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	100	⊢
2	instantiation	4, 15, 5, 72, 6	⊢
	: , : , :
3	instantiation	7, 12, 8, 9, 10, 13	⊢
	: , : , :
4	theorem		⊢
	proveit.numbers.division.strong_div_from_denom_bound__all_pos
5	instantiation	11, 12, 13	⊢
	:
6	instantiation	14, 72	⊢
	:
7	theorem		⊢
	proveit.numbers.division.weak_div_from_numer_bound__pos_denom
8	instantiation	291, 207, 15	⊢
	: , : , :
9	instantiation	16, 17	⊢
	:
10	instantiation	18, 19, 20, 21*	⊢
	:
11	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
12	instantiation	22, 104, 255, 24	⊢
	: , : , :
13	instantiation	23, 104, 255, 24	⊢
	: , : , :
14	theorem		⊢
	proveit.trigonometry.sine_linear_bound_by_arg_pos
15	instantiation	25, 94, 95, 26, 27, 208	⊢
	: , :
16	theorem		⊢
	proveit.trigonometry.real_closure
17	instantiation	139, 28, 29	⊢
	: , : , :
18	theorem		⊢
	proveit.trigonometry.sine_linear_bound_nonneg
19	instantiation	30, 94, 58, 31, 32, 33	⊢
	: , :
20	instantiation	116, 34, 35	⊢
	: , : , :
21	instantiation	241, 36, 37	⊢
	: , : , :
22	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
23	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
24	instantiation	38, 39	⊢
	:
25	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure
26	instantiation	291, 40, 240	⊢
	: , : , :
27	instantiation	291, 40, 41	⊢
	: , : , :
28	instantiation	168, 42, 181	⊢
	: , :
29	instantiation	143, 199, 293, 286, 200, 43, 204, 157, 158	⊢
	: , : , : , : , : , :
30	theorem		⊢
	proveit.numbers.multiplication.mult_real_nonneg_closure
31	instantiation	291, 44, 45	⊢
	: , : , :
32	instantiation	291, 46, 206	⊢
	: , : , :
33	instantiation	291, 46, 208	⊢
	: , : , :
34	instantiation	116, 102, 47	⊢
	: , : , :
35	instantiation	48, 157, 277, 256, 49*	⊢
	: , :
36	instantiation	182, 50	⊢
	: , : , :
37	instantiation	241, 51, 52	⊢
	: , : , :
38	theorem		⊢
	proveit.trigonometry.sine_pos_interval
39	instantiation	53, 104, 180, 54, 55	⊢
	: , : , :
40	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
41	instantiation	291, 258, 266	⊢
	: , : , :
42	instantiation	168, 231, 180	⊢
	: , :
43	instantiation	274	⊢
	: , :
44	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
45	instantiation	291, 56, 247	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
47	instantiation	92, 199, 286, 200, 204, 157, 158	⊢
	: , : , : , : , : , : , :
48	theorem		⊢
	proveit.numbers.division.div_as_mult
49	instantiation	241, 57, 124	⊢
	: , : , :
50	instantiation	143, 286, 94, 199, 58, 200, 277, 204, 157, 158	⊢
	: , : , : , : , : , :
51	instantiation	241, 59, 60	⊢
	: , : , :
52	instantiation	61, 70	⊢
	:
53	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOO
54	instantiation	291, 207, 72	⊢
	: , : , :
55	instantiation	62, 63, 64	⊢
	: , :
56	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
57	instantiation	182, 65	⊢
	: , : , :
58	instantiation	113	⊢
	: , : , :
59	instantiation	182, 66	⊢
	: , : , :
60	instantiation	67, 68, 69, 70, 71*	⊢
	: , : , :
61	theorem		⊢
	proveit.numbers.division.frac_one_denom
62	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
63	instantiation	167, 72	⊢
	:
64	instantiation	73, 74, 75	⊢
	: , : , :
65	instantiation	76, 77, 257, 197*	⊢
	: , :
66	instantiation	241, 78, 79	⊢
	: , : , :
67	theorem		⊢
	proveit.numbers.division.frac_cancel_left
68	instantiation	291, 90, 80	⊢
	: , : , :
69	instantiation	291, 90, 81	⊢
	: , : , :
70	instantiation	139, 82, 83	⊢
	: , : , :
71	instantiation	276, 157	⊢
	:
72	instantiation	84, 206, 208	⊢
	: , :
73	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
74	instantiation	139, 85, 118	⊢
	: , : , :
75	instantiation	151, 148, 86, 87, 88*, 89*	⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
77	instantiation	291, 90, 91	⊢
	: , : , :
78	instantiation	92, 199, 293, 286, 200, 99, 277, 204, 157, 158	⊢
	: , : , : , : , : , : , :
79	instantiation	93, 286, 94, 199, 95, 200, 157, 277, 204, 158	⊢
	: , : , : , : , : , :
80	instantiation	291, 96, 206	⊢
	: , : , :
81	instantiation	291, 111, 97	⊢
	: , : , :
82	instantiation	291, 283, 98	⊢
	: , : , :
83	instantiation	143, 199, 293, 286, 200, 99, 277, 204, 158	⊢
	: , : , : , : , : , :
84	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
85	instantiation	100, 101, 102	⊢
	: , : , :
86	instantiation	168, 169, 104	⊢
	: , :
87	instantiation	103, 169, 104, 180, 142, 105	⊢
	: , : , :
88	instantiation	241, 106, 107	⊢
	: , : , :
89	instantiation	108, 109, 110*	⊢
	: , :
90	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
91	instantiation	291, 111, 112	⊢
	: , : , :
92	theorem		⊢
	proveit.numbers.multiplication.leftward_commutation
93	theorem		⊢
	proveit.numbers.multiplication.association
94	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
95	instantiation	113	⊢
	: , : , :
96	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
97	instantiation	291, 133, 114	⊢
	: , : , :
98	instantiation	168, 115, 181	⊢
	: , :
99	instantiation	274	⊢
	: , :
100	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
101	instantiation	116, 117, 118	⊢
	: , : , :
102	instantiation	119, 180, 120, 236, 121, 122, 123*, 124*	⊢
	: , : , :
103	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
104	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
105	instantiation	125, 218	⊢
	:
106	instantiation	182, 126	⊢
	: , : , :
107	instantiation	127, 128	⊢
	:
108	theorem		⊢
	proveit.logic.equality.equals_reversal
109	instantiation	129, 199, 293, 286, 200, 130, 147, 157	⊢
	: , : , : , : , : , :
110	instantiation	241, 131, 132	⊢
	: , : , :
111	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
112	instantiation	291, 133, 134	⊢
	: , : , :
113	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
114	instantiation	291, 150, 257	⊢
	: , : , :
115	instantiation	168, 284, 231	⊢
	: , :
116	theorem		⊢
	proveit.logic.equality.sub_left_side_into
117	instantiation	135, 286, 206, 208, 136, 137*	⊢
	: , : , : , : , : , :
118	instantiation	182, 138	⊢
	: , : , :
119	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
120	instantiation	168, 231, 181	⊢
	: , :
121	instantiation	139, 140, 141	⊢
	: , : , :
122	instantiation	215, 142	⊢
	: , :
123	instantiation	143, 286, 293, 199, 144, 200, 157, 204, 158	⊢
	: , : , : , : , : , :
124	instantiation	145, 157, 147	⊢
	: , :
125	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
126	instantiation	146, 147	⊢
	:
127	theorem		⊢
	proveit.numbers.addition.elim_zero_left
128	instantiation	291, 283, 148	⊢
	: , : , :
129	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
130	instantiation	274	⊢
	: , :
131	instantiation	182, 149	⊢
	: , : , :
132	instantiation	275, 157	⊢
	:
133	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
134	instantiation	291, 150, 273	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_factor_bound
136	instantiation	151, 230, 284, 152, 153, 154*, 155*	⊢
	: , : , :
137	instantiation	156, 286, 157, 158	⊢
	: , : , : , :
138	instantiation	241, 159, 160	⊢
	: , : , :
139	theorem		⊢
	proveit.logic.equality.sub_right_side_into
140	instantiation	161, 162, 163	⊢
	: , :
141	instantiation	164, 273, 165, 204, 252, 166*	⊢
	: , :
142	instantiation	167, 206	⊢
	:
143	theorem		⊢
	proveit.numbers.multiplication.disassociation
144	instantiation	274	⊢
	: , :
145	theorem		⊢
	proveit.numbers.multiplication.commutation
146	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
147	instantiation	291, 283, 236	⊢
	: , : , :
148	instantiation	168, 169, 180	⊢
	: , :
149	instantiation	170, 282, 290, 171*	⊢
	: , : , : , :
150	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
151	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
152	instantiation	291, 287, 172	⊢
	: , : , :
153	instantiation	173, 284, 231, 255, 174, 175	⊢
	: , : , :
154	instantiation	176, 210, 277, 177	⊢
	: , : , :
155	instantiation	241, 178, 179	⊢
	: , : , :
156	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
157	instantiation	291, 283, 180	⊢
	: , : , :
158	instantiation	291, 283, 181	⊢
	: , : , :
159	instantiation	182, 183	⊢
	: , : , :
160	instantiation	184, 252	⊢
	:
161	theorem		⊢
	proveit.numbers.absolute_value.weak_upper_bound
162	instantiation	185, 213, 236, 214	⊢
	: , : , :
163	instantiation	215, 186	⊢
	: , :
164	theorem		⊢
	proveit.numbers.absolute_value.abs_prod
165	instantiation	274	⊢
	: , :
166	instantiation	187, 188	⊢
	:
167	theorem		⊢
	proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos
168	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
169	instantiation	291, 287, 189	⊢
	: , : , :
170	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
171	instantiation	241, 190, 191	⊢
	: , : , :
172	instantiation	291, 289, 192	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
174	instantiation	193, 284, 255, 194, 195, 196, 197*	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
176	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
177	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
178	instantiation	198, 199, 293, 286, 200, 201, 204, 210, 202	⊢
	: , : , : , : , : , :
179	instantiation	203, 210, 204, 205	⊢
	: , : , :
180	instantiation	291, 207, 206	⊢
	: , : , :
181	instantiation	291, 207, 208	⊢
	: , : , :
182	axiom		⊢
	proveit.logic.equality.substitution
183	instantiation	209, 210, 252, 211*	⊢
	: , :
184	theorem		⊢
	proveit.numbers.absolute_value.abs_even
185	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
186	instantiation	212, 213, 236, 214	⊢
	: , : , :
187	theorem		⊢
	proveit.numbers.absolute_value.abs_non_neg_elim
188	instantiation	215, 216	⊢
	: , :
189	instantiation	291, 217, 218	⊢
	: , : , :
190	instantiation	259, 293, 219, 220, 221, 222	⊢
	: , : , : , :
191	instantiation	223, 224, 225	⊢
	:
192	instantiation	291, 292, 226	⊢
	: , : , :
193	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
194	instantiation	249, 250, 228	⊢
	: , : , :
195	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
196	instantiation	227, 228	⊢
	:
197	instantiation	229, 277	⊢
	:
198	theorem		⊢
	proveit.numbers.addition.disassociation
199	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
200	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
201	instantiation	274	⊢
	: , :
202	instantiation	291, 283, 230	⊢
	: , : , :
203	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
204	instantiation	291, 283, 231	⊢
	: , : , :
205	instantiation	232	⊢
	:
206	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
207	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
208	instantiation	233, 234	⊢
	:
209	theorem		⊢
	proveit.numbers.multiplication.mult_neg_left
210	instantiation	291, 283, 255	⊢
	: , : , :
211	instantiation	275, 252	⊢
	:
212	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
213	instantiation	235, 236	⊢
	:
214	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval
215	theorem		⊢
	proveit.numbers.ordering.relax_less
216	instantiation	237, 266	⊢
	:
217	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
218	instantiation	238, 239, 240	⊢
	: , :
219	instantiation	274	⊢
	: , :
220	instantiation	274	⊢
	: , :
221	instantiation	241, 242, 243	⊢
	: , : , :
222	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
223	theorem		⊢
	proveit.numbers.division.frac_cancel_complete
224	instantiation	291, 283, 244	⊢
	: , : , :
225	instantiation	272, 245	⊢
	:
226	instantiation	246, 247, 286	⊢
	: , :
227	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
228	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
229	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
230	instantiation	291, 287, 248	⊢
	: , : , :
231	instantiation	249, 250, 266	⊢
	: , : , :
232	axiom		⊢
	proveit.logic.equality.equals_reflexivity
233	theorem		⊢
	proveit.numbers.absolute_value.abs_nonzero_closure
234	instantiation	251, 252, 253	⊢
	:
235	theorem		⊢
	proveit.numbers.negation.real_closure
236	instantiation	254, 255, 284, 256	⊢
	: , :
237	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
238	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
239	instantiation	291, 258, 257	⊢
	: , : , :
240	instantiation	291, 258, 273	⊢
	: , : , :
241	axiom		⊢
	proveit.logic.equality.equals_transitivity
242	instantiation	259, 293, 260, 261, 262, 263	⊢
	: , : , : , :
243	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
244	instantiation	291, 287, 264	⊢
	: , : , :
245	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
246	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
247	instantiation	291, 265, 266	⊢
	: , : , :
248	instantiation	291, 289, 267	⊢
	: , : , :
249	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
250	instantiation	268, 269	⊢
	: , :
251	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
252	instantiation	291, 283, 270	⊢
	: , : , :
253	assumption		⊢
254	theorem		⊢
	proveit.numbers.division.div_real_closure
255	instantiation	291, 287, 271	⊢
	: , : , :
256	instantiation	272, 273	⊢
	:
257	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
258	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
259	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
260	instantiation	274	⊢
	: , :
261	instantiation	274	⊢
	: , :
262	instantiation	275, 277	⊢
	:
263	instantiation	276, 277	⊢
	:
264	instantiation	291, 289, 278	⊢
	: , : , :
265	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
266	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
267	instantiation	279, 282	⊢
	:
268	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
269	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
270	instantiation	280, 281	⊢
	:
271	instantiation	291, 289, 282	⊢
	: , : , :
272	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
273	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
274	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
275	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
276	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
277	instantiation	291, 283, 284	⊢
	: , : , :
278	instantiation	291, 292, 285	⊢
	: , : , :
279	theorem		⊢
	proveit.numbers.negation.int_closure
280	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
281	theorem		⊢
	proveit.physics.quantum.QPE._best_round_is_int
282	instantiation	291, 292, 286	⊢
	: , : , :
283	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
284	instantiation	291, 287, 288	⊢
	: , : , :
285	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
286	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
287	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
288	instantiation	291, 289, 290	⊢
	: , : , :
289	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
290	instantiation	291, 292, 293	⊢
	: , : , :
291	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
292	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
293	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements