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, 7, 8, 9	⊢
	: , : , : , : , : , :
1	reference	104	⊢
2	reference	105	⊢
3	reference	197	⊢
4	reference	226	⊢
5	reference	106	⊢
6	instantiation	173	⊢
	: , :
7	instantiation	224, 183, 10	⊢
	: , : , :
8	reference	175	⊢
9	reference	53	⊢
10	instantiation	11, 172, 12	⊢
	: , :
11	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
12	instantiation	13, 14, 15	⊢
	:
13	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
14	instantiation	16, 17, 18	⊢
	: , :
15	instantiation	19, 20	⊢
	: , :
16	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
17	instantiation	224, 21, 113	⊢
	: , : , :
18	instantiation	224, 22, 23	⊢
	: , : , :
19	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
20	instantiation	62, 66, 172, 24, 25, 26*, 27*	⊢
	: , : , :
21	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
22	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
23	instantiation	28, 169	⊢
	:
24	instantiation	86, 30, 172	⊢
	: , :
25	instantiation	29, 172, 30, 31, 145	⊢
	: , : , :
26	instantiation	148, 32, 33	⊢
	: , : , :
27	instantiation	148, 34, 35	⊢
	: , : , :
28	theorem		⊢
	proveit.numbers.negation.int_neg_closure
29	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
30	instantiation	86, 88, 132	⊢
	: , :
31	instantiation	46, 36, 37	⊢
	: , : , :
32	instantiation	104, 226, 197, 105, 52, 106, 161, 110, 53	⊢
	: , : , : , : , : , :
33	instantiation	109, 161, 110, 71	⊢
	: , : , :
34	instantiation	118, 38	⊢
	: , : , :
35	instantiation	39, 40, 41, 42	⊢
	: , : , : , :
36	instantiation	43, 223, 60, 44, 45*	⊢
	: , :
37	instantiation	46, 47, 48	⊢
	: , : , :
38	instantiation	104, 105, 197, 226, 106, 70, 73, 108, 161	⊢
	: , : , : , : , : , :
39	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
40	instantiation	104, 105, 50, 226, 106, 51, 73, 108, 161, 49	⊢
	: , : , : , : , : , :
41	instantiation	104, 50, 197, 105, 51, 52, 106, 73, 108, 161, 110, 53	⊢
	: , : , : , : , : , :
42	instantiation	148, 54, 55	⊢
	: , : , :
43	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
44	instantiation	56, 206, 82, 57, 198, 58*	⊢
	: , : , :
45	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
46	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
47	instantiation	59, 60, 179, 61	⊢
	: , :
48	instantiation	62, 132, 63, 88, 64, 65*	⊢
	: , : , :
49	instantiation	224, 183, 66	⊢
	: , : , :
50	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
51	instantiation	67	⊢
	: , : , :
52	instantiation	173	⊢
	: , :
53	instantiation	68, 161	⊢
	:
54	instantiation	69, 197, 226, 105, 70, 106, 73, 108, 161, 110, 71	⊢
	: , : , : , : , : , : , : , :
55	instantiation	72, 110, 73, 112	⊢
	: , : , :
56	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less
57	instantiation	74, 172, 75, 76, 77, 78*, 79*	⊢
	: , : , :
58	instantiation	80, 206	⊢
	:
59	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
60	instantiation	184, 206, 82, 186	⊢
	: , :
61	instantiation	81, 206, 82, 185, 83, 198	⊢
	: , : , :
62	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
63	instantiation	86, 153, 133	⊢
	: , :
64	axiom		⊢
	proveit.physics.quantum.QPE._t_req
65	instantiation	148, 84, 85	⊢
	: , : , :
66	instantiation	86, 153, 87	⊢
	: , :
67	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
68	theorem		⊢
	proveit.numbers.negation.complex_closure
69	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
70	instantiation	173	⊢
	: , :
71	instantiation	134	⊢
	:
72	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
73	instantiation	224, 183, 88	⊢
	: , : , :
74	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
75	instantiation	89, 155, 217	⊢
	: , :
76	instantiation	224, 220, 90	⊢
	: , : , :
77	instantiation	91, 155, 217, 218, 92, 93	⊢
	: , : , :
78	instantiation	148, 94, 95	⊢
	: , : , :
79	instantiation	148, 96, 97	⊢
	: , : , :
80	theorem		⊢
	proveit.numbers.logarithms.log_eq_1
81	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
82	instantiation	193, 98, 206, 123	⊢
	: , :
83	instantiation	99, 172, 100, 101, 102, 103*	⊢
	: , : , :
84	instantiation	104, 105, 197, 226, 106, 107, 110, 111, 108	⊢
	: , : , : , : , : , :
85	instantiation	109, 110, 111, 112	⊢
	: , : , :
86	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
87	instantiation	152, 172	⊢
	:
88	instantiation	167, 168, 113	⊢
	: , : , :
89	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
90	instantiation	114, 171, 221	⊢
	: , :
91	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
92	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
93	instantiation	115, 181	⊢
	:
94	instantiation	118, 116	⊢
	: , : , :
95	instantiation	117, 161	⊢
	:
96	instantiation	118, 119	⊢
	: , : , :
97	instantiation	128, 223, 188, 129*, 120*, 146*	⊢
	: , : , : , :
98	instantiation	224, 208, 121	⊢
	: , : , :
99	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
100	instantiation	122, 218, 172, 123	⊢
	: , :
101	instantiation	224, 124, 190	⊢
	: , : , :
102	instantiation	125, 126, 204, 206, 127	⊢
	: , : , :
103	instantiation	128, 188, 223, 129*, 130*, 131*	⊢
	: , : , : , :
104	theorem		⊢
	proveit.numbers.addition.disassociation
105	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
106	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
107	instantiation	173	⊢
	: , :
108	instantiation	224, 183, 132	⊢
	: , : , :
109	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
110	instantiation	224, 183, 153	⊢
	: , : , :
111	instantiation	224, 183, 133	⊢
	: , : , :
112	instantiation	134	⊢
	:
113	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
114	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
115	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
116	instantiation	135, 136	⊢
	:
117	theorem		⊢
	proveit.numbers.addition.elim_zero_left
118	axiom		⊢
	proveit.logic.equality.substitution
119	instantiation	174, 136	⊢
	:
120	instantiation	148, 137, 138	⊢
	: , : , :
121	instantiation	224, 213, 139	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.division.div_real_closure
123	instantiation	140, 214	⊢
	:
124	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
125	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
126	instantiation	224, 141, 142	⊢
	: , : , :
127	instantiation	143, 172, 211, 218, 144, 145, 146*	⊢
	: , : , :
128	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
129	instantiation	147, 161	⊢
	:
130	instantiation	148, 149, 150	⊢
	: , : , :
131	instantiation	151, 161	⊢
	:
132	instantiation	152, 153	⊢
	:
133	instantiation	224, 220, 154	⊢
	: , : , :
134	axiom		⊢
	proveit.logic.equality.equals_reflexivity
135	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
136	instantiation	224, 183, 155	⊢
	: , : , :
137	instantiation	162, 197, 156, 157, 166, 165	⊢
	: , : , : , :
138	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
139	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
140	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
141	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
142	instantiation	224, 158, 226	⊢
	: , : , :
143	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
144	instantiation	159, 217, 218, 219	⊢
	: , : , :
145	instantiation	160, 197	⊢
	:
146	instantiation	174, 161	⊢
	:
147	theorem		⊢
	proveit.numbers.division.frac_one_denom
148	axiom		⊢
	proveit.logic.equality.equals_transitivity
149	instantiation	162, 197, 163, 164, 165, 166	⊢
	: , : , : , :
150	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
151	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
152	theorem		⊢
	proveit.numbers.negation.real_closure
153	instantiation	167, 168, 169	⊢
	: , : , :
154	instantiation	224, 222, 170	⊢
	: , : , :
155	instantiation	224, 220, 171	⊢
	: , : , :
156	instantiation	173	⊢
	: , :
157	instantiation	173	⊢
	: , :
158	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
159	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
160	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
161	instantiation	224, 183, 172	⊢
	: , : , :
162	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
163	instantiation	173	⊢
	: , :
164	instantiation	173	⊢
	: , :
165	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
166	instantiation	174, 175	⊢
	:
167	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
168	instantiation	176, 177	⊢
	: , :
169	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
170	instantiation	178, 179	⊢
	:
171	instantiation	224, 180, 181	⊢
	: , : , :
172	instantiation	224, 220, 182	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
174	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
175	instantiation	224, 183, 218	⊢
	: , : , :
176	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
177	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
178	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
179	instantiation	184, 206, 185, 186	⊢
	: , :
180	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
181	instantiation	187, 199, 209	⊢
	: , :
182	instantiation	224, 222, 188	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
184	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
185	instantiation	189, 206, 190	⊢
	: , :
186	instantiation	191, 192	⊢
	: , :
187	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
188	instantiation	224, 225, 197	⊢
	: , : , :
189	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
190	instantiation	193, 194, 204, 195	⊢
	: , :
191	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
192	instantiation	196, 226, 197, 198	⊢
	: , :
193	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
194	instantiation	224, 208, 199	⊢
	: , : , :
195	instantiation	200, 201	⊢
	:
196	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
197	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
198	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
199	instantiation	224, 213, 202	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
201	instantiation	224, 203, 204	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
203	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
204	instantiation	205, 206, 207	⊢
	: , :
205	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
206	instantiation	224, 208, 209	⊢
	: , : , :
207	instantiation	210, 211, 212	⊢
	:
208	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
209	instantiation	224, 213, 214	⊢
	: , : , :
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
211	instantiation	215, 217, 218, 219	⊢
	: , : , :
212	instantiation	216, 217, 218, 219	⊢
	: , : , :
213	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
214	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
215	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
216	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
217	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
218	instantiation	224, 220, 221	⊢
	: , : , :
219	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
220	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
221	instantiation	224, 222, 223	⊢
	: , : , :
222	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
223	instantiation	224, 225, 226	⊢
	: , : , :
224	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
225	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
226	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements