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