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	217	⊢
2	reference	176	⊢
3	instantiation	4, 165, 5	⊢
	: , :
4	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
5	instantiation	6, 7, 8	⊢
	:
6	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
7	instantiation	9, 10, 11	⊢
	: , :
8	instantiation	12, 13	⊢
	: , :
9	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
10	instantiation	217, 14, 106	⊢
	: , : , :
11	instantiation	217, 15, 16	⊢
	: , : , :
12	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
13	instantiation	55, 59, 165, 17, 18, 19*, 20*	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
15	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
16	instantiation	21, 162	⊢
	:
17	instantiation	79, 23, 165	⊢
	: , :
18	instantiation	22, 165, 23, 24, 138	⊢
	: , : , :
19	instantiation	141, 25, 26	⊢
	: , : , :
20	instantiation	141, 27, 28	⊢
	: , : , :
21	theorem		⊢
	proveit.numbers.negation.int_neg_closure
22	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
23	instantiation	79, 81, 125	⊢
	: , :
24	instantiation	39, 29, 30	⊢
	: , : , :
25	instantiation	97, 219, 190, 98, 45, 99, 154, 103, 46	⊢
	: , : , : , : , : , :
26	instantiation	102, 154, 103, 64	⊢
	: , : , :
27	instantiation	111, 31	⊢
	: , : , :
28	instantiation	32, 33, 34, 35	⊢
	: , : , : , :
29	instantiation	36, 216, 53, 37, 38*	⊢
	: , :
30	instantiation	39, 40, 41	⊢
	: , : , :
31	instantiation	97, 98, 190, 219, 99, 63, 66, 101, 154	⊢
	: , : , : , : , : , :
32	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
33	instantiation	97, 98, 43, 219, 99, 44, 66, 101, 154, 42	⊢
	: , : , : , : , : , :
34	instantiation	97, 43, 190, 98, 44, 45, 99, 66, 101, 154, 103, 46	⊢
	: , : , : , : , : , :
35	instantiation	141, 47, 48	⊢
	: , : , :
36	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
37	instantiation	49, 199, 75, 50, 191, 51*	⊢
	: , : , :
38	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
39	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
40	instantiation	52, 53, 172, 54	⊢
	: , :
41	instantiation	55, 125, 56, 81, 57, 58*	⊢
	: , : , :
42	instantiation	217, 176, 59	⊢
	: , : , :
43	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
44	instantiation	60	⊢
	: , : , :
45	instantiation	166	⊢
	: , :
46	instantiation	61, 154	⊢
	:
47	instantiation	62, 190, 219, 98, 63, 99, 66, 101, 154, 103, 64	⊢
	: , : , : , : , : , : , : , :
48	instantiation	65, 103, 66, 105	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less
50	instantiation	67, 165, 68, 69, 70, 71*, 72*	⊢
	: , : , :
51	instantiation	73, 199	⊢
	:
52	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
53	instantiation	177, 199, 75, 179	⊢
	: , :
54	instantiation	74, 199, 75, 178, 76, 191	⊢
	: , : , :
55	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
56	instantiation	79, 146, 126	⊢
	: , :
57	axiom		⊢
	proveit.physics.quantum.QPE._t_req
58	instantiation	141, 77, 78	⊢
	: , : , :
59	instantiation	79, 146, 80	⊢
	: , :
60	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
61	theorem		⊢
	proveit.numbers.negation.complex_closure
62	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
63	instantiation	166	⊢
	: , :
64	instantiation	127	⊢
	:
65	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
66	instantiation	217, 176, 81	⊢
	: , : , :
67	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
68	instantiation	82, 148, 210	⊢
	: , :
69	instantiation	217, 213, 83	⊢
	: , : , :
70	instantiation	84, 148, 210, 211, 85, 86	⊢
	: , : , :
71	instantiation	141, 87, 88	⊢
	: , : , :
72	instantiation	141, 89, 90	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.logarithms.log_eq_1
74	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
75	instantiation	186, 91, 199, 116	⊢
	: , :
76	instantiation	92, 165, 93, 94, 95, 96*	⊢
	: , : , :
77	instantiation	97, 98, 190, 219, 99, 100, 103, 104, 101	⊢
	: , : , : , : , : , :
78	instantiation	102, 103, 104, 105	⊢
	: , : , :
79	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
80	instantiation	145, 165	⊢
	:
81	instantiation	160, 161, 106	⊢
	: , : , :
82	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
83	instantiation	107, 164, 214	⊢
	: , :
84	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
85	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
86	instantiation	108, 174	⊢
	:
87	instantiation	111, 109	⊢
	: , : , :
88	instantiation	110, 154	⊢
	:
89	instantiation	111, 112	⊢
	: , : , :
90	instantiation	121, 216, 181, 122*, 113*, 139*	⊢
	: , : , : , :
91	instantiation	217, 201, 114	⊢
	: , : , :
92	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
93	instantiation	115, 211, 165, 116	⊢
	: , :
94	instantiation	217, 117, 183	⊢
	: , : , :
95	instantiation	118, 119, 197, 199, 120	⊢
	: , : , :
96	instantiation	121, 181, 216, 122*, 123*, 124*	⊢
	: , : , : , :
97	theorem		⊢
	proveit.numbers.addition.disassociation
98	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
99	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
100	instantiation	166	⊢
	: , :
101	instantiation	217, 176, 125	⊢
	: , : , :
102	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
103	instantiation	217, 176, 146	⊢
	: , : , :
104	instantiation	217, 176, 126	⊢
	: , : , :
105	instantiation	127	⊢
	:
106	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
107	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
108	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
109	instantiation	128, 129	⊢
	:
110	theorem		⊢
	proveit.numbers.addition.elim_zero_left
111	axiom		⊢
	proveit.logic.equality.substitution
112	instantiation	167, 129	⊢
	:
113	instantiation	141, 130, 131	⊢
	: , : , :
114	instantiation	217, 206, 132	⊢
	: , : , :
115	theorem		⊢
	proveit.numbers.division.div_real_closure
116	instantiation	133, 207	⊢
	:
117	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
118	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
119	instantiation	217, 134, 135	⊢
	: , : , :
120	instantiation	136, 165, 204, 211, 137, 138, 139*	⊢
	: , : , :
121	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
122	instantiation	140, 154	⊢
	:
123	instantiation	141, 142, 143	⊢
	: , : , :
124	instantiation	144, 154	⊢
	:
125	instantiation	145, 146	⊢
	:
126	instantiation	217, 213, 147	⊢
	: , : , :
127	axiom		⊢
	proveit.logic.equality.equals_reflexivity
128	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
129	instantiation	217, 176, 148	⊢
	: , : , :
130	instantiation	155, 190, 149, 150, 159, 158	⊢
	: , : , : , :
131	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
132	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
133	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
134	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
135	instantiation	217, 151, 219	⊢
	: , : , :
136	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
137	instantiation	152, 210, 211, 212	⊢
	: , : , :
138	instantiation	153, 190	⊢
	:
139	instantiation	167, 154	⊢
	:
140	theorem		⊢
	proveit.numbers.division.frac_one_denom
141	axiom		⊢
	proveit.logic.equality.equals_transitivity
142	instantiation	155, 190, 156, 157, 158, 159	⊢
	: , : , : , :
143	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
144	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
145	theorem		⊢
	proveit.numbers.negation.real_closure
146	instantiation	160, 161, 162	⊢
	: , : , :
147	instantiation	217, 215, 163	⊢
	: , : , :
148	instantiation	217, 213, 164	⊢
	: , : , :
149	instantiation	166	⊢
	: , :
150	instantiation	166	⊢
	: , :
151	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
152	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
153	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
154	instantiation	217, 176, 165	⊢
	: , : , :
155	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
156	instantiation	166	⊢
	: , :
157	instantiation	166	⊢
	: , :
158	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
159	instantiation	167, 168	⊢
	:
160	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
161	instantiation	169, 170	⊢
	: , :
162	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
163	instantiation	171, 172	⊢
	:
164	instantiation	217, 173, 174	⊢
	: , : , :
165	instantiation	217, 213, 175	⊢
	: , : , :
166	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
167	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
168	instantiation	217, 176, 211	⊢
	: , : , :
169	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
170	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
171	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
172	instantiation	177, 199, 178, 179	⊢
	: , :
173	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
174	instantiation	180, 192, 202	⊢
	: , :
175	instantiation	217, 215, 181	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
177	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
178	instantiation	182, 199, 183	⊢
	: , :
179	instantiation	184, 185	⊢
	: , :
180	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
181	instantiation	217, 218, 190	⊢
	: , : , :
182	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
183	instantiation	186, 187, 197, 188	⊢
	: , :
184	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
185	instantiation	189, 219, 190, 191	⊢
	: , :
186	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
187	instantiation	217, 201, 192	⊢
	: , : , :
188	instantiation	193, 194	⊢
	:
189	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
190	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
191	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
192	instantiation	217, 206, 195	⊢
	: , : , :
193	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
194	instantiation	217, 196, 197	⊢
	: , : , :
195	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
196	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
197	instantiation	198, 199, 200	⊢
	: , :
198	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
199	instantiation	217, 201, 202	⊢
	: , : , :
200	instantiation	203, 204, 205	⊢
	:
201	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
202	instantiation	217, 206, 207	⊢
	: , : , :
203	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
204	instantiation	208, 210, 211, 212	⊢
	: , : , :
205	instantiation	209, 210, 211, 212	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
207	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
208	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
209	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
211	instantiation	217, 213, 214	⊢
	: , : , :
212	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
213	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
214	instantiation	217, 215, 216	⊢
	: , : , :
215	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
216	instantiation	217, 218, 219	⊢
	: , : , :
217	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
218	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
219	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements