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.number_sets.integers.nonneg_int_is_natural
2	instantiation	4, 5, 6	⊢
	: , :
3	instantiation	7, 8	⊢
	: , :
4	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
5	instantiation	210, 9, 93	⊢
	: , : , :
6	instantiation	210, 10, 11	⊢
	: , : , :
7	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
8	instantiation	36, 12, 13	⊢
	: , : , :
9	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
10	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
11	instantiation	14, 200	⊢
	:
12	axiom		⊢
	proveit.physics.quantum.QPE._n_ge_two
13	instantiation	51, 55, 157, 15, 16, 17*, 18*	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.negation.int_neg_closure
15	instantiation	70, 20, 157	⊢
	: , :
16	instantiation	19, 157, 20, 21, 127	⊢
	: , : , :
17	instantiation	138, 22, 23	⊢
	: , : , :
18	instantiation	138, 24, 25	⊢
	: , : , :
19	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
20	instantiation	70, 72, 110	⊢
	: , :
21	instantiation	36, 26, 27	⊢
	: , : , :
22	instantiation	84, 212, 183, 85, 42, 86, 144, 90, 43	⊢
	: , : , : , : , : , :
23	instantiation	89, 144, 90, 60	⊢
	: , : , :
24	instantiation	117, 28	⊢
	: , : , :
25	instantiation	29, 30, 31, 32	⊢
	: , : , : , :
26	instantiation	33, 209, 49, 34, 35*	⊢
	: , :
27	instantiation	36, 37, 38	⊢
	: , : , :
28	instantiation	84, 85, 183, 212, 86, 59, 62, 88, 144	⊢
	: , : , : , : , : , :
29	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
30	instantiation	84, 85, 40, 212, 86, 41, 62, 88, 144, 39	⊢
	: , : , : , : , : , :
31	instantiation	84, 40, 183, 85, 41, 42, 86, 62, 88, 144, 90, 43	⊢
	: , : , : , : , : , :
32	instantiation	138, 44, 45	⊢
	: , : , :
33	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
34	instantiation	46, 192, 66, 47	⊢
	: , :
35	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
36	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
37	instantiation	48, 49, 161, 50	⊢
	: , :
38	instantiation	51, 110, 52, 72, 53, 54*	⊢
	: , : , :
39	instantiation	210, 172, 55	⊢
	: , : , :
40	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
41	instantiation	56	⊢
	: , : , :
42	instantiation	163	⊢
	: , :
43	instantiation	57, 144	⊢
	:
44	instantiation	58, 183, 212, 85, 59, 86, 62, 88, 144, 90, 60	⊢
	: , : , : , : , : , : , : , :
45	instantiation	61, 90, 62, 92	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
47	instantiation	63, 184, 64	⊢
	: , :
48	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
49	instantiation	167, 192, 66, 169	⊢
	: , :
50	instantiation	65, 192, 66, 168, 67, 184	⊢
	: , : , :
51	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
52	instantiation	70, 133, 111	⊢
	: , :
53	axiom		⊢
	proveit.physics.quantum.QPE._t_req
54	instantiation	138, 68, 69	⊢
	: , : , :
55	instantiation	70, 133, 71	⊢
	: , :
56	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
57	theorem		⊢
	proveit.numbers.negation.complex_closure
58	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
59	instantiation	163	⊢
	: , :
60	instantiation	112	⊢
	:
61	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
62	instantiation	210, 172, 72	⊢
	: , : , :
63	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
64	instantiation	73, 157, 74, 75, 76, 77*, 78*	⊢
	: , : , :
65	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
66	instantiation	210, 194, 79	⊢
	: , : , :
67	instantiation	80, 157, 151, 81, 82, 83*	⊢
	: , : , :
68	instantiation	84, 85, 183, 212, 86, 87, 90, 91, 88	⊢
	: , : , : , : , : , :
69	instantiation	89, 90, 91, 92	⊢
	: , : , :
70	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
71	instantiation	132, 157	⊢
	:
72	instantiation	147, 148, 93	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
74	instantiation	94, 151, 203	⊢
	: , :
75	instantiation	210, 206, 95	⊢
	: , : , :
76	instantiation	96, 151, 203, 204, 97, 98	⊢
	: , : , :
77	instantiation	138, 99, 100	⊢
	: , : , :
78	instantiation	138, 101, 102	⊢
	: , : , :
79	instantiation	178, 103, 195	⊢
	: , :
80	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
81	instantiation	210, 104, 175	⊢
	: , : , :
82	instantiation	105, 106, 190, 192, 107	⊢
	: , : , :
83	instantiation	119, 173, 209, 120*, 108*, 109*	⊢
	: , : , : , :
84	theorem		⊢
	proveit.numbers.addition.disassociation
85	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
86	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
87	instantiation	163	⊢
	: , :
88	instantiation	210, 172, 110	⊢
	: , : , :
89	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
90	instantiation	210, 172, 133	⊢
	: , : , :
91	instantiation	210, 172, 111	⊢
	: , : , :
92	instantiation	112	⊢
	:
93	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
94	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
95	instantiation	113, 162, 207	⊢
	: , :
96	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
97	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
98	instantiation	114, 171	⊢
	:
99	instantiation	117, 115	⊢
	: , : , :
100	instantiation	116, 144	⊢
	:
101	instantiation	117, 118	⊢
	: , : , :
102	instantiation	119, 209, 173, 120*, 121*, 128*	⊢
	: , : , : , :
103	instantiation	210, 199, 122	⊢
	: , : , :
104	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
105	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
106	instantiation	210, 123, 124	⊢
	: , : , :
107	instantiation	125, 157, 197, 204, 126, 127, 128*	⊢
	: , : , :
108	instantiation	138, 129, 130	⊢
	: , : , :
109	instantiation	131, 144	⊢
	:
110	instantiation	132, 133	⊢
	:
111	instantiation	210, 206, 134	⊢
	: , : , :
112	axiom		⊢
	proveit.logic.equality.equals_reflexivity
113	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
114	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
115	instantiation	135, 136	⊢
	:
116	theorem		⊢
	proveit.numbers.addition.elim_zero_left
117	axiom		⊢
	proveit.logic.equality.substitution
118	instantiation	164, 136	⊢
	:
119	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
120	instantiation	137, 144	⊢
	:
121	instantiation	138, 139, 140	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
123	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
124	instantiation	210, 141, 212	⊢
	: , : , :
125	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
126	instantiation	142, 203, 204, 205	⊢
	: , : , :
127	instantiation	143, 183	⊢
	:
128	instantiation	164, 144	⊢
	:
129	instantiation	152, 183, 145, 146, 156, 155	⊢
	: , : , : , :
130	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
131	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
132	theorem		⊢
	proveit.numbers.negation.real_closure
133	instantiation	147, 148, 149	⊢
	: , : , :
134	instantiation	210, 208, 150	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
136	instantiation	210, 172, 151	⊢
	: , : , :
137	theorem		⊢
	proveit.numbers.division.frac_one_denom
138	axiom		⊢
	proveit.logic.equality.equals_transitivity
139	instantiation	152, 183, 153, 154, 155, 156	⊢
	: , : , : , :
140	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
141	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
142	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
143	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
144	instantiation	210, 172, 157	⊢
	: , : , :
145	instantiation	163	⊢
	: , :
146	instantiation	163	⊢
	: , :
147	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
148	instantiation	158, 159	⊢
	: , :
149	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
150	instantiation	160, 161	⊢
	:
151	instantiation	210, 206, 162	⊢
	: , : , :
152	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
153	instantiation	163	⊢
	: , :
154	instantiation	163	⊢
	: , :
155	instantiation	164, 165	⊢
	:
156	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
157	instantiation	210, 206, 166	⊢
	: , : , :
158	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
159	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
160	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
161	instantiation	167, 192, 168, 169	⊢
	: , :
162	instantiation	210, 170, 171	⊢
	: , : , :
163	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
164	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
165	instantiation	210, 172, 204	⊢
	: , : , :
166	instantiation	210, 208, 173	⊢
	: , : , :
167	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
168	instantiation	174, 192, 175	⊢
	: , :
169	instantiation	176, 177	⊢
	: , :
170	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
171	instantiation	178, 185, 195	⊢
	: , :
172	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
173	instantiation	210, 211, 183	⊢
	: , : , :
174	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
175	instantiation	179, 180, 190, 181	⊢
	: , :
176	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
177	instantiation	182, 212, 183, 184	⊢
	: , :
178	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
179	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
180	instantiation	210, 194, 185	⊢
	: , : , :
181	instantiation	186, 187	⊢
	:
182	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
183	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
184	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
185	instantiation	210, 199, 188	⊢
	: , : , :
186	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
187	instantiation	210, 189, 190	⊢
	: , : , :
188	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
189	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
190	instantiation	191, 192, 193	⊢
	: , :
191	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
192	instantiation	210, 194, 195	⊢
	: , : , :
193	instantiation	196, 197, 198	⊢
	:
194	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
195	instantiation	210, 199, 200	⊢
	: , : , :
196	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
197	instantiation	201, 203, 204, 205	⊢
	: , : , :
198	instantiation	202, 203, 204, 205	⊢
	: , : , :
199	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
200	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
201	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
202	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
203	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
204	instantiation	210, 206, 207	⊢
	: , : , :
205	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
206	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
207	instantiation	210, 208, 209	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
209	instantiation	210, 211, 212	⊢
	: , : , :
210	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
211	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
212	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements