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	⊢
	: , :
1	theorem		⊢
	proveit.numbers.negation.negated_weak_bound
2	instantiation	7, 251, 5, 6	⊢
	: , :
3	instantiation	7, 251, 8, 9	⊢
	: , :
4	instantiation	10, 11, 28, 12, 13	⊢
	: , : , :
5	instantiation	16, 21, 23	⊢
	: , :
6	instantiation	14, 15	⊢
	:
7	theorem		⊢
	proveit.numbers.division.div_real_closure
8	instantiation	16, 21, 22	⊢
	: , :
9	instantiation	179, 17	⊢
	:
10	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
11	instantiation	270, 18, 19	⊢
	: , : , :
12	instantiation	270, 257, 27	⊢
	: , : , :
13	instantiation	20, 21, 22, 23, 24, 25	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_if_in_rational_nonzero
15	instantiation	270, 26, 27	⊢
	: , : , :
16	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
17	instantiation	270, 189, 28	⊢
	: , : , :
18	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
19	instantiation	270, 29, 272	⊢
	: , : , :
20	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
21	instantiation	270, 260, 30	⊢
	: , : , :
22	instantiation	31, 47, 269	⊢
	: , :
23	instantiation	31, 92, 269	⊢
	: , :
24	instantiation	32, 242, 47, 92, 33, 94	⊢
	: , : , :
25	instantiation	34, 49	⊢
	:
26	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
27	instantiation	35, 45, 36	⊢
	: , :
28	instantiation	212, 37, 38	⊢
	: , :
29	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
30	instantiation	270, 267, 39	⊢
	: , : , :
31	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
32	theorem		⊢
	proveit.numbers.exponentiation.exp_pos_lesseq
33	instantiation	40, 41, 42	⊢
	: , :
34	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
35	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
36	instantiation	43, 44, 263	⊢
	: , :
37	instantiation	270, 257, 45	⊢
	: , : , :
38	instantiation	46, 47, 48	⊢
	:
39	instantiation	270, 271, 49	⊢
	: , : , :
40	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
41	instantiation	50, 51	⊢
	:
42	instantiation	81, 235, 52, 140, 53, 54*	⊢
	: , : , :
43	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_pos_closure
44	instantiation	55, 104, 56	⊢
	:
45	instantiation	270, 265, 57	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.exponentiation.sqrd_pos_closure
47	instantiation	97, 251, 103	⊢
	: , :
48	instantiation	179, 58	⊢
	:
49	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
50	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg
51	instantiation	270, 59, 68	⊢
	: , : , :
52	instantiation	270, 233, 144	⊢
	: , : , :
53	instantiation	60, 242, 130, 61, 215, 62, 63*	⊢
	: , : , :
54	instantiation	193, 64, 65	⊢
	: , : , :
55	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.pos_rational_is_rational_pos
56	instantiation	66, 67	⊢
	: , :
57	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
58	instantiation	270, 189, 68	⊢
	: , : , :
59	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
60	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
61	instantiation	97, 84, 82	⊢
	: , :
62	instantiation	69, 70, 71	⊢
	: , : , :
63	instantiation	72, 245, 144, 185	⊢
	: , :
64	instantiation	132, 207, 269, 272, 208, 73, 232, 89, 225	⊢
	: , : , : , : , : , :
65	instantiation	193, 74, 75	⊢
	: , : , :
66	theorem		⊢
	proveit.numbers.addition.subtraction.pos_difference
67	instantiation	76, 235, 242, 77, 78, 142*, 79*	⊢
	: , : , :
68	instantiation	196, 244, 155	⊢
	: , :
69	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
70	instantiation	80, 130	⊢
	:
71	instantiation	81, 82, 83, 84, 85, 86*	⊢
	: , : , :
72	theorem		⊢
	proveit.numbers.exponentiation.exponent_log_with_same_base
73	instantiation	221	⊢
	: , :
74	instantiation	87, 272, 207, 208, 232, 89, 225	⊢
	: , : , : , : , : , : , :
75	instantiation	126, 207, 269, 272, 208, 88, 232, 225, 89, 142*	⊢
	: , : , : , : , : , :
76	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
77	instantiation	270, 260, 90	⊢
	: , : , :
78	instantiation	91, 242, 92, 93, 94	⊢
	: , : , :
79	instantiation	193, 95, 96	⊢
	: , : , :
80	theorem		⊢
	proveit.numbers.rounding.ceil_x_ge_x
81	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
82	instantiation	247, 157	⊢
	:
83	instantiation	97, 157, 98	⊢
	: , :
84	instantiation	162, 163, 186	⊢
	: , : , :
85	axiom		⊢
	proveit.physics.quantum.QPE._t_req
86	instantiation	99, 100, 101, 102	⊢
	: , : , : , :
87	theorem		⊢
	proveit.numbers.addition.leftward_commutation
88	instantiation	221	⊢
	: , :
89	instantiation	270, 241, 103	⊢
	: , : , :
90	instantiation	113, 104, 255	⊢
	: , :
91	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
92	instantiation	270, 260, 104	⊢
	: , : , :
93	theorem		⊢
	proveit.physics.quantum.QPE._e_value_ge_two
94	instantiation	105, 266	⊢
	:
95	instantiation	191, 106	⊢
	: , : , :
96	instantiation	193, 107, 108	⊢
	: , : , :
97	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
98	instantiation	270, 260, 109	⊢
	: , : , :
99	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
100	instantiation	193, 110, 111	⊢
	: , : , :
101	instantiation	149	⊢
	:
102	instantiation	112, 131	⊢
	: , :
103	instantiation	270, 233, 155	⊢
	: , : , :
104	instantiation	113, 151, 114	⊢
	: , :
105	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
106	instantiation	193, 115, 116	⊢
	: , : , :
107	instantiation	132, 207, 269, 272, 208, 117, 128, 205, 225	⊢
	: , : , : , : , : , :
108	instantiation	118, 205, 128, 119	⊢
	: , : , :
109	instantiation	270, 267, 120	⊢
	: , : , :
110	instantiation	191, 121	⊢
	: , : , :
111	instantiation	193, 122, 123	⊢
	: , : , :
112	theorem		⊢
	proveit.logic.equality.equals_reversal
113	theorem		⊢
	proveit.numbers.addition.add_rational_closure_bin
114	instantiation	270, 267, 124	⊢
	: , : , :
115	instantiation	132, 207, 269, 272, 208, 125, 128, 225, 232	⊢
	: , : , : , : , : , :
116	instantiation	126, 272, 269, 207, 127, 208, 128, 225, 232, 129*	⊢
	: , : , : , : , : , :
117	instantiation	221	⊢
	: , :
118	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
119	instantiation	149	⊢
	:
120	instantiation	173, 130	⊢
	:
121	instantiation	191, 131	⊢
	: , : , :
122	instantiation	132, 207, 269, 272, 208, 133, 147, 136, 134	⊢
	: , : , : , : , : , :
123	instantiation	135, 147, 136, 137	⊢
	: , : , :
124	instantiation	270, 138, 139	⊢
	: , : , :
125	instantiation	221	⊢
	: , :
126	theorem		⊢
	proveit.numbers.addition.association
127	instantiation	221	⊢
	: , :
128	instantiation	270, 241, 140	⊢
	: , : , :
129	instantiation	141, 142, 143	⊢
	: , : , :
130	instantiation	183, 245, 144, 185	⊢
	: , :
131	instantiation	191, 145	⊢
	: , : , :
132	theorem		⊢
	proveit.numbers.addition.disassociation
133	instantiation	221	⊢
	: , :
134	instantiation	146, 147	⊢
	:
135	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
136	instantiation	270, 241, 148	⊢
	: , : , :
137	instantiation	149	⊢
	:
138	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
139	instantiation	150, 264	⊢
	:
140	instantiation	270, 260, 151	⊢
	: , : , :
141	theorem		⊢
	proveit.logic.equality.sub_right_side_into
142	instantiation	152, 205, 232, 153	⊢
	: , : , :
143	instantiation	154, 232, 225	⊢
	: , :
144	instantiation	196, 245, 155	⊢
	: , :
145	instantiation	191, 156	⊢
	: , : , :
146	theorem		⊢
	proveit.numbers.negation.complex_closure
147	instantiation	270, 241, 157	⊢
	: , : , :
148	instantiation	270, 260, 158	⊢
	: , : , :
149	axiom		⊢
	proveit.logic.equality.equals_reflexivity
150	theorem		⊢
	proveit.numbers.negation.int_neg_closure
151	instantiation	270, 159, 160	⊢
	: , : , :
152	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
153	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
154	theorem		⊢
	proveit.numbers.addition.commutation
155	instantiation	243, 244, 190, 169	⊢
	: , :
156	instantiation	191, 161	⊢
	: , : , :
157	instantiation	162, 163, 200	⊢
	: , : , :
158	instantiation	270, 267, 164	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_nonzero_within_rational
160	instantiation	165, 240, 166	⊢
	: , :
161	instantiation	167, 205, 168, 169, 170*	⊢
	: , :
162	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
163	instantiation	171, 172	⊢
	: , :
164	instantiation	173, 174	⊢
	:
165	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_nonzero_closure
166	instantiation	175, 176, 177	⊢
	: , :
167	theorem		⊢
	proveit.numbers.division.div_as_mult
168	instantiation	270, 241, 178	⊢
	: , : , :
169	instantiation	179, 180	⊢
	:
170	instantiation	193, 181, 182	⊢
	: , : , :
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	183, 245, 184, 185	⊢
	: , :
175	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
176	instantiation	270, 199, 186	⊢
	: , : , :
177	instantiation	187, 188	⊢
	:
178	instantiation	270, 233, 190	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
180	instantiation	270, 189, 190	⊢
	: , : , :
181	instantiation	191, 192	⊢
	: , : , :
182	instantiation	193, 194, 195	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
184	instantiation	196, 245, 197	⊢
	: , :
185	instantiation	216, 198	⊢
	: , :
186	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
187	theorem		⊢
	proveit.numbers.negation.int_closure
188	instantiation	270, 199, 200	⊢
	: , : , :
189	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
190	instantiation	212, 245, 227	⊢
	: , :
191	axiom		⊢
	proveit.logic.equality.substitution
192	instantiation	201, 232, 224, 235, 246, 202, 203*	⊢
	: , : , :
193	axiom		⊢
	proveit.logic.equality.equals_transitivity
194	instantiation	204, 272, 269, 207, 209, 208, 205, 210, 211	⊢
	: , : , : , : , : , :
195	instantiation	206, 207, 269, 208, 209, 210, 211	⊢
	: , : , : , :
196	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
197	instantiation	212, 234, 213	⊢
	: , :
198	instantiation	214, 272, 269, 215	⊢
	: , :
199	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
200	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
201	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
202	instantiation	216, 217	⊢
	: , :
203	instantiation	218, 219, 264, 220*	⊢
	: , :
204	theorem		⊢
	proveit.numbers.multiplication.disassociation
205	instantiation	270, 241, 251	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
207	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
208	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
209	instantiation	221	⊢
	: , :
210	instantiation	270, 241, 222	⊢
	: , : , :
211	instantiation	223, 224, 225	⊢
	: , :
212	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
213	instantiation	226, 227, 235	⊢
	: , :
214	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
215	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
216	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
217	instantiation	228, 250, 237, 238	⊢
	: , :
218	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
219	instantiation	270, 229, 230	⊢
	: , : , :
220	instantiation	231, 232	⊢
	:
221	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
222	instantiation	270, 233, 234	⊢
	: , : , :
223	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
224	instantiation	270, 241, 237	⊢
	: , : , :
225	instantiation	270, 241, 235	⊢
	: , : , :
226	theorem		⊢
	proveit.numbers.exponentiation.exp_real_pos_closure
227	instantiation	236, 237, 238	⊢
	:
228	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
229	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
230	instantiation	270, 239, 240	⊢
	: , : , :
231	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
232	instantiation	270, 241, 242	⊢
	: , : , :
233	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
234	instantiation	243, 244, 245, 246	⊢
	: , :
235	instantiation	247, 251	⊢
	:
236	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
237	instantiation	248, 250, 251, 252	⊢
	: , : , :
238	instantiation	249, 250, 251, 252	⊢
	: , : , :
239	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
240	instantiation	270, 253, 254	⊢
	: , : , :
241	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
242	instantiation	270, 260, 255	⊢
	: , : , :
243	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
244	instantiation	270, 257, 256	⊢
	: , : , :
245	instantiation	270, 257, 258	⊢
	: , : , :
246	instantiation	259, 266	⊢
	:
247	theorem		⊢
	proveit.numbers.negation.real_closure
248	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
249	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
250	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
251	instantiation	270, 260, 261	⊢
	: , : , :
252	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
253	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
254	instantiation	270, 262, 266	⊢
	: , : , :
255	instantiation	270, 267, 263	⊢
	: , : , :
256	instantiation	270, 265, 264	⊢
	: , : , :
257	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
258	instantiation	270, 265, 266	⊢
	: , : , :
259	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
260	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
261	instantiation	270, 267, 268	⊢
	: , : , :
262	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
263	instantiation	270, 271, 269	⊢
	: , : , :
264	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
265	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
266	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
267	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
268	instantiation	270, 271, 272	⊢
	: , : , :
269	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
270	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
271	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
272	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements