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	reference	199	⊢
2	instantiation	119, 4, 5	⊢
	: , : , :
3	instantiation	6, 324, 321, 257, 7, 258, 284, 255, 137, 8*, 9*	⊢
	: , : , : , : , : , :
4	instantiation	13, 321, 257, 10, 258, 303, 11, 12	⊢
	: , : , : , : , : , :
5	instantiation	13, 324, 303, 14, 15	⊢
	: , : , : , : , : , :
6	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
7	instantiation	273	⊢
	: , :
8	instantiation	16, 284	⊢
	:
9	instantiation	199, 17, 18	⊢
	: , : , :
10	instantiation	273	⊢
	: , :
11	instantiation	299, 24	⊢
	:
12	instantiation	22, 21, 19, 20	⊢
	: , :
13	theorem		⊢
	proveit.numbers.addition.weak_bound_via_term_bound
14	instantiation	299, 21	⊢
	:
15	instantiation	22, 23, 24, 25	⊢
	: , :
16	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
17	instantiation	199, 26, 27	⊢
	: , : , :
18	instantiation	242, 28, 29	⊢
	: , : , :
19	instantiation	84, 303, 30, 31	⊢
	: , :
20	instantiation	40, 41, 70, 32, 33	⊢
	: , : , :
21	instantiation	84, 303, 34, 35	⊢
	: , :
22	theorem		⊢
	proveit.numbers.negation.negated_weak_bound
23	instantiation	84, 303, 36, 37	⊢
	: , :
24	instantiation	84, 303, 38, 39	⊢
	: , :
25	instantiation	40, 41, 78, 42, 43	⊢
	: , : , :
26	instantiation	44, 255, 271, 45, 46	⊢
	: , : , : , : , :
27	instantiation	240, 47	⊢
	: , : , :
28	instantiation	240, 48	⊢
	: , : , :
29	instantiation	94, 49	⊢
	:
30	instantiation	57, 54, 51	⊢
	: , :
31	instantiation	228, 50	⊢
	:
32	instantiation	322, 309, 75	⊢
	: , : , :
33	instantiation	63, 54, 51, 55, 52, 53	⊢
	: , : , :
34	instantiation	57, 54, 55	⊢
	: , :
35	instantiation	58, 56	⊢
	:
36	instantiation	57, 294, 151	⊢
	: , :
37	instantiation	58, 59	⊢
	:
38	instantiation	322, 285, 78	⊢
	: , : , :
39	instantiation	228, 60	⊢
	:
40	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
41	instantiation	322, 61, 62	⊢
	: , : , :
42	instantiation	322, 309, 77	⊢
	: , : , :
43	instantiation	63, 294, 97, 151, 89, 64	⊢
	: , : , :
44	theorem		⊢
	proveit.numbers.division.mult_frac_cancel_numer_left
45	instantiation	322, 281, 65	⊢
	: , : , :
46	instantiation	322, 281, 66	⊢
	: , : , :
47	instantiation	242, 67, 68	⊢
	: , : , :
48	instantiation	83, 255	⊢
	:
49	instantiation	322, 293, 69	⊢
	: , : , :
50	instantiation	322, 238, 70	⊢
	: , : , :
51	instantiation	74, 97, 321	⊢
	: , :
52	instantiation	71, 294, 97, 151, 72, 153	⊢
	: , : , :
53	instantiation	80, 104	⊢
	:
54	instantiation	322, 312, 73	⊢
	: , : , :
55	instantiation	74, 151, 321	⊢
	: , :
56	instantiation	322, 76, 75	⊢
	: , : , :
57	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
58	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_if_in_rational_nonzero
59	instantiation	322, 76, 77	⊢
	: , : , :
60	instantiation	322, 238, 78	⊢
	: , : , :
61	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
62	instantiation	322, 79, 324	⊢
	: , : , :
63	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
64	instantiation	80, 321	⊢
	:
65	instantiation	322, 291, 81	⊢
	: , : , :
66	instantiation	322, 238, 279	⊢
	: , : , :
67	instantiation	240, 82	⊢
	: , : , :
68	instantiation	83, 284	⊢
	:
69	instantiation	84, 303, 289, 252	⊢
	: , :
70	instantiation	262, 85, 86	⊢
	: , :
71	theorem		⊢
	proveit.numbers.exponentiation.exp_pos_lesseq
72	instantiation	87, 88, 89	⊢
	: , :
73	instantiation	322, 319, 90	⊢
	: , : , :
74	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
75	instantiation	92, 95, 91	⊢
	: , :
76	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
77	instantiation	92, 310, 106	⊢
	: , :
78	instantiation	262, 297, 118	⊢
	: , :
79	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
80	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
81	instantiation	322, 305, 93	⊢
	: , : , :
82	instantiation	94, 284	⊢
	:
83	theorem		⊢
	proveit.numbers.division.frac_one_denom
84	theorem		⊢
	proveit.numbers.division.div_real_closure
85	instantiation	322, 309, 95	⊢
	: , : , :
86	instantiation	96, 97, 98	⊢
	:
87	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
88	instantiation	99, 100	⊢
	:
89	instantiation	129, 287, 101, 198, 102, 103*	⊢
	: , : , :
90	instantiation	322, 323, 104	⊢
	: , : , :
91	instantiation	105, 106, 315	⊢
	: , :
92	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
93	instantiation	322, 314, 316	⊢
	: , : , :
94	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
95	instantiation	322, 317, 107	⊢
	: , : , :
96	theorem		⊢
	proveit.numbers.exponentiation.sqrd_pos_closure
97	instantiation	142, 303, 148	⊢
	: , :
98	instantiation	228, 108	⊢
	:
99	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg
100	instantiation	322, 109, 118	⊢
	: , : , :
101	instantiation	322, 285, 190	⊢
	: , : , :
102	instantiation	110, 294, 176, 111, 265, 112, 113*	⊢
	: , : , :
103	instantiation	242, 114, 115	⊢
	: , : , :
104	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
105	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_pos_closure
106	instantiation	116, 160, 117	⊢
	:
107	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
108	instantiation	322, 238, 118	⊢
	: , : , :
109	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
110	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
111	instantiation	142, 132, 130	⊢
	: , :
112	instantiation	119, 120, 121	⊢
	: , : , :
113	instantiation	122, 297, 190, 234	⊢
	: , :
114	instantiation	184, 257, 321, 324, 258, 123, 284, 137, 277	⊢
	: , : , : , : , : , :
115	instantiation	242, 124, 125	⊢
	: , : , :
116	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.pos_rational_is_rational_pos
117	instantiation	126, 127	⊢
	: , :
118	instantiation	245, 296, 202	⊢
	: , :
119	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
120	instantiation	128, 176	⊢
	:
121	instantiation	129, 130, 131, 132, 133, 134*	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.exponentiation.exponent_log_with_same_base
123	instantiation	273	⊢
	: , :
124	instantiation	135, 324, 257, 258, 284, 137, 277	⊢
	: , : , : , : , : , : , :
125	instantiation	186, 257, 321, 324, 258, 136, 284, 277, 137, 200*	⊢
	: , : , : , : , : , :
126	theorem		⊢
	proveit.numbers.addition.subtraction.pos_difference
127	instantiation	138, 287, 294, 139, 140, 200*, 141*	⊢
	: , : , :
128	theorem		⊢
	proveit.numbers.rounding.ceil_x_ge_x
129	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
130	instantiation	299, 204	⊢
	:
131	instantiation	142, 204, 143	⊢
	: , :
132	instantiation	212, 213, 248	⊢
	: , : , :
133	axiom		⊢
	proveit.physics.quantum.QPE._t_req
134	instantiation	144, 145, 146, 147	⊢
	: , : , : , :
135	theorem		⊢
	proveit.numbers.addition.leftward_commutation
136	instantiation	273	⊢
	: , :
137	instantiation	322, 293, 148	⊢
	: , : , :
138	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
139	instantiation	322, 312, 149	⊢
	: , : , :
140	instantiation	150, 294, 151, 152, 153	⊢
	: , : , :
141	instantiation	242, 154, 155	⊢
	: , : , :
142	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
143	instantiation	322, 312, 156	⊢
	: , : , :
144	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
145	instantiation	242, 157, 158	⊢
	: , : , :
146	instantiation	195	⊢
	:
147	instantiation	159, 177	⊢
	: , :
148	instantiation	322, 285, 202	⊢
	: , : , :
149	instantiation	169, 160, 307	⊢
	: , :
150	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
151	instantiation	322, 312, 160	⊢
	: , : , :
152	theorem		⊢
	proveit.physics.quantum.QPE._e_value_ge_two
153	instantiation	161, 318	⊢
	:
154	instantiation	240, 162	⊢
	: , : , :
155	instantiation	242, 163, 164	⊢
	: , : , :
156	instantiation	322, 319, 165	⊢
	: , : , :
157	instantiation	240, 166	⊢
	: , : , :
158	instantiation	242, 167, 168	⊢
	: , : , :
159	theorem		⊢
	proveit.logic.equality.equals_reversal
160	instantiation	169, 207, 170	⊢
	: , :
161	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
162	instantiation	242, 171, 172	⊢
	: , : , :
163	instantiation	184, 257, 321, 324, 258, 173, 188, 255, 277	⊢
	: , : , : , : , : , :
164	instantiation	174, 255, 188, 175	⊢
	: , : , :
165	instantiation	223, 176	⊢
	:
166	instantiation	240, 177	⊢
	: , : , :
167	instantiation	184, 257, 321, 324, 258, 178, 193, 181, 179	⊢
	: , : , : , : , : , :
168	instantiation	180, 193, 181, 182	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.addition.add_rational_closure_bin
170	instantiation	322, 319, 183	⊢
	: , : , :
171	instantiation	184, 257, 321, 324, 258, 185, 188, 277, 284	⊢
	: , : , : , : , : , :
172	instantiation	186, 324, 321, 257, 187, 258, 188, 277, 284, 189*	⊢
	: , : , : , : , : , :
173	instantiation	273	⊢
	: , :
174	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
175	instantiation	195	⊢
	:
176	instantiation	232, 297, 190, 234	⊢
	: , :
177	instantiation	240, 191	⊢
	: , : , :
178	instantiation	273	⊢
	: , :
179	instantiation	192, 193	⊢
	:
180	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
181	instantiation	322, 293, 194	⊢
	: , : , :
182	instantiation	195	⊢
	:
183	instantiation	322, 196, 197	⊢
	: , : , :
184	theorem		⊢
	proveit.numbers.addition.disassociation
185	instantiation	273	⊢
	: , :
186	theorem		⊢
	proveit.numbers.addition.association
187	instantiation	273	⊢
	: , :
188	instantiation	322, 293, 198	⊢
	: , : , :
189	instantiation	199, 200, 201	⊢
	: , : , :
190	instantiation	245, 297, 202	⊢
	: , :
191	instantiation	240, 203	⊢
	: , : , :
192	theorem		⊢
	proveit.numbers.negation.complex_closure
193	instantiation	322, 293, 204	⊢
	: , : , :
194	instantiation	322, 312, 205	⊢
	: , : , :
195	axiom		⊢
	proveit.logic.equality.equals_reflexivity
196	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
197	instantiation	206, 316	⊢
	:
198	instantiation	322, 312, 207	⊢
	: , : , :
199	theorem		⊢
	proveit.logic.equality.sub_right_side_into
200	instantiation	208, 255, 284, 209	⊢
	: , : , :
201	instantiation	210, 284, 277	⊢
	: , :
202	instantiation	295, 296, 239, 219	⊢
	: , :
203	instantiation	240, 211	⊢
	: , : , :
204	instantiation	212, 213, 267	⊢
	: , : , :
205	instantiation	322, 319, 214	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.negation.int_neg_closure
207	instantiation	322, 215, 216	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
209	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
210	theorem		⊢
	proveit.numbers.addition.commutation
211	instantiation	217, 255, 218, 219, 220*	⊢
	: , :
212	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
213	instantiation	221, 222	⊢
	: , :
214	instantiation	223, 224	⊢
	:
215	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_nonzero_within_rational
216	instantiation	225, 292, 226	⊢
	: , :
217	theorem		⊢
	proveit.numbers.division.div_as_mult
218	instantiation	322, 293, 227	⊢
	: , : , :
219	instantiation	228, 229	⊢
	:
220	instantiation	242, 230, 231	⊢
	: , : , :
221	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
222	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
223	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
224	instantiation	232, 297, 233, 234	⊢
	: , :
225	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_nonzero_closure
226	instantiation	235, 236, 237	⊢
	: , :
227	instantiation	322, 285, 239	⊢
	: , : , :
228	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
229	instantiation	322, 238, 239	⊢
	: , : , :
230	instantiation	240, 241	⊢
	: , : , :
231	instantiation	242, 243, 244	⊢
	: , : , :
232	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
233	instantiation	245, 297, 246	⊢
	: , :
234	instantiation	268, 247	⊢
	: , :
235	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
236	instantiation	322, 266, 248	⊢
	: , : , :
237	instantiation	249, 250	⊢
	:
238	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
239	instantiation	262, 297, 279	⊢
	: , :
240	axiom		⊢
	proveit.logic.equality.substitution
241	instantiation	251, 284, 276, 287, 298, 252, 253*	⊢
	: , : , :
242	axiom		⊢
	proveit.logic.equality.equals_transitivity
243	instantiation	254, 324, 321, 257, 259, 258, 255, 260, 261	⊢
	: , : , : , : , : , :
244	instantiation	256, 257, 321, 258, 259, 260, 261	⊢
	: , : , : , :
245	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
246	instantiation	262, 286, 263	⊢
	: , :
247	instantiation	264, 324, 321, 265	⊢
	: , :
248	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
249	theorem		⊢
	proveit.numbers.negation.int_closure
250	instantiation	322, 266, 267	⊢
	: , : , :
251	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
252	instantiation	268, 269	⊢
	: , :
253	instantiation	270, 271, 316, 272*	⊢
	: , :
254	theorem		⊢
	proveit.numbers.multiplication.disassociation
255	instantiation	322, 293, 303	⊢
	: , : , :
256	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
257	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
258	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
259	instantiation	273	⊢
	: , :
260	instantiation	322, 293, 274	⊢
	: , : , :
261	instantiation	275, 276, 277	⊢
	: , :
262	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
263	instantiation	278, 279, 287	⊢
	: , :
264	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
265	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
266	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
267	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
268	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
269	instantiation	280, 302, 289, 290	⊢
	: , :
270	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
271	instantiation	322, 281, 282	⊢
	: , : , :
272	instantiation	283, 284	⊢
	:
273	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
274	instantiation	322, 285, 286	⊢
	: , : , :
275	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
276	instantiation	322, 293, 289	⊢
	: , : , :
277	instantiation	322, 293, 287	⊢
	: , : , :
278	theorem		⊢
	proveit.numbers.exponentiation.exp_real_pos_closure
279	instantiation	288, 289, 290	⊢
	:
280	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
281	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
282	instantiation	322, 291, 292	⊢
	: , : , :
283	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
284	instantiation	322, 293, 294	⊢
	: , : , :
285	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
286	instantiation	295, 296, 297, 298	⊢
	: , :
287	instantiation	299, 303	⊢
	:
288	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
289	instantiation	300, 302, 303, 304	⊢
	: , : , :
290	instantiation	301, 302, 303, 304	⊢
	: , : , :
291	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
292	instantiation	322, 305, 306	⊢
	: , : , :
293	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
294	instantiation	322, 312, 307	⊢
	: , : , :
295	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
296	instantiation	322, 309, 308	⊢
	: , : , :
297	instantiation	322, 309, 310	⊢
	: , : , :
298	instantiation	311, 318	⊢
	:
299	theorem		⊢
	proveit.numbers.negation.real_closure
300	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
301	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
302	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
303	instantiation	322, 312, 313	⊢
	: , : , :
304	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
305	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
306	instantiation	322, 314, 318	⊢
	: , : , :
307	instantiation	322, 319, 315	⊢
	: , : , :
308	instantiation	322, 317, 316	⊢
	: , : , :
309	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
310	instantiation	322, 317, 318	⊢
	: , : , :
311	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
312	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
313	instantiation	322, 319, 320	⊢
	: , : , :
314	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
315	instantiation	322, 323, 321	⊢
	: , : , :
316	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
317	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
318	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
319	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
320	instantiation	322, 323, 324	⊢
	: , : , :
321	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
322	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
323	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
324	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements