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