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