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, 4, 5, 6*	⊢
	: , : , :
1	reference	64	⊢
2	instantiation	145, 217	⊢
	:
3	reference	217	⊢
4	instantiation	7, 170, 13	⊢
	: , :
5	instantiation	8, 9	⊢
	:
6	instantiation	10, 178, 11	⊢
	: , :
7	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
8	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
9	instantiation	12, 196, 13	⊢
	: , :
10	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
11	instantiation	125	⊢
	:
12	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
13	instantiation	14, 15, 16	⊢
	:
14	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
15	instantiation	17, 18, 19	⊢
	: , :
16	instantiation	20, 21	⊢
	: , :
17	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
18	instantiation	223, 22, 106	⊢
	: , : , :
19	instantiation	223, 23, 24	⊢
	: , : , :
20	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
21	instantiation	49, 25, 26	⊢
	: , : , :
22	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
23	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
24	instantiation	27, 213	⊢
	:
25	axiom		⊢
	proveit.physics.quantum.QPE._n_ge_two
26	instantiation	64, 68, 170, 28, 29, 30*, 31*	⊢
	: , : , :
27	theorem		⊢
	proveit.numbers.negation.int_neg_closure
28	instantiation	83, 33, 170	⊢
	: , :
29	instantiation	32, 170, 33, 34, 140	⊢
	: , : , :
30	instantiation	151, 35, 36	⊢
	: , : , :
31	instantiation	151, 37, 38	⊢
	: , : , :
32	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
33	instantiation	83, 85, 123	⊢
	: , :
34	instantiation	49, 39, 40	⊢
	: , : , :
35	instantiation	97, 225, 196, 98, 55, 99, 157, 103, 56	⊢
	: , : , : , : , : , :
36	instantiation	102, 157, 103, 73	⊢
	: , : , :
37	instantiation	130, 41	⊢
	: , : , :
38	instantiation	42, 43, 44, 45	⊢
	: , : , : , :
39	instantiation	46, 222, 62, 47, 48*	⊢
	: , :
40	instantiation	49, 50, 51	⊢
	: , : , :
41	instantiation	97, 98, 196, 225, 99, 72, 75, 101, 157	⊢
	: , : , : , : , : , :
42	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
43	instantiation	97, 98, 53, 225, 99, 54, 75, 101, 157, 52	⊢
	: , : , : , : , : , :
44	instantiation	97, 53, 196, 98, 54, 55, 99, 75, 101, 157, 103, 56	⊢
	: , : , : , : , : , :
45	instantiation	151, 57, 58	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
47	instantiation	59, 205, 79, 60	⊢
	: , :
48	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
49	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
50	instantiation	61, 62, 174, 63	⊢
	: , :
51	instantiation	64, 123, 65, 85, 66, 67*	⊢
	: , : , :
52	instantiation	223, 185, 68	⊢
	: , : , :
53	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
54	instantiation	69	⊢
	: , : , :
55	instantiation	176	⊢
	: , :
56	instantiation	70, 157	⊢
	:
57	instantiation	71, 196, 225, 98, 72, 99, 75, 101, 157, 103, 73	⊢
	: , : , : , : , : , : , : , :
58	instantiation	74, 103, 75, 105	⊢
	: , : , :
59	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
60	instantiation	76, 197, 77	⊢
	: , :
61	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
62	instantiation	180, 205, 79, 182	⊢
	: , :
63	instantiation	78, 205, 79, 181, 80, 197	⊢
	: , : , :
64	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
65	instantiation	83, 146, 124	⊢
	: , :
66	axiom		⊢
	proveit.physics.quantum.QPE._t_req
67	instantiation	151, 81, 82	⊢
	: , : , :
68	instantiation	83, 146, 84	⊢
	: , :
69	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
70	theorem		⊢
	proveit.numbers.negation.complex_closure
71	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
72	instantiation	176	⊢
	: , :
73	instantiation	125	⊢
	:
74	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
75	instantiation	223, 185, 85	⊢
	: , : , :
76	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
77	instantiation	86, 170, 87, 88, 89, 90*, 91*	⊢
	: , : , :
78	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
79	instantiation	223, 207, 92	⊢
	: , : , :
80	instantiation	93, 170, 164, 94, 95, 96*	⊢
	: , : , :
81	instantiation	97, 98, 196, 225, 99, 100, 103, 104, 101	⊢
	: , : , : , : , : , :
82	instantiation	102, 103, 104, 105	⊢
	: , : , :
83	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
84	instantiation	145, 170	⊢
	:
85	instantiation	160, 161, 106	⊢
	: , : , :
86	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
87	instantiation	107, 164, 216	⊢
	: , :
88	instantiation	223, 219, 108	⊢
	: , : , :
89	instantiation	109, 164, 216, 217, 110, 111	⊢
	: , : , :
90	instantiation	151, 112, 113	⊢
	: , : , :
91	instantiation	151, 114, 115	⊢
	: , : , :
92	instantiation	191, 116, 208	⊢
	: , :
93	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
94	instantiation	223, 117, 188	⊢
	: , : , :
95	instantiation	118, 119, 203, 205, 120	⊢
	: , : , :
96	instantiation	132, 186, 222, 133*, 121*, 122*	⊢
	: , : , : , :
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	176	⊢
	: , :
101	instantiation	223, 185, 123	⊢
	: , : , :
102	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
103	instantiation	223, 185, 146	⊢
	: , : , :
104	instantiation	223, 185, 124	⊢
	: , : , :
105	instantiation	125	⊢
	:
106	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
107	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
108	instantiation	126, 175, 220	⊢
	: , :
109	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
110	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
111	instantiation	127, 184	⊢
	:
112	instantiation	130, 128	⊢
	: , : , :
113	instantiation	129, 157	⊢
	:
114	instantiation	130, 131	⊢
	: , : , :
115	instantiation	132, 222, 186, 133*, 134*, 141*	⊢
	: , : , : , :
116	instantiation	223, 212, 135	⊢
	: , : , :
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	223, 136, 137	⊢
	: , : , :
120	instantiation	138, 170, 210, 217, 139, 140, 141*	⊢
	: , : , :
121	instantiation	151, 142, 143	⊢
	: , : , :
122	instantiation	144, 157	⊢
	:
123	instantiation	145, 146	⊢
	:
124	instantiation	223, 219, 147	⊢
	: , : , :
125	axiom		⊢
	proveit.logic.equality.equals_reflexivity
126	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
127	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
128	instantiation	148, 149	⊢
	:
129	theorem		⊢
	proveit.numbers.addition.elim_zero_left
130	axiom		⊢
	proveit.logic.equality.substitution
131	instantiation	177, 149	⊢
	:
132	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
133	instantiation	150, 157	⊢
	:
134	instantiation	151, 152, 153	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
136	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
137	instantiation	223, 154, 225	⊢
	: , : , :
138	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
139	instantiation	155, 216, 217, 218	⊢
	: , : , :
140	instantiation	156, 196	⊢
	:
141	instantiation	177, 157	⊢
	:
142	instantiation	165, 196, 158, 159, 169, 168	⊢
	: , : , : , :
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	223, 221, 163	⊢
	: , : , :
148	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
149	instantiation	223, 185, 164	⊢
	: , : , :
150	theorem		⊢
	proveit.numbers.division.frac_one_denom
151	axiom		⊢
	proveit.logic.equality.equals_transitivity
152	instantiation	165, 196, 166, 167, 168, 169	⊢
	: , : , : , :
153	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
154	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
155	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
156	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
157	instantiation	223, 185, 170	⊢
	: , : , :
158	instantiation	176	⊢
	: , :
159	instantiation	176	⊢
	: , :
160	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
161	instantiation	171, 172	⊢
	: , :
162	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
163	instantiation	173, 174	⊢
	:
164	instantiation	223, 219, 175	⊢
	: , : , :
165	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
166	instantiation	176	⊢
	: , :
167	instantiation	176	⊢
	: , :
168	instantiation	177, 178	⊢
	:
169	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
170	instantiation	223, 219, 179	⊢
	: , : , :
171	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
172	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
173	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
174	instantiation	180, 205, 181, 182	⊢
	: , :
175	instantiation	223, 183, 184	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
177	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
178	instantiation	223, 185, 217	⊢
	: , : , :
179	instantiation	223, 221, 186	⊢
	: , : , :
180	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
181	instantiation	187, 205, 188	⊢
	: , :
182	instantiation	189, 190	⊢
	: , :
183	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
184	instantiation	191, 198, 208	⊢
	: , :
185	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
186	instantiation	223, 224, 196	⊢
	: , : , :
187	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
188	instantiation	192, 193, 203, 194	⊢
	: , :
189	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
190	instantiation	195, 225, 196, 197	⊢
	: , :
191	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
192	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
193	instantiation	223, 207, 198	⊢
	: , : , :
194	instantiation	199, 200	⊢
	:
195	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
196	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
197	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
198	instantiation	223, 212, 201	⊢
	: , : , :
199	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
200	instantiation	223, 202, 203	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
202	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
203	instantiation	204, 205, 206	⊢
	: , :
204	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
205	instantiation	223, 207, 208	⊢
	: , : , :
206	instantiation	209, 210, 211	⊢
	:
207	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
208	instantiation	223, 212, 213	⊢
	: , : , :
209	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
210	instantiation	214, 216, 217, 218	⊢
	: , : , :
211	instantiation	215, 216, 217, 218	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
213	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
214	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
215	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
216	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
217	instantiation	223, 219, 220	⊢
	: , : , :
218	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
219	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
220	instantiation	223, 221, 222	⊢
	: , : , :
221	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
222	instantiation	223, 224, 225	⊢
	: , : , :
223	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
224	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
225	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements