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