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