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