	generalization	23	⊢
13	instantiation	29	⊢
	: , : , :
14	generalization	24	⊢
15	instantiation	27	⊢
	: , : , :
16	generalization	25	⊢
17	instantiation	27	⊢
	: , : , :
18	generalization	26	⊢
19	instantiation	27	⊢
	: , : , :
20	generalization	28	⊢
21	instantiation	29	⊢
	: , : , :
22	generalization	30	⊢
23	instantiation	62, 31, 239	,  ⊢
	: , :
24	instantiation	39, 32, 108	,  ⊢
	:
25	instantiation	36, 218, 33, 34	,  ⊢
	: , :
26	instantiation	62, 35, 239	,  ⊢
	: , :
27	theorem		⊢
	proveit.numbers.summation.summation_real_closure
28	instantiation	36, 218, 37, 38	,  ⊢
	: , :
29	theorem		⊢
	proveit.numbers.summation.weak_summation_from_summands_bound
30	instantiation	39, 40, 113	,  ⊢
	:
31	instantiation	240, 45, 41	,  ⊢
	: , : , :
32	instantiation	53, 150, 223, 119, 42	,  ⊢
	: , : , :
33	instantiation	122, 47, 43	,  ⊢
	: , :
34	instantiation	49, 239, 50, 51, 44	,  ⊢
	: , :
35	instantiation	240, 45, 46	,  ⊢
	: , : , :
36	theorem		⊢
	proveit.numbers.division.div_real_closure
37	instantiation	122, 47, 48	,  ⊢
	: , :
38	instantiation	49, 239, 50, 51, 52	,  ⊢
	: , :
39	theorem		⊢
	proveit.physics.quantum.QPE._alpha_sqrd_upper_bound
40	instantiation	53, 150, 223, 141, 54	,  ⊢
	: , : , :
41	instantiation	59, 55	,  ⊢
	:
42	instantiation	67, 56, 57	,  ⊢
	: , :
43	instantiation	62, 85, 239	,  ⊢
	: , :
44	instantiation	65, 58, 192	,  ⊢
	: , :
45	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_nonneg_within_real
46	instantiation	59, 60	,  ⊢
	:
47	instantiation	240, 227, 61	⊢
	: , : , :
48	instantiation	62, 90, 239	,  ⊢
	: , :
49	theorem		⊢
	proveit.numbers.multiplication.mult_not_eq_zero
50	instantiation	63	⊢
	: , :
51	instantiation	240, 208, 64	⊢
	: , : , :
52	instantiation	65, 66, 192	,  ⊢
	: , :
53	theorem		⊢
	proveit.numbers.number_sets.integers.in_interval
54	instantiation	67, 68, 69	,  ⊢
	: , :
55	instantiation	74, 70	,  ⊢
	:
56	instantiation	211, 150, 151, 152	,  ⊢
	: , : , :
57	instantiation	81, 71	,  ⊢
	: , :
58	instantiation	78, 72, 73	,  ⊢
	:
59	theorem		⊢
	proveit.numbers.absolute_value.abs_complex_closure
60	instantiation	74, 75	,  ⊢
	:
61	instantiation	240, 235, 76	⊢
	: , : , :
62	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
63	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
64	instantiation	240, 220, 77	⊢
	: , : , :
65	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_nonzero_closure
66	instantiation	78, 79, 80	,  ⊢
	:
67	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
68	instantiation	81, 82	,  ⊢
	: , :
69	instantiation	149, 167, 223, 168	,  ⊢
	: , : , :
70	instantiation	87, 140, 119	,  ⊢
	: , :
71	instantiation	83, 121, 84	,  ⊢
	: , : , :
72	instantiation	240, 217, 85	,  ⊢
	: , : , :
73	instantiation	91, 86	,  ⊢
	: , :
74	theorem		⊢
	proveit.physics.quantum.QPE._alpha_are_complex
75	instantiation	87, 140, 141	,  ⊢
	: , :
76	instantiation	240, 241, 88	⊢
	: , : , :
77	instantiation	240, 229, 89	⊢
	: , : , :
78	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
79	instantiation	240, 217, 90	,  ⊢
	: , : , :
80	instantiation	91, 92	,  ⊢
	: , :
81	theorem		⊢
	proveit.numbers.ordering.relax_less
82	instantiation	136, 93, 142	,  ⊢
	: , : , :
83	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
84	instantiation	165, 232	⊢
	:
85	instantiation	97, 94, 99	,  ⊢
	: , :
86	instantiation	100, 95	,  ⊢
	: , :
87	theorem		⊢
	proveit.physics.quantum.QPE._mod_add_closure
88	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
89	instantiation	240, 237, 96	⊢
	: , : , :
90	instantiation	97, 98, 99	,  ⊢
	: , :
91	theorem		⊢
	proveit.numbers.addition.subtraction.nonzero_difference_if_different
92	instantiation	100, 101	,  ⊢
	: , :
93	instantiation	102, 128, 218, 103, 104, 105*, 106*	⊢
	: , : , :
94	instantiation	240, 227, 107	,  ⊢
	: , : , :
95	instantiation	111, 112, 119, 108	,  ⊢
	: , :
96	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
97	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
98	instantiation	240, 227, 109	,  ⊢
	: , : , :
99	instantiation	206, 110	⊢
	:
100	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
101	instantiation	111, 112, 141, 113	,  ⊢
	: , :
102	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
103	instantiation	214, 215, 232	⊢
	: , : , :
104	instantiation	114, 232	⊢
	:
105	instantiation	188, 204, 115	⊢
	: , :
106	instantiation	129, 116, 117	⊢
	: , : , :
107	instantiation	240, 235, 119	,  ⊢
	: , : , :
108	instantiation	118, 119, 120, 121	,  ⊢
	: , :
109	instantiation	240, 235, 141	,  ⊢
	: , : , :
110	instantiation	122, 162, 123	⊢
	: , :
111	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_not_eq_nonzeroInt
112	instantiation	124, 125, 242, 126	⊢
	: , : , : , : , :
113	instantiation	207, 127	,  ⊢
	:
114	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
115	instantiation	240, 217, 128	⊢
	: , : , :
116	instantiation	129, 130, 131	⊢
	: , : , :
117	instantiation	132, 133, 134	⊢
	: , :
118	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_int
119	instantiation	240, 135, 152	,  ⊢
	: , : , :
120	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
121	instantiation	136, 137, 138	,  ⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
123	instantiation	139, 140	⊢
	:
124	theorem		⊢
	proveit.logic.sets.enumeration.in_enumerated_set
125	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
126	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
127	instantiation	185, 141, 142	,  ⊢
	:
128	instantiation	240, 227, 143	⊢
	: , : , :
129	axiom		⊢
	proveit.logic.equality.equals_transitivity
130	instantiation	182, 158	⊢
	: , : , :
131	instantiation	182, 144	⊢
	: , : , :
132	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
133	instantiation	145, 146, 147	⊢
	: , :
134	instantiation	148	⊢
	:
135	instantiation	210, 150, 151	⊢
	: , :
136	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
137	instantiation	149, 150, 151, 152	,  ⊢
	: , : , :
138	instantiation	153, 154	⊢
	:
139	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
140	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
141	instantiation	240, 155, 168	,  ⊢
	: , : , :
142	instantiation	199, 156, 157	,  ⊢
	: , : , :
143	instantiation	240, 235, 163	⊢
	: , : , :
144	instantiation	182, 158	⊢
	: , : , :
145	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
146	instantiation	159, 204, 160, 161	⊢
	: , :
147	instantiation	240, 217, 162	⊢
	: , : , :
148	axiom		⊢
	proveit.logic.equality.equals_reflexivity
149	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
150	instantiation	222, 163, 236	⊢
	: , :
151	instantiation	233, 167	⊢
	:
152	assumption		⊢
153	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
154	instantiation	164, 166	⊢
	:
155	instantiation	210, 167, 223	⊢
	: , :
156	instantiation	165, 166	⊢
	:
157	instantiation	211, 167, 223, 168	,  ⊢
	: , : , :
158	instantiation	169, 170, 171, 172	⊢
	: , : , : , :
159	theorem		⊢
	proveit.numbers.division.div_complex_closure
160	instantiation	173, 192, 204	⊢
	: , :
161	instantiation	174, 194, 175	⊢
	: , : , :
162	instantiation	214, 215, 176	⊢
	: , : , :
163	instantiation	233, 223	⊢
	:
164	theorem		⊢
	proveit.numbers.negation.int_neg_closure
165	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
166	instantiation	177, 178, 197	⊢
	: , :
167	instantiation	222, 186, 236	⊢
	: , :
168	assumption		⊢
169	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
170	instantiation	182, 179	⊢
	: , : , :
171	instantiation	180, 181	⊢
	: , :
172	instantiation	182, 183	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
174	theorem		⊢
	proveit.logic.equality.sub_left_side_into
175	instantiation	184, 192	⊢
	:
176	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
177	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
178	instantiation	185, 186, 187	⊢
	:
179	instantiation	188, 189, 190	⊢
	: , :
180	theorem		⊢
	proveit.logic.equality.equals_reversal
181	instantiation	191, 192, 193, 202, 194	⊢
	: , : , :
182	axiom		⊢
	proveit.logic.equality.substitution
183	instantiation	195, 196, 197	⊢
	: , :
184	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
185	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
186	instantiation	240, 198, 213	⊢
	: , : , :
187	instantiation	199, 200, 201	⊢
	: , : , :
188	theorem		⊢
	proveit.numbers.addition.commutation
189	instantiation	240, 217, 202	⊢
	: , : , :
190	instantiation	203, 204	⊢
	:
191	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
192	instantiation	240, 217, 205	⊢
	: , : , :
193	instantiation	206, 218	⊢
	:
194	instantiation	207, 238	⊢
	:
195	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
196	instantiation	240, 208, 209	⊢
	: , : , :
197	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
198	instantiation	210, 236, 212	⊢
	: , :
199	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
200	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
201	instantiation	211, 236, 212, 213	⊢
	: , : , :
202	instantiation	214, 215, 216	⊢
	: , : , :
203	theorem		⊢
	proveit.numbers.negation.complex_closure
204	instantiation	240, 217, 218	⊢
	: , : , :
205	instantiation	240, 227, 219	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.negation.real_closure
207	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
208	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
209	instantiation	240, 220, 221	⊢
	: , : , :
210	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
211	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
212	instantiation	222, 223, 224	⊢
	: , :
213	assumption		⊢
214	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
215	instantiation	225, 226	⊢
	: , :
216	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
217	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
218	instantiation	240, 227, 228	⊢
	: , : , :
219	instantiation	240, 235, 234	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
221	instantiation	240, 229, 230	⊢
	: , : , :
222	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
223	instantiation	240, 231, 232	⊢
	: , : , :
224	instantiation	233, 234	⊢
	:
225	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
226	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
227	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
228	instantiation	240, 235, 236	⊢
	: , : , :
229	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
230	instantiation	240, 237, 238	⊢
	: , : , :
231	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
232	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
233	theorem		⊢
	proveit.numbers.negation.int_closure
234	instantiation	240, 241, 239	⊢
	: , : , :
235	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
236	instantiation	240, 241, 242	⊢
	: , : , :
237	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
238	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
239	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
240	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
241	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
242	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements